2 Main results

We consider an operator-valued probability space $(\mathcal A, E, \mathcal B)$ . That is, $\mathcal A$ is a unital von Neumann algebra, $\mathcal B \subset \mathcal A$ is a von Neumann subalgebra with the same unit and $E \colon \mathcal A \to \mathcal B$ is a positive conditional expectation, that is, $E[b] = b$ for all $b \in \mathcal B$ , $E[b_1 \mathfrak a b_2] = b_1 E[\mathfrak a] b_2$ for all $\mathfrak a \in \mathcal A$ and $b_1$ , $b_2 \in \mathcal B$ as well as $E[\mathfrak a]$ is positive for all positive $\mathfrak a \in \mathcal A$ . We also assume that $\mathcal B$ is equipped with a faithful tracial state $\langle \mspace {2 mu}\cdot \mspace {2 mu} \rangle $ such that $(\mathcal A, \langle E[\,\cdot \,] \rangle )$ is a tracial $W^*$ -probability space. We call an element $ \mathfrak {c} \in \mathcal {A}$ a $\mathcal {B}$ -valued circular element if it is centred, that is, $E[\mathfrak {c}]=0$ , all mixed free $\mathcal {B}$ -cumulantsFootnote ² of $\mathfrak {c}$ and $\mathfrak {c}^*$ of any order larger than $2$ vanish and if the second order cumulants satisfy

(2.1) $$ \begin{align} { E[\mathfrak{c}b\mathfrak{c}] =0\,, \qquad E[\mathfrak{c}^*b\mathfrak{c}^*] =0 } \end{align} $$

for all $b \in \mathcal {B}$ . Define the operators $S \colon \mathcal B \to \mathcal B$ and $S^* \colon \mathcal B \to \mathcal B$ through

(2.2) $$ \begin{align} S b := E[\mathfrak c b \mathfrak c^*] , \qquad S^* b:= E[\mathfrak c^* b \mathfrak c] \end{align} $$

for all $b \in \mathcal B$ . Note that $S^*b = (S(b^*))^*$ for all $b \in \mathcal B$ . We call $a+\mathfrak {c}$ with some $a \in \mathcal {B}$ a deformed $\mathcal {B}$ -valued circular element with deformation a and covariance S.

For $\mathfrak a \in \mathcal A$ the Brown measure $\sigma _{\mathfrak a}$ of $\mathfrak a$ is the unique compactly supported probability measure on $\mathbb {C}$ with

(2.3) $$ \begin{align} \int_{\mathbb{C}} \log \lvert \zeta - \xi \rvert\sigma_{\mathfrak a}(\mathrm{d} \xi) = \log D(\mathfrak a - \zeta) \end{align} $$

for all $\zeta \in \mathbb {C}$ . Here, $D(\mathfrak a - \zeta )$ denotes the Fuglede-Kadison determinant of $\mathfrak a - \zeta $ . The Fuglede-Kadison determinant of an arbitrary $\mathfrak b \in \mathcal A$ is defined by

(2.4) $$ \begin{align} D(\mathfrak b) := \lim_{\varepsilon \downarrow 0} \exp( \langle E[\log (\mathfrak b^* \mathfrak b + \varepsilon)^{1/2}] \rangle ) \in [0,\infty). \end{align} $$

Originally introduced in [Reference Brown22], the Brown measure was revived in [Reference Haagerup and Larsen36]. An introduction to the Brown measure and the Fuglede-Kadison determinant can be found for example in [Reference Mingo and Speicher47, Chapter 11].

In the following we will consider the case when $\mathcal B$ is commutative. Therefore we may assume that $\mathcal B= L^\infty (\mathfrak X, \Sigma , \mu )$ is the space of $\mu $ -almost everywhere bounded functions with respect to a probability measure $\mu $ on $(\mathfrak X, \Sigma )$ and $\langle b \rangle = \int _{\mathfrak X} b \mspace {2 mu}\mathrm {d}\mu $ is the trace of $b \in \mathcal {B}$ [Reference Takesaki55, Chapter III, Theorem 1.18].

We study the Brown measure of a deformed $\mathcal {B}$ -valued circular element. To that end, fix $a \in \mathcal {B}$ and let $\mathfrak {c} \in \mathcal {A}$ be a $\mathcal {B}$ -valued circular element that satisfies (2.1) and (2.2). Throughout this work we will assume that S and $S^*$ from (2.2) are represented by an integral kernel, that is, that there is a measurable function $s \colon \mathfrak X \times \mathfrak X \to [0,\infty )$ such that

(2.5) $$ \begin{align} (S u)(x) = \int_{\mathfrak X} s(x,y) u(y) \mu(\mathrm{d} y), \qquad (S^* u)(x) = \int_{\mathfrak X} s(y,x) u(y) \mu(\mathrm{d} y) \end{align} $$

for $x \in \mathfrak X$ and $u \in \mathcal {B}$ . Since S and $S^*$ are positive operators and therefore, bounded, after possibly changing s on a $\mu \otimes \mu $ -nullset, we assume

(2.6) $$ \begin{align} \sup_{x \in \mathfrak X} \int_{\mathfrak X} s(x,y) \mu(\mathrm{d} y) < \infty, \qquad \qquad \sup_{y \in \mathfrak X} \int_{\mathfrak X} s(x,y) \mu(\mathrm{d} x) < \infty. \end{align} $$

Under the following regularity assumptions on a and s, we prove detailed information about the Brown measure $\sigma = \sigma _{a + \mathfrak c}$ of the deformed $\mathcal B$ -valued circular operator $a + \mathfrak c$ . From here on and throughout the paper, we will impose some of the following assumptions.

A1

Block-primitivity of S: There is a constant $C>0$ , a primitive matrix $Z=(z_{ij})_{i,j=1}^K \in \{0,1\}^{K \times K}$ with $z_{ii}=1$ for each i and a measurable partition $(I_i)_{i=1}^K$ of $\mathfrak X$ with $\min _i\mu (I_i)\ge \frac {1}{C}$ , such that

for all x, $y \in \mathfrak X$ .

We recall that a matrix $Z \in [0,\infty )^{K\times K}$ with nonnegative entries is called primitive if there is an $L \in \mathbb {N}$ such that all entries of the power $Z^L$ are strictly positive. For a and s as above, we define the function $\Gamma _{a,s} \colon (0,\infty ) \to (0,\infty )$ similarly to [Reference Alt, Erdős and Krüger6, Section 9] as

(2.7) $$ \begin{align} \Gamma_{a,s}(\tau) := \bigg({\operatorname{\mbox{ess inf}}_{x \in \mathfrak X} \int_{\mathfrak X} \frac{1}{(\tau^{-1} + \lvert a(x) - a(y) \rvert + d_s(x,y))^2} \mu(\mathrm{d} y)}\bigg)^{1/2}, \end{align} $$

where $d_s(x,y) := \big ({\int _{\mathfrak X} ( \lvert s(x,q) - s(y,q) \rvert ^2 + \lvert s(q,x) - s(q,y) \rvert ^2 )\mu (\mathrm {d} q)}\big )^{1/2}$ . Note that $\Gamma _{a,s}$ is strictly monotonically increasing.

A2

Data regularity: The data a and s satisfy the regularity assumption

$$\begin{align*}\lim_{\tau \to \infty} \Gamma_{a,s}(\tau) = \infty \,.\end{align*}$$

To classify the support of $\sigma $ we introduce the operator $B\equiv B_\zeta : \mathcal {B} \to \mathcal {B}$ given by

(2.8) $$ \begin{align} B_\zeta:= D_{\lvert a-\zeta \rvert^2}-S\,, \end{align} $$

where $D_u \colon \mathcal {B} \to \mathcal {B}, x \mapsto ux$ denotes the multiplication operator on $\mathcal {B}$ with $u \in \mathcal B$ . Since B maps real-valued functions to real-valued functions, we obtain a function $\beta \colon \mathbb {C} \to \mathbb {R}$ defined through

(2.9) $$ \begin{align} \beta(\zeta) := \inf_{x \in \mathcal B_+} \sup_{y \in \mathcal B_+} \frac{\langle{x} \mspace{2mu}, {B_\zeta \mspace{2 mu}y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle}\end{align} $$

for $\zeta \in \mathbb {C}$ , where the infimum and supremum are taken over $\mathcal B_+:=\{x \in \mathcal {B}: x>0\}$ and $ \langle {u_1} \mspace {2mu}, {u_2}\rangle := \langle \overline {u}_1 u_2 \rangle $ for all $u_1$ , $u_2 \in \mathcal B$ . The definition of $\beta $ is motivated by the Birkhoff-Varga formula for the spectral radius of a matrix with positive entries [Reference Birkhoff and Varga16]. In terms of $\beta $ we define the set

(2.10) $$ \begin{align} \mathbb S := \{ \zeta \in \mathbb{C} \colon \beta(\zeta ) < 0 \}\,, \end{align} $$

whose closure coincides with $\operatorname {\mathrm {supp}} \sigma $ , as stated in the next theorem. We will see in Proposition 5.16 (i) that $\beta $ is a continuous function and therefore $\mathbb {S}$ is an open set. Moreover, $\beta $ is real analytic in a neighbourhood of $\partial \mathbb {S}$ , see Corollary 5.20 below.

The proof of Theorem 2.2 is given at the end of Section 6 below.

Our next results describe in detail the edge behaviour of $\sigma $ at points $\zeta _0 \in \partial \mathbb {S}$ . Typically we observe a sharp drop of the density $\sigma $ at the edge $\partial \overline {\mathbb {S}}$ . In fact, by Theorem 2.2 (v) this is the case if and only if $\partial _\zeta \beta (\zeta _0)$ does not vanish.

To give a short statement of the classification of the singular points for $\sigma $ we introduce the following notion.

Definition 2.4 (Singularity types).

Let $\alpha : U \to \mathbb {C}$ be a real analytic function on an open set $U \subset \mathbb {C}$ and $\zeta _0 \in U$ . We say that $\alpha $ is of singularity type $p(x,y)$ at $\zeta _0$ for a polynomial $p: \mathbb {R}^2 \to \mathbb {C}$ if there is a real analytic diffeomorphism $\Phi : V \to U_0$ from an open neighbourhood V of $0 \in \mathbb {C}$ to an open neighbourhood $U_0$ of $\zeta _0 =\Phi (0)$ , such that

$$\begin{align*}\alpha(\Phi(x + \mathrm{i} y)) = p(x,y) \,, \qquad x + \mathrm{i} y \in V\,. \end{align*}$$

In the sense above $\beta $ is of singularity type x at any regular edge point. By Theorem 2.2 the edge $\partial \operatorname {\mathrm {supp}} \sigma $ is locally analytically diffeomorphic to a line and $\sigma $ has a jump discontinuity along this line.

Theorem 2.5 (Classification of edge and internal singularities).

Let $a \in \mathcal B$ and s satisfy A1 and A2. The set of singular points for $\sigma $ admits a partition

(2.11) $$ \begin{align} { \mathrm{Sing}= \bigcup_{k=1}^\infty\big({\mathrm{Sing}_{k}^{\mathrm{int}}\cup \mathrm{Sing}_{k}^{\mathrm{edge}}}\big)\cup \mathrm{Sing}^{\mathrm{int}}_{\infty}\,. } \end{align} $$

Here $\mathrm {Sing}\setminus \mathrm {Sing}_{\infty }$ is a finite set, while $\mathrm {Sing}_{\infty }$ is either empty or a finite disjoint union of closed real analytic paths without self-intersections. The different singularity types are characterised by the following properties:

1. 1) Internal singularities of order k: Any $\zeta \in \mathrm {Sing}_{k}^{\mathrm {int}}$ is an isolated point of $\mathbb {C} \setminus \mathbb S$ and $\sigma $ is of singularity type $x^2 + y^{2k}$ at $\zeta $ .

2. 2) Edge singularities of order k: Any $\zeta \in \mathrm {Sing}_{k}^{\mathrm {edge}}$ lies in $\partial \overline {\mathbb S}$ and $\beta $ is of singularity type $ y^{1+k}-x^2$ at $\zeta $ .

3. 3) Internal singularities of order $\infty $ : Any $\zeta \in \mathrm {Sing}^{\mathrm {int}}_{\infty }$ lies in the interior of $\overline {\mathbb S}$ and $\sigma $ is of singularity type $x^2$ at $\zeta $ .

The proof of Theorem 2.5 is presented at the end of Section 6 below.

Remark 2.6. The internal singularities $\mathrm {Sing}_{k}^{\mathrm {int}}\cup \mathrm {Sing}^{\mathrm {int}}_{\infty }$ are classified by their local behaviour of the density $\sigma $ . The edge singularities $\mathrm {Sing}_{k}^{\mathrm {edge}} $ are equivalently characterised by the local shape of the spectral edge $\partial \operatorname {\mathrm {supp}} \sigma $ , namely for $\zeta \in \mathrm {Sing}_{k}^{\mathrm {edge}}$ there is a local analytic diffeomorphism $\Psi \colon U \to V$ from an open neighbourhood V of $\zeta _0$ to an open $U \subset \mathbb {C}$ with $\Psi (\zeta _0)= 0$ such that

$$\begin{align*}\Psi( \operatorname{\mathrm{supp}} \sigma \cap U) = \{x+\mathrm{i} y \in \mathbb{C}: x^2 -y^{1+k}\ge 0\}\cap V \end{align*}$$

Finally we show that all singularities allowed by the classification in Theorem 2.5 do occur in the simple case when $\mathfrak {c}$ is a standard circular element and $\mathfrak {X}=[0,1]$ with the Lebesgue measure. A standard circular element satisfies $E[\mathfrak {c}^* b \mathfrak {c}]=E[\mathfrak {c} b \mathfrak {c}^*] = \langle b \rangle $ for all $b \in \mathcal B$ .

The proof of Theorem 2.7 can be found just before Section 7.1.

Remark 2.8 (Circular element with general deformation).

Although not directly covered by our results stated above, the classification of singularities from Theorem 2.5 also holds for the Brown measure of $a + \mathfrak c$ , where a is a general operator in a tracial von Neumann algebra and $\mathfrak c$ is a standard circular element, which is $\ast $ -free from a. We note that a can be non-normal in general. We define $\mathbb {S} = \{ \zeta \in \mathbb {C} \colon f(\zeta )> 1\}$ with

$$\begin{align*}f(\zeta):= \lim_{\eta \downarrow 0} \bigg\langle \frac{1}{(a-\zeta)(a-\zeta)^* + \eta^2} \bigg\rangle\, \end{align*}$$

for $\zeta \in \mathbb {C}$ . Then the support of the Brown measure $\sigma $ is $\overline {\mathbb S}$ [Reference Zhong64] and the measure has a density on $\mathbb {C}$ [Reference Belinschi, Yin and Zhong13], which is real analytic and strictly positive inside $\mathbb {S}$ [Reference Zhong64]. Under the additional assumption

(2.12) $$ \begin{align} \operatorname{\mathrm{Spec}}(a) \subset \mathbb{S}, \end{align} $$

the dichotomy between regular edge points, at which the density has a jump discontinuity, and singular points of $\sigma $ was established in [Reference Erdős and Ji32]. The singular points $\zeta \in \mathrm {Sing}$ are called quadratic edges in [Reference Erdős and Ji32] and are classified by satisfying $f(\zeta )=1$ and $\partial _\zeta f(\zeta )=0$ , while regular edge points $\zeta \in \mathrm {Reg}$ satisfy $f(\zeta )=1$ and $\partial _\zeta f(\zeta )\ne 0$ . Here we provide a classification of the singular points $\zeta \in \mathrm {Sing}$ by showing that $\mathrm {Sing}$ is a disjoint union of the sets $ \mathrm {Sing}_{k}^{\mathrm {int}}$ , $ \mathrm {Sing}_{k}^{\mathrm {edge}}$ with $k \in \mathbb {N}$ and $ \mathrm {Sing}^{\mathrm {int}}_{\infty }$ as in (2.11), where these sets are defined as in Theorem 2.5 with $\beta := 1 - f$ . We provide the proof for this classification in Section 8 below.

2.1 Notations

We now introduce some notations used throughout. We write for $n \in \mathbb {N}$ . For $r>0$ , we denote by $\mathbb {D}_r := \{ z \in \mathbb {C} \colon \lvert z \rvert < r \}$ the disk of radius r around the origin in $\mathbb {C}$ and by $\mathrm{dist} (x,A) :=\inf \{ \lvert x-y \rvert \colon y \in A \}$ the Euclidean distance of a point $x \in \mathbb {C}$ from a set $A \subset \mathbb {C}$ .

We use the convention that c and C denote generic constants that may depend on the model parameters, but are otherwise uniform in all other parameters, for example, n, $\zeta $ , etc.. For two real scalars f and g we write $f \lesssim g$ and $g \gtrsim f$ if $f \leq C g$ for such a constant $C>0$ . In case $f \lesssim g$ and $f \gtrsim g$ both hold, we write $f \sim g$ . If the constant C depends on a parameter $\delta $ that is not a model parameter, we write $\lesssim _\delta $ , $\gtrsim _\delta $ and $\sim _\delta $ , respectively. The notation for inequality up to constant is also used for self-adjoint matrices/operators f and g, where $f \leq C g$ is interpreted in the sense of quadratic forms. For complex f and $g \geq 0$ we write $f = O(g)$ in case $\lvert f \rvert \lesssim g$ . Analogously $f = O_\delta (g)$ expresses the fact $\lvert f \rvert \lesssim _\delta g$ .

In addition to $L^\infty $ , we introduce the usual $L^p$ spaces on $(\mathfrak X, \mathcal A, \mu )$ . We denote them by $L^p:=L^p(\mathfrak X, \mathcal {A}, \mu )$ and the corresponding norms by $\lVert \mspace {2 mu}\cdot \mspace {2 mu} \rVert _p$ . For functions $u_1$ , $u_2 \in L^2$ , we define their scalar product as

$$\begin{align*}\langle{u_1} \mspace{2mu}, {u_2}\rangle := \langle \overline{u}_1 u_2 \rangle\,. \end{align*}$$

3 Examples

In this section, we present several examples that illustrate some of the different singularity types appearing in Theorem 2.5. Throughout this section, we choose $\mathfrak X = [0,1]$ , $\mu $ the Lebesgue-measure on $[0,1]$ and s constant on $[0,1]^2$ . If $s \equiv t$ is constant on $[0,1]^2$ then it is easy to see that $\beta (\zeta ) = \frac {1}{t} - \int _0^1 \frac {\mathrm {d} x}{\lvert a(x) - \zeta \rvert ^2}$ for all $\zeta \in \mathbb {C}$ and therefore, as has already been derived in [Reference Khoruzhenko42, Reference Tao, Vu and Krishnapur57, Reference Bordenave, Caputo and Chafaï18],

(3.1) $$ \begin{align} \mathbb{S} = \bigg\{ \zeta \in \mathbb{C} \colon \int_0^1 \frac{\mathrm{d} x}{\lvert a(x)-\zeta \rvert^2}> \frac{1}{t} \bigg\}. \end{align} $$

Example 3.1 (Simplest edge singularity of $\partial \mathbb {S}$ ).

We choose $s \equiv 1$ and $a \colon [0,1] \to \mathbb {C}$ with $a(x_1) =1$ and $a(x_2) = -1$ for all $x_1 \in [0,1/2)$ and $x_2 \in [1/2,1]$ . In that case, we obtain from (3.1) that

$$\begin{align*}\mathbb{S} = \{ \zeta \in \mathbb{C} \colon \lvert 1-\zeta \rvert^{-2} + \lvert 1+ \zeta \rvert^{-2}>1 \}. \end{align*}$$

As we will see in Lemma 7.3 (a) below, this yields an edge singularity of order $1$ , that is, in $\mathrm {Sing}_1^{\mathrm {edge}}$ at zero. The boundary of $\mathbb {S}$ and the eigenvalues of a sampled corresponding random matrix model are depicted in Figure 1a. This example also appeared in [Reference Biane and Lehner15, Example 5.2].

Figure 1

The solid black lines in subfigures (1a) and (1b) show the boundary of $\mathbb {S}$ from Examples 3.1 and 3.2, respectively. The black dots are the eigenvalues of a sample of $A + X/\sqrt {n}$ , where X is an $n\times n$ matrix with i.i.d. $N(0,1)$ standard real normal distributed entries, $n=10000$ and $A=(\operatorname {\mathrm {diag}}(a(i/n))\delta _{ij})_{i,j=1}^n$ is a diagonal matrix and a is chosen as in Examples 3.1 and 3.2, respectively.

Example 3.2 (Simplest internal singularity of $\sigma $ ).

With the choices $s \equiv 1/4$ and $a \colon [0,1] \to \mathbb {C}$ with $a(x_1) = (1 + \mathrm {i})/\sqrt {2}$ , $a(x_2) = (1 - \mathrm {i})/\sqrt {2}$ , $a(x_3) = (-1 + \mathrm {i})/\sqrt {2}$ and $a(x_4) = -(1 + \mathrm {i})/\sqrt {2}$ for all $x_1 \in [0,1/4)$ , $x_2 \in [1/4,1/2)$ , $x_3 \in [1/2,3/4)$ and $x_4 \in [3/4,1]$ , (3.1) yields

$$\begin{align*}\mathbb{S} = \{ \zeta \in \mathbb{C} \colon \lvert \zeta -(1 + \mathrm{i})/\sqrt{2} \rvert^{-2} + \lvert \zeta -(1 - \mathrm{i})/\sqrt{2} \rvert^{-2} + \lvert \zeta +(1 - \mathrm{i})/\sqrt{2} \rvert^{-2} + \lvert \zeta +(1 + \mathrm{i})/\sqrt{2} \rvert^{-2}> 1 \}.\end{align*}$$

Lemma 7.3 (a) below will demonstrate that this gives rise to an internal singularity of order $1$ , that is, in $\mathrm {Sing}_2^{\mathrm {int}}$ at zero. Figure 1b shows the boundary of $\mathbb {S}$ and the eigenvalues of a sampled corresponding random matrix model. We note that a similar example was given in [Reference Erdős and Ji32, Example 3.1(d)].

Example 3.3 (One-sided edge cusp of $\partial \mathbb {S}$ ).

Let $s \equiv t:= \frac {2}{3}(20-7\sqrt {7})$ be constant on $[0,1]^2$ , $\delta = (-17 + 7\sqrt {7})/8$ and $a \colon [0,1] \to \mathbb {C}$ with $a(x_1) = 1$ , $a(x_2) = -1$ and $a(x_3) = \mathrm {i}$ for all $x_1 \in [0,1/(2+ \delta ))$ , $x_2 \in [1/(2 + \delta ),2/(2+\delta ))$ and $x_3 \in [2/(2+\delta ),1]$ . We note that (3.1) yields

$$\begin{align*}\mathbb{S} = \bigg\{ \zeta \in \mathbb{C} \colon \frac{1}{\lvert 1-\zeta \rvert^2} + \frac{1}{\lvert 1+ \zeta \rvert^2} + \frac{\delta}{\lvert \zeta- \mathrm{i} \rvert^2}> \frac{2 + \delta}{t} \bigg\}. \end{align*}$$

With $y_0 = (\sqrt {7} - 2)/3$ , a simple calculation shows that $\beta $ is of singularity type $x^2 + y^3$ at $\mathrm {i} y_0$ and, thus, $\mathrm {i} y_0$ is an edge singularity of order $2$ . For this example, a plot analogous to the previous examples is presented in Figure 2a.

Figure 2

The solid black lines in subfigures (2a) and (2b) show the boundary of $\mathbb {S}$ from Examples 3.3 and 3.4, respectively. The black dots are the eigenvalues of a sample of $A + X/\sqrt {n}$ , where X is an $n\times n$ matrix with i.i.d. $N(0,1)$ standard real normal distributed entries, $n=10000$ and $A=(\operatorname {\mathrm {diag}}(a(i/n))\delta _{ij})_{i,j=1}^n$ is a diagonal matrix and a is chosen as in Examples 3.3 and 3.4, respectively.

Example 3.4 (Two-sided edge cusp).

We set $s \equiv t := 4$ to be constant on $[0,1]^2$ and $a\colon [0,1] \to \mathbb {C}$ , with $a(x_1) = \sqrt {3} + \mathrm {i}$ , $a(x_2) = \sqrt {3} - \mathrm {i}$ , $a(x_3) = - \sqrt {3} + \mathrm {i}$ and $a(x_4) = - \sqrt {3} - \mathrm {i}$ for all $x_1 \in [0,1/4)$ , $x_2 \in [1/4,1/2)$ , $x_3 \in [1/2,3/4)$ and $x_4 \in [3/4,1]$ . We conclude from (3.1) that

$$\begin{align*}\mathbb{S} = \bigg\{ \zeta \in \mathbb{C} \colon \frac{1}{\lvert \sqrt{3} + \mathrm{i}-\zeta \rvert^2} + \frac{1}{\lvert \sqrt{3} - \mathrm{i}-\zeta \rvert^2} + \frac{1}{\lvert - \sqrt{3} + \mathrm{i}-\zeta \rvert^2} + \frac{1}{\lvert - \sqrt{3} - \mathrm{i}-\zeta \rvert^2}> 1 \bigg\}. \end{align*}$$

Using Lemma 7.2 below, $-\beta $ and hence the boundary of $ \mathbb {S}$ has a singularity of type $x^2 - y^4$ at zero, making this an edge singularity of order $2$ . Figure 2b shows a plot of this example analogous to the previous figures.

We refer to [Reference Alt, Erdős, Krüger and Nemish7, Example 2.6 and Figure 1] for more examples in the spirit of the previous examples.

Example 3.5 (Internal singularity of type $\mathrm {Sing}^{\mathrm {int}}_{\infty }$ ).

We choose $s \equiv 1$ on $[0,1]^2$ and

$$\begin{align*}a \colon [0,1] \to \mathbb{C}, \qquad x \mapsto \begin{cases} \sqrt{2} \mathrm{e}^{4\pi \mathrm{i} x} & \text{ if } x \in [0,1/2], \\ 0 & \text{ if } x \in (1/2,1]. \end{cases} \end{align*}$$

As will be proved in Section 7.3 below, in this setup

(3.2) $$ \begin{align} \mathbb S = \big\{ \zeta \in \mathbb{C} \colon 0 \leq \lvert \zeta \rvert^2 < 1 \text{ or } 1 < \lvert \zeta \rvert^2 < {(3 + \sqrt{5})/2} \big\} \end{align} $$

and each point of $\partial \mathbb {D}_1$ belongs to $\mathrm {Sing}_\infty ^{\mathrm {int}}$ . Moreover, it is shown in Section 7.3 that, for each $\zeta \in \mathbb {C}$ ,

(3.3) $$ \begin{align} \sigma(\zeta) = \frac{1}{\pi} \bigg( 1 - \frac{2}{2+x + \frac{x}{(2x-1)^3}} \bigg( 1 + \frac{1}{x^2} \bigg) \bigg) \mathbf{1} (\zeta \in \mathbb{S}), \end{align} $$

where $x =x(\lvert \zeta \rvert ^2) \in (0,\infty )$ is the unique positive solution of $\frac {1}{x} + \frac {1}{\sqrt {1 + 4x+x^2-8\lvert \zeta \rvert ^2 + 3}}=2$ . The boundary of $\mathbb {S}$ and the sampled eigenvalues are depicted in Figure 3.

Figure 3

The solid lines in this figure show the boundary of $\mathbb {S}$ from Example 3.5. The black dots show the eigenvalues of a sample of $A + X/\sqrt {n}$ , where X is an $n\times n$ matrix with i.i.d. $N(0,1)$ standard real normal distributed entries, $n=10000$ and $A=(\operatorname {\mathrm {diag}}(a(i/n))\delta _{ij})_{i,j=1}^n$ is a diagonal matrix and a is chosen as in Example 3.5.

4 Brown measure, Hermitisation and Dyson equation

In this section, we derive the Dyson equation for the Cauchy transform of the Hermitisation of the deformed operator-valued circular element $a + \mathfrak c$ . This Hermitisation itself is a family of operator-valued semicircular elements indexed by the spectral parameter $\zeta \in \mathbb {C}$ and the associated Dyson equation is our main tool for analysing the Brown measure of $a + \mathfrak c$ .

Throughout the following, let $(\mathcal A, E, \mathcal B)$ be an operator-valued probability space with positive conditional expectation $E \colon \mathcal A \to \mathcal B$ such that $(\mathcal A, \langle E[\,\cdot \,] \rangle )$ is a tracial $W^*$ -probability space. We assume that $\mathcal B \subset \mathcal A$ is a commutative von Neumann subalgebra with trace $\langle \mspace {2 mu}\cdot \mspace {2 mu} \rangle $ containing the unit of $\mathcal A$ . We will also work on the operator-valued probability space $(\mathcal A^{2\times 2}, \mathrm {id} \otimes E, \mathcal B^{2\times 2})$ , for which we employ the identifications $\mathcal A^{2\times 2}= \mathbb {C}^{2 \times 2} \otimes \mathcal A$ and $\mathcal B^{2\times 2}= \mathbb {C}^{2 \times 2} \otimes \mathcal B$ .

For any $\mathfrak c \in \mathcal A$ , $a \in \mathcal B$ and $\zeta \in \mathbb {C}$ , we hermitise $\mathfrak c + a - \zeta $ and apply $\mathrm {id} \otimes E$ to the resolvent of the Hermitisation at $w \in \mathbb {C}$ with $\mathrm {Im}\, w>0$ . We define

(4.1) $$ \begin{align} M(\zeta,w ) = \mathrm{id} \otimes E \bigg[\begin{pmatrix} - w & \mathfrak c + a - \zeta \\ (\mathfrak c +a - \zeta)^* & - w \end{pmatrix}^{-1} \bigg]. \end{align} $$

Here, w and $\zeta $ are interpreted as the constant functions on $\mathfrak X$ with the respective value. Note that the inverse is well defined as $\mathrm {Im}\, w>0$ and the operator whose inverse is taken in (4.1) is self-adjoint for $w = 0$ . Furthermore, the imaginary part $\mathrm {Im}\, M(\zeta ,w) =\frac {1}{2\mathrm {i}} ( M(\zeta ,w) - M(\zeta ,w)^*)$ is positive definite.

For the next lemma, we define $\Sigma \colon \mathcal B^{2\times 2} \to \mathcal B^{2\times 2}$ through

(4.2) $$ \begin{align} \Sigma \bigg[ \begin{pmatrix} r_{11} & r_{12} \\ r_{21} & r_{22} \end{pmatrix} \bigg] = \begin{pmatrix} S r_{22} & 0 \\ 0 & S^* r_{11} \end{pmatrix} \end{align} $$

for all $r_{11}$ , $r_{12}$ , $r_{21}$ , $r_{22} \in \mathcal B$ , where S and $S^*$ are as in (2.2).

Lemma 4.1. Let $\mathfrak c \in \mathcal A$ be a $\mathcal {B}$ -valued circular element satisfying (2.1) and (2.2). Then $\begin {pmatrix} 0 & \mathfrak {c} \\ \mathfrak {c}^* &0 \end {pmatrix}$ is an operator-valued semicircular element in $(\mathcal A^{2\times 2}, \mathrm {id} \otimes E, \mathcal B^{2\times 2})$ with covariance $\Sigma $ from (4.2). Moreover, $M(\zeta ,w)$ from (4.1) with such $\mathfrak c$ satisfies

(4.3) $$ \begin{align} - M(\zeta,w)^{-1} = \begin{pmatrix} w & \zeta - a \\ \overline{\zeta - a} & w \end{pmatrix} + \Sigma[M(\zeta,w)] \end{align} $$

for any $a \in \mathcal B$ , $w \in \mathbb {C}$ with $\mathrm {Im}\, w>0$ and $\zeta \in \mathbb {C}$ .

The identity (4.3) is called Dyson equation. Under the constraint that $\mathrm {Im}\, M(\zeta , w) $ is positive definite, the Dyson equation has a unique solution [Reference Helton, Rashidi Far and Speicher38, Theorem 2.1]. In particular, $M(\zeta ,w)$ from (4.1) coincides with this solution. For constant function s the identity (4.3) has appeared in [Reference Bordenave, Caputo and Chafaï18] and its analog for elliptic operators in [Reference Zhong64].

We now express the Brown measure of $\mathfrak c + a$ through M from (4.1) or, equivalently, the solution to (4.3). To that end, it suffices to consider $w = \mathrm {i} \eta $ with $\eta \in (0,\infty )$ and we see that the Dyson equation can be simplified in this case. We consider two coupled equations for functions $v_1, v_2 \in \mathcal {B}$ with $v_1>0$ and $v_2>0$ , namely

(4.4a) $$ \begin{align} \frac{1}{ v_1} & = \eta +Sv_2+ \frac{\lvert \zeta-a \rvert^2}{\eta +S^*v_1} \,, \end{align} $$

(4.4b) $$ \begin{align} \frac{1}{ v_2} & = \eta +S^*v_1+ \frac{\lvert \zeta-a \rvert^2}{\eta +Sv_2}\,, \end{align} $$

for all $\eta>0$ and $\zeta \in \mathbb {C}$ . We also call (4.4) Dyson equation. Given $v_1$ and $v_2$ from (4.4), we introduce y defined by

(4.5) $$ \begin{align} y:=\frac{v_1\mspace{2 mu}( \bar a-\bar \zeta)}{\eta + S^*v_1}=\frac{v_2\mspace{2 mu}(\bar a-\bar \zeta)}{\eta + Sv_2}\,, \end{align} $$

where the second step follows from

(4.6) $$ \begin{align} v_2(\eta + S^*v_1)=v_1(\eta + Sv_2)\,. \end{align} $$

Indeed, for the proof of the last identity, we multiply (4.4a) by $v_1 v_2 (\eta + S^*v_1)$ and (4.4b) by $v_1 v_2(\eta + Sv_2)$ and conclude (4.6) from the resulting relations.

The next lemma relates the solutions to (4.4) and (4.3) and implies existence and uniqueness to the one of (4.4).

When evaluated on the imaginary axis at $w=\mathrm {i} \eta $ with $\eta>0$ , the representation (4.1) of the solution to the Dyson equation (4.3) with positive definite imaginary part implies the trivial bound

(4.8) $$ \begin{align} \lVert M(\zeta,\mathrm{i} \eta) \rVert \leq \eta^{-1}. \end{align} $$

Indeed, we note that the argument of $ \mathrm {id} \otimes E$ in (4.1) with $w = \mathrm {i} \eta $ is bounded in norm by $\eta ^{-1}$ as the resolvent of a self-adjoint operator. Therefore, $\lVert M(\zeta ,\mathrm {i}\eta ) \rVert \leq \eta ^{-1}$ follows from $\lVert E \rVert = \lVert E(1) \rVert = 1$ due to the positivity of E and $E(1) =1$ . This completes the proof of (4.8).

Using the solution to (4.4) we characterise the Brown measure in the following proposition as the Laplacian on $\mathbb {C}$ of the function $-\frac {1}{2\pi } W$ defined through

(4.9) $$ \begin{align} W (\zeta) := \int_0^\infty \bigg(\langle v_1(\zeta, \eta) \rangle - \frac{1}{1 + \eta} \bigg) \mathrm{d} \eta \end{align} $$

for each $\zeta \in \mathbb {C}$ , where $v_1$ is the first function in the solution pair $(v_1,v_2)$ of the Dyson equation (4.4). If A1 holds then the integral in (4.9) exists in the Lebesgue sense, which follows from Lemma 6.1 below. The proposition below therefore connects the Brown measure of $a+ \mathfrak c$ to $v_1$ .

The proof of Proposition 4.3 is presented at the end of Section 6.1 below. We conclude this section with a short overview of the contents of the remaining paper. The next section provides a detailed analysis of the solution $(v_1,v_2)$ of the Dyson equation as well as a characterisation of $\mathbb {S}$ from (2.10). Section 6 is devoted to the translation of these insights to the analysis of the Brown measure. We establish the existence of all singularity types in Section 7.

5 The Dyson equation and its solution

In this section we analyse the solution $v_1, v_2$ to the Dyson equation (4.4). We begin by establishing bounds on the solution. Then we determine stability and regularity properties. Finally we use these insights to characterise the set $\mathbb {S}$ from (2.10) and expand $v_1$ and $v_2$ around edge points $\zeta _0 \in \partial \mathbb {S}$ . We start with some useful identities and a priori estimates.

As S and $S^*$ both act on $\mathcal B = L^\infty (\mathfrak X,\mu )$ as integral operators in the form (2.5), the identity (4.6) implies $\langle v_2 S^* v_1 \rangle = \langle v_1 Sv_2 \rangle $ and thus

(5.1) $$ \begin{align} \langle v_1 \rangle =\langle v_2 \rangle\,. \end{align} $$

We conclude from (4.4) and (4.5) that

(5.2) $$ \begin{align} v_1 \leq \eta^{-1}, \qquad v_2 \leq \eta^{-1}, \qquad \lvert y \rvert \leq \lvert a - \zeta \rvert \eta^{-2} \end{align} $$

for all $\zeta \in \mathbb {C}$ and $\eta>0$ . Furthermore, we have the identity

(5.3) $$ \begin{align} y= \frac{v_1(\bar a-\bar \zeta)}{\eta +S^*v_1} = \frac{1}{a-\zeta}(1- v_1( \eta +Sv_2)) =\frac{1}{a-\zeta}- \frac{v_1v_2}{\overline{y}} \end{align} $$

for any $\zeta \in \mathbb {C}\setminus \operatorname {\mathrm {Spec}}(D_a)$ . Here, $\operatorname {\mathrm {Spec}}(D_a)$ denotes the spectrum of $D_a$ considered as multiplication operator $\mathcal B \to \mathcal B$ , which coincides with the essential range of a.

Throughout the remainder of this section, we assume that s satisfies A1. This implies

(5.4)

for all $w \in \mathcal B$ with $w \geq 0$ and all .

5.1 Bounds on the solution

This subsection contains bounds on $v_1$ and $v_2$ under varying assumptions on s and a.

Lower bound on $v_1$ and $v_2$

We start with some preliminary bounds that require $a \in \mathcal B$ and the boundedness of the function s.

Lemma 5.1 (Preliminary bounds on $v_1$ and $v_2$ ).

If for some constant $C>0$ , $s(x,y) \leq C$ for all x, $y \in \mathfrak X$ then

(5.5) $$ \begin{align} \frac{\eta}{\eta^2 + 1 +\lvert \zeta \rvert^2} \lesssim_C v_i \leq \frac{1}{\eta} \end{align} $$

uniformly for all $i=1$ , 2, $\zeta \in \mathbb {C}$ and $\eta>0$ .

Proof. The positivity of $v_i$ and $\eta $ in (4.4) imply $1/v_i \geq \eta $ . Hence, the upper bounds in (5.5) follow.

As $s(x,y) \leq C$ for all x, $y \in \mathfrak X$ , we conclude from (4.4) and the upper bound $v_2 \leq \eta ^{-1}$ that

$$\begin{align*}\frac{1}{v_1} \leq \eta + C\eta^{-1} + \eta^{-1} 2 ( \lVert a \rVert_\infty^2 + \lvert \zeta \rvert^2), \end{align*}$$

which implies the missing lower bound in (5.5).

We continue by showing that $v_1$ and $v_2$ are bounded from below by their average.

Lemma 5.2. If s satisfies A1 then

(5.6) $$ \begin{align} v_1 \gtrsim \langle v_1 \rangle \,, \qquad v_2 \gtrsim \langle v_2 \rangle \end{align} $$

uniformly for all $\zeta \in \mathbb {C}$ and $\eta>0$ . As a consequence $S^*$ and S act averaging on $v_1$ and $v_2$ , respectively, that is,

(5.7) $$ \begin{align} S^*v_1 \sim \langle v_1 \rangle \,, \qquad Sv_2 \sim \langle v_2 \rangle \,. \end{align} $$

The following proof is based on identifying $(v_1,v_2)$ as solution to a variational problem. In a similar context, such a strategy was also used in [Reference Ajanki, Erdős and Krüger2, Section 6.2 and Appendix A.4] and [Reference Alt, Erdős and Krüger4, Lemmas 3.11 and 3.17].

Proof. For the proof we rephrase (4.4) as a variational problem. This will help us to make use of the assumption A1. For $\eta>0$ , we define $J_{\eta }\colon \mathcal {B}_+ \times \mathcal {B}_+ \to \mathbb {R} \cup \{ + \infty \}$ , a functional which is minimised by $(v_1,v_2)$ , by

$$\begin{align*}J_\eta(x_1,x_2):= \langle x_1 {S}x_2 \rangle +\eta \langle x_1+x_2 \rangle- \big\langle ({1-u_\eta})\log x_1 \big\rangle -\big\langle ({1-u_\eta})\log x_2 \big\rangle \,, \end{align*}$$

where

$$\begin{align*}u_\eta:= \frac{|a-\zeta|^2\mspace{1 mu}v_1}{\eta+S^* v_1} =\frac{|a-\zeta|^2\mspace{1 mu}v_2}{\eta +S v_2}\, \end{align*}$$

by (4.6). Note that $1-u_\eta = v_1(\eta + Sv_2) \in \mathcal {B}_+$ by (4.4).

Next, we prove that $(x_1,x_2)=(v_1(\eta ),v_2(\eta ))$ is a minimiser for $J_\eta $ . Indeed, the functional $J_\eta $ satisfies the lower bounds

with and . Here, in the second step, we applied Jensen’s inequality with the expectation , that is, . Since $\langle \log x_i \rangle \le \log \langle x_i \rangle $ by Jensen’s inequality, the functional $J_\eta $ is bounded from below for any $\eta>0$ . Therefore, $\iota _\eta = \inf _{(x_1,x_2) \in \mathcal B_+ \times \mathcal B_+} J_\eta (x_1,x_2)> -\infty $ . Moreover, $(v_1,v_2)$ satisfies the corresponding Euler-Lagrange equations and $J_\eta $ is strictly convex due to $1-u_\eta = v_1(\eta + Sv_2)$ and (5.5). Hence, $J_\eta (v_1,v_2) = \iota _\eta $ .

In the regime $\eta \gg 1$ , that is, there is a constant $\eta _0>0$ depending only on the model parameters such that for all $\eta \geq \eta _0$ , we deduce from (4.4) and the bound $v_i \leq 1/\eta $ in Lemma 5.1 that

$$\begin{align*}v_i \sim \frac{\eta}{|\zeta|^2 + \eta^2}, \end{align*}$$

which implies $v_i \sim \langle v_i \rangle $ . Thus, we restrict to $\eta \lesssim 1$ . In case $\eta \lesssim 1$ and $|\zeta |\gg 1 $ we get

$$\begin{align*}v_i \sim \frac{\eta}{|\zeta|^2}, \end{align*}$$

and therefore $v_i \sim \langle v_i \rangle $ . Hence, we restrict to $\eta \lesssim 1$ and $|\zeta |\lesssim 1$ in the following. As $(v_1,v_2)$ is a minimiser of $J_\eta $ , the value $J_\eta (v_1,v_2)$ is bounded by

$$\begin{align*}J_\eta(v_1,v_2)\le J_\eta(1,1) = 2\eta +\langle {S}1 \rangle\,. \end{align*}$$

With the notations $c_i:= \mu (I_i)$ , and , we see by Jensen’s inequality that

$$ \begin{align*} 1 &\gtrsim J_{\eta}(v_1,v_2) \\ &= \langle v_1 {S}v_2 \rangle - \sum_{j}c_j\big\langle ({1-u})\log v_1 \big\rangle_j -\sum_{j}c_j\big\langle ({1-u})\log v_2 \big\rangle_j+ \eta \langle v_1 + v_2 \rangle \\ &\ge c\mspace{1 mu}\sum_{i,j} z_{ij}\langle v_1 \rangle_i\langle v_2 \rangle_j - \sum_{j}c_j\langle 1-u \rangle_j\log\frac{ \langle (1-u)v_1 \rangle_j\langle (1-u)v_2 \rangle_j}{\langle 1-u \rangle_j^2} \\ &\ge c\mspace{1 mu}\sum_{i,j} z_{ij}\langle v_1 \rangle_i\langle v_2 \rangle_j - \sum_{j}c_j\langle 1-u \rangle_j\log \big({\langle v_1 \rangle_j \langle v_2 \rangle_j}\big)+2\sum_{j}c_j\langle 1-u \rangle_j\log\langle 1-u \rangle_j \\ &\ge \sum_{j} \varphi_j(\chi_j)-\frac{2}{\mathrm{e}}\sum_{j}c_j\,, \end{align*} $$

where in the last step we used $z_{jj} = 1$ for all and defined

for all and $\chi \in (0,\infty )$ . For all , we infer $\chi _j \lesssim 1$ and

(5.8) $$ \begin{align}{ \chi_j \ge \exp\bigg({-\frac{C}{1- \langle u \rangle_j}}\bigg) =\exp\bigg({-\frac{C}{\langle v_1(\eta+Sv_2) \rangle_j}}\bigg) } \end{align} $$

for some positive constant $C \sim 1$ . We rewrite the Dyson equation (4.4) in the form

(5.9) $$ \begin{align} { v_1 = \frac{\eta +S^*v_1}{|a-\zeta|^2+(\eta +S^*v_1)(\eta+Sv_2)}\,, \qquad v_2 = \frac{\eta +Sv_2}{|a-\zeta|^2+(\eta+S^*v_1)(\eta+Sv_2)}\,. } \end{align} $$

Now we set $x:=(\eta +S^*v_1)(\eta +Sv_2)$ . In particular, we have

(5.10)

due to (5.4). Suppose now that $\langle x \rangle _i \gg 1$ for an . Then by taking the $\langle \mspace {2 mu}\cdot \mspace {2 mu} \rangle _i$ -average of (5.9), multiplying the two equations and using (5.4) and (5.10) as well as $\langle x \rangle _i \gg 1$ we get

$$\begin{align*}\chi_i \sim \frac{1}{\langle x \rangle_i}\ll 1\,. \end{align*}$$

On the other hand by multiplying the first equation in (5.9) with $\eta +Sv_2$ we find

$$\begin{align*}\langle v_1(\eta +Sv_2) \rangle_i \sim 1\, \end{align*}$$

as $\langle x \rangle _i \gg 1$ . This contradicts (5.8) and $\chi _i \ll 1$ and we conclude $\langle x \rangle _i \lesssim 1$ for all . With (5.9) and (5.10), this implies

$$\begin{align*}v_1 \gtrsim S^*v_1\,, \qquad v_2 \gtrsim Sv_2\,. \end{align*}$$

Iterating these inequalities yields $v_1 \gtrsim \langle v_1 \rangle $ and $v_2 \gtrsim \langle v_2 \rangle $ , that is, (5.6), since $(z_{ij})_{i,j}$ is primitive by A1. Since $z_{ii}=1$ for all , we have $S1 \gtrsim 1$ and $S^* 1 \gtrsim 1$ . This implies $Sv_2 \gtrsim \langle v_2 \rangle $ and $S^* v_1 \gtrsim \langle v_1 \rangle $ , respectively. The upper bounds on $Sv_2$ and $S^*v_1$ in (5.7) follow from the upper bound on s in A1.

Bound in $L^2$ -norm:

Now we show a bound with respect to the norm on $L^2$ .

Lemma 5.3. If s satisfies A1 then

(5.11) $$ \begin{align} \langle v_1^2 \rangle + \langle v_2^2 \rangle + \langle \lvert y \rvert^2 \rangle \lesssim 1 \end{align} $$

uniformly for all $\zeta \in \mathbb {C}$ and $\eta>0$ .

Proof. We multiply the first relation in (4.4a) by $v_1^2$ and estimate $v_1 \geq v_1^2 S v_2 \gtrsim v_1^2 \langle v_2 \rangle = v_1^2 \langle v_1 \rangle $ due to (5.7) and (5.1). Averaging this estimate and using $\langle v_1 \rangle>0$ yields $\langle v_1^2 \rangle \lesssim 1$ . The bound $\langle v_2^2 \rangle \lesssim 1$ is proved analogously. From (4.5), we conclude

(5.12) $$ \begin{align} \lvert y \rvert^2 = \frac{v_1^2 \lvert a - \zeta \rvert^2}{(\eta + S^* v_1)^2} \leq \frac{v_1^2 \lvert a - \zeta \rvert^2}{(\eta + S^* v_1)^2} + \frac{v_1^2(\eta + S v_2)}{\eta + S^* v_1} = \frac{v_1}{\eta + S^* v_1}, \end{align} $$

where we used (4.4) in the last step. Hence, (5.7) implies

$$\begin{align*}\langle \lvert y \rvert^2 \rangle \lesssim \frac{\langle v_1 \rangle}{\eta + \langle v_1 \rangle} \leq 1.\\[-43pt]\end{align*}$$

Corollary 5.4. Let $a \in \mathcal B$ and s satisfy A1. Then

$$\begin{align*}\frac{\eta + \langle v_i \rangle}{1+\eta^2 + \lvert \zeta \rvert^2 } \lesssim v_i \lesssim \frac{1}{\eta + \langle v_i \rangle}\,, \qquad i=1,2\,. \end{align*}$$

Proof. We only consider the case $i = 1$ . The case $i=2$ follows analogously. For the lower bound, we start from (4.4) and get

$$\begin{align*}v_1 =\frac{\eta + S^*v_1}{(\eta + S^*v_1)(\eta + Sv_2) + \lvert \zeta -a \rvert^2} \gtrsim \frac{\eta + \langle v_1 \rangle}{\eta^2 + \langle v_1 \rangle^2 + \lvert \zeta \rvert^2 + \lVert a \rVert_\infty^2} \gtrsim \frac{\eta + \langle v_1 \rangle}{1+\eta^2 + \lvert \zeta \rvert^2 }\,, \end{align*}$$

using $\langle v_1 \rangle =\langle v_2 \rangle $ by (5.1), (5.11) and (5.7). For the upper bound, (4.4), $\langle v_1 \rangle =\langle v_2 \rangle $ and (5.7) imply

$$\begin{align*}v_1 \le \frac{1}{\eta +Sv_2} \sim \frac{1}{\eta +\langle v_1 \rangle}\,.\\[-43pt] \end{align*}$$

Bound in supremum norm:

Under stronger assumptions on s and a, we can also get a bound on $v_i$ in the $L^\infty $ -norm. Let $\Gamma _{a,s}$ be as defined in (2.7).

Lemma 5.5. Assuming the upper bound $\max \big \{{\lVert v_1 \rVert _2,\lVert v_2 \rVert _2}\big \} \le \Lambda $ for some $\Lambda \ge 1$ and $4\Lambda ^2 < \lim _{\tau \to \infty }\Gamma _{a,s}(\tau )$ , then we have the bound

$$\begin{align*}\max\big\{{\lVert v_1 \rVert_\infty,\lVert v_2 \rVert_\infty}\big\} \le \frac{\Gamma_{a,s}^{-1}(4\Lambda^2)}{\Lambda}\,. \end{align*}$$

We remark that if s satisfies A1, then by Lemma 5.3 a $\Lambda \ge 1$ with $\Lambda \sim 1$ exists such that $\max \big \{{\lVert v_1 \rVert _2,\lVert v_2 \rVert _2}\big \} \le \Lambda $ holds uniformly for all $\eta>0$ and $\zeta \in \mathbb {C}$ .

Proof. We specialise the proof of [Reference Alt, Erdős and Krüger6, Lemma 9.1] to our setting.

Corollary 5.6. Let $a \in \mathcal B$ . If s and a satisfy A1 and A2 then

$$\begin{align*}\lVert v_1 \rVert_\infty + \lVert v_2 \rVert_\infty \lesssim 1\end{align*}$$

uniformly for all $\zeta \in \mathbb {C}$ and $\eta>0$ .

Scaling relations

Lemma 5.7. Let $a \in \mathcal B$ and s satisfy A1 and A2. Then the following holds.

1. (i) Uniformly for all $\zeta \in \mathbb {C}$ and $\eta>0$ , we have

   $$\begin{align*}v_1 \sim \langle v_1 \rangle = \langle v_2 \rangle \sim v_2, \qquad \lvert y \rvert \lesssim 1. \end{align*}$$

2. (ii) For any sufficiently small positive constant $c\sim 1$ the inequalities $\eta + \langle v_1(\zeta ,\eta ) \rangle \leq c$ and $\lvert \zeta \rvert \le 1/c$ imply $\lvert \zeta - a \rvert \sim 1$ and $\lvert y \rvert \sim 1$ .

Before going into the proof of Lemma 5.7, we remark that if $a \in \mathcal B$ and s satisfies A1 then

(5.13) $$ \begin{align} \lVert v_1(\zeta, \eta) - (1 + \eta)^{-1} \rVert \lesssim (1 + \lvert \zeta \rvert)\eta^{-2} \end{align} $$

uniformly for $\eta>0$ and $\zeta \in \mathbb {C}$ . Indeed, for $\eta \in (0,1]$ , (5.13) is a trivial consequence of (5.2). For $\eta \geq 1$ , (5.13) follows by inverting (4.4a), subtracting $(1 + \eta )^{-1}$ on both sides and estimating the right hand side using (5.7) and Lemma 5.3.

Proof. We first prove that $v_1 \sim \langle v_1 \rangle = \langle v_2 \rangle \sim v_2$ . As s satisfies A1, equation (4.4), $v_1$ , $v_2> 0$ and (5.7) imply

(5.14) $$ \begin{align} v_1 \sim v_2 \,. \end{align} $$

Hence, it suffices to show $v_1 \sim \langle v_1 \rangle $ due to (5.1).

From (5.13), we conclude that $v_1 \sim (1+ \eta )^{-1}$ and $\langle v_1 \rangle \sim (1 + \eta )^{-1}$ uniformly for $\eta \gtrsim 1$ and $\lvert \zeta \rvert \lesssim 1$ . This proves Lemma 5.7 in that regime. If, on the other hand, $\lvert \zeta \rvert \geq \lVert a \rVert _\infty + 1$ then $\lvert \zeta - a \rvert \sim \lvert \zeta \rvert $ . Hence, for such $\zeta $ , we conclude from (4.4a) and (5.7) that

$$\begin{align*}\frac{1}{v_1} \sim \eta + \langle v_2 \rangle + \frac{\lvert \zeta \rvert^2}{\eta + \langle v_1 \rangle}. \end{align*}$$

As the right hand side is a constant function on $\mathfrak X$ , we obtain $v_1 \sim \langle v_1 \rangle $ if $\lvert \zeta \rvert \geq \lVert a \rVert _\infty + 1$ .

Hence, it remains to consider $\lvert \zeta \rvert \lesssim 1$ and $\eta \lesssim 1$ . In particular, $\lvert \zeta - a \rvert \lesssim 1$ as $a \in \mathcal B$ . Thus, (4.4a), (5.7) and (5.1) imply

(5.15) $$ \begin{align} \eta + \langle v_1 \rangle \sim v_1 ((\eta + \langle v_1 \rangle)^2 + \lvert \zeta - a \rvert^2). \end{align} $$

Together with Lemma 5.3, this yields $\eta + \langle v_1 \rangle \lesssim v_1$ . We conclude $v_1 \gtrsim \langle v_1 \rangle $ and $v_1 \gtrsim \eta $ as well as $\langle v_1 \rangle \gtrsim \eta $ . If $\lvert \zeta - a \rvert \geq c$ for any $c \sim 1$ then $v_1 \lesssim \eta + \langle v_1 \rangle \sim \langle v_1 \rangle $ by (5.15). Therefore we conclude $v_1 \sim \langle v_1 \rangle $ if $\lvert \zeta - a \rvert \geq c$ . What remains is the case $\lvert \zeta - a \rvert \leq c$ and $\eta \leq c$ for some constant $c \sim 1$ . As $v_1\lesssim 1$ by A2 and Lemma 5.5, we conclude from (5.15) that $1 \gtrsim \eta + \langle v_1 \rangle $ or $1 \gtrsim \frac { \lvert \zeta - a \rvert ^2}{\eta + \langle v_1 \rangle }$ . In the second case, (5.15) implies $ \langle v_1 \rangle \lesssim \eta + \langle v_1 \rangle \sim v_1 ( \lvert \zeta - a \rvert ^4 + \lvert \zeta - a \rvert ^2)$ . Using $\lvert \zeta - a \rvert \leq c$ , choosing $c \sim 1$ sufficiently small and averaging $\langle v_1 \rangle \lesssim v_1 ( \lvert \zeta - a \rvert ^4 + \lvert \zeta - a \rvert ^2)$ yield a contradiction as $\langle v_1 \rangle> 0$ . Hence, $\langle v_1 \rangle \gtrsim 1$ and, thus, $\langle v_1 \rangle \sim 1$ by Lemma 5.3 as well as $1 \gtrsim v_1$ by (5.15) as $\eta \leq c$ for some small enough $c \sim 1$ . Since $v_1 \lesssim 1$ by Lemma 5.5, this completes the proof of $v_1 \sim \langle v_1 \rangle $ uniformly for $\zeta \in \mathbb {C}$ and $\eta>0$ . This completes the proof of (i).

For the proof of (ii), from $v_1 \sim \langle v_1 \rangle $ , (5.7) and (5.12), we conclude $\lvert y \rvert \lesssim 1$ uniformly for $\zeta \in \mathbb {C}$ and $\eta>0$ . Owing to (5.15) and $v_1 \sim \langle v_1 \rangle $ , we have $\eta + \langle v_1 \rangle \sim \langle v_1 \rangle (\eta + \langle v_1 \rangle )^2 + \langle v_1 \rangle \lvert \zeta - a \rvert ^2 $ . As $\eta + \langle v_1 \rangle \leq c$ , by choosing $c\sim 1$ sufficiently small, we can incorporate $\langle v_1 \rangle (\eta + \langle v_1 \rangle )^2$ into the left-hand side and obtain $\langle v_1 \rangle \lesssim \lvert \zeta - a \rvert ^2 \langle v_1 \rangle $ . Hence, $1 \lesssim \lvert \zeta - a \rvert $ as $\langle v_1 \rangle>0$ for $\eta>0$ . The bound $\lvert y \rvert \sim 1$ follows from (5.3). This proves the additional statement and completes the proof of Lemma 5.7.

5.2 A relation between the derivatives of M

In this subsection, we restrict the Dyson equation (4.3) to the imaginary axis $w=\mathrm {i} \eta $ for $\eta>0$ , that is, we consider the solution $M=M(\zeta , \mathrm {i} \eta )$ of

(5.16) $$ \begin{align} - M^{-1} = \begin{pmatrix} \mathrm{i} \eta & \zeta - a \\ \overline{\zeta- a} & \mathrm{i} \eta \end{pmatrix} + \Sigma[M]. \end{align} $$

We take derivatives of M with respect to $\eta $ , $\zeta $ and $\bar \zeta $ . We establish a useful relation between these derivatives in the next lemma. Here, we use the notation

$$\begin{align*}\begin{aligned} \langle R \rangle: = \frac{1}{2}(\langle r_{11} \rangle + \langle r_{22} \rangle) \,, \qquad R= \begin{pmatrix} r_{11} & r_{12}\\ r_{21} & r_{22} \end{pmatrix} \in \mathcal{B}^{2 \times 2}\,. \end{aligned} \end{align*}$$

Lemma 5.8. In the setup of Lemma 4.1 we have

(5.17) $$ \begin{align} \langle \partial_\eta M_{21}(\zeta, \mathrm{i} \eta) \rangle= 2\mathrm{i} \langle \partial_\zeta M(\zeta, \mathrm{i}\eta) \rangle, \qquad \langle \partial_\eta M_{12}(\zeta, \mathrm{i}\eta) \rangle = 2 \mathrm{i} \langle \partial_{\bar \zeta} M(\zeta, \mathrm{i}\eta) \rangle \end{align} $$

for every $\zeta \in \mathbb {C}$ and $\eta>0$ , where we decomposed

$$\begin{align*}M(\zeta,\mathrm{i}\eta) = \begin{pmatrix} M_{11}(\zeta,\mathrm{i}\eta) & M_{12}(\zeta,\mathrm{i}\eta) \\ M_{21}(\zeta,\mathrm{i}\eta) & M_{22}(\zeta,\mathrm{i}\eta) \end{pmatrix}. \end{align*}$$

Here, the derivatives of M are taken entrywise. Before proving Lemma 5.8, we note that the Schur complement formula implies

(5.18) $$ \begin{align} \begin{pmatrix} - w & \mathfrak d \\ \mathfrak d^* & - w\end{pmatrix} ^{-1} = \begin{pmatrix} w (\mathfrak d\mathfrak d^* - w^2)^{-1} & \mathfrak d (\mathfrak d^* \mathfrak d - w^2)^{-1} \\ (\mathfrak d^* \mathfrak d -w^2)^{-1} \mathfrak d^* & w (\mathfrak d^* \mathfrak d -w^2)^{-1} \end{pmatrix} \end{align} $$

for any $\mathfrak d \in \mathcal A$ and any $w \in \mathbb {C}$ with $\mathrm {Im}\, w>0$ .

Proof of Lemma 5.8.

Throughout the following, we set $\lvert \mathfrak d \rvert = \sqrt {\mathfrak d^* \mathfrak d}$ for any $\mathfrak d \in \mathcal A$ . Let $\eta \in (0,\infty )$ and $\zeta \in \mathbb {C}$ . Owing to (2.4) and the invertibility of $\lvert a + \mathfrak c -\zeta \rvert ^2 + \eta ^2$ , we have

$$ \begin{align*} \log D( \lvert a+ \mathfrak c - \zeta \rvert^2 + \eta^2) = \langle E[ \log ( \lvert a + \mathfrak c -\zeta \rvert^2 + \eta^2)] \rangle. \end{align*} $$

In particular, the right-hand side is infinitely often differentiable with respect to $\zeta $ , $\overline {\zeta }$ and $\eta $ if $\eta>0$ . Differentiating with respect to $\zeta $ , $\overline {\zeta }$ and $\eta $ yield

(5.19a) $$ \begin{align} \partial_\zeta \log D( \lvert a+ \mathfrak c - \zeta \rvert^2 + \eta^2) & = - \langle E[ ( \lvert a + \mathfrak c - \zeta \rvert^2 + \eta^2)^{-1} (a + \mathfrak c - \zeta)^*] \rangle = - \langle M_{21}(\zeta,\eta) \rangle , \end{align} $$

(5.19b) $$ \begin{align} \partial_{\bar \zeta} \log D( \lvert a+ \mathfrak c - \zeta \rvert^2 + \eta^2) & = - \langle E[ ( \lvert a + \mathfrak c - \zeta \rvert^2 + \eta^2)^{-1} (a + \mathfrak c - \zeta)] \rangle = - \langle M_{12}(\zeta,\eta) \rangle, \end{align} $$

(5.19c) $$ \begin{align} \partial_\eta \log D( \lvert a+ \mathfrak c - \zeta \rvert^2 + \eta^2) & = 2 \eta \langle E[(\lvert a + \mathfrak c - \zeta \rvert^2 + \eta^2)^{-1}] \rangle = - 2 \mathrm{i} \langle M_{22}(\zeta,\eta) \rangle. \end{align} $$

Here, we used in the third steps that, for all $\eta>0$ and $\zeta \in \mathbb {C}$ , (4.1) and (5.18) imply

$$\begin{align*}M(\zeta,\mathrm{i}\eta) = \begin{pmatrix} M_{11}(\zeta,\mathrm{i}\eta) & M_{12}(\zeta,\mathrm{i}\eta) \\ M_{21}(\zeta,\mathrm{i}\eta) & M_{22}(\zeta,\mathrm{i}\eta) \end{pmatrix} = \begin{pmatrix} \mathrm{i} \eta E[(\mathfrak d\mathfrak d^* + \eta^2)^{-1}] & E[\mathfrak d (\mathfrak d^* \mathfrak d + \eta^2)^{-1}] \\ E[(\mathfrak d^* \mathfrak d + \eta^2)^{-1} \mathfrak d^*] & \mathrm{i} \eta E[(\mathfrak d^* \mathfrak d + \eta^2)^{-1}] \end{pmatrix} \end{align*}$$

with $\mathfrak d = a + \mathfrak c - \zeta $ . As $\langle M_{22} \rangle = \langle v_2 \rangle = \langle v_1 \rangle = \langle M_{11} \rangle $ by (5.1), Lemma 5.8 follows from (5.19) due to the exchangebility of the derivatives.

Let $\mathcal L$ be the stability operator of (5.16), defined as

(5.20) $$ \begin{align} \mathcal L \colon \mathcal B^{2\times 2} \to \mathcal B^{2\times 2}, \quad R \mapsto \mathcal L[R] := M^{-1} R M^{-1} - \Sigma[R]. \end{align} $$

This operator is invertible for any $\zeta \in \mathbb {C}$ and $\eta>0$ due to Lemma B.1 below. Therefore, the implicit function theorem applied to (5.16) and simple computations show that

(5.21) $$ \begin{align} \partial_\zeta M = \mathcal L^{-1} [E_{12}], \qquad \partial_{\bar \zeta} M = \mathcal L^{-1} [E_{21}], \qquad \partial_{\eta} M = \mathrm{i} \mathcal L^{-1} [E_+] \end{align} $$

for all $\eta>0$ and $\zeta \in \mathbb {C}$ , where we used the notations $E_{12}$ , $E_{21}$ and $E_+$ for the elements of $\mathcal B^{2\times 2}$ defined through

(5.22) $$ \begin{align} E_{12} := \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \qquad E_{21} := \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \qquad E_+ := \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \end{align} $$

5.3 Stability of Dyson equation and analyticity of its solution

In this section we show how the solution $v_1, v_2$ of (4.4) can be extended to $\eta =0$ . The characterisation of the Brown measure in Proposition 4.3 in combination with Lemma 4.2 shows how the Brown measure $\sigma $ relates to the solution of (4.4) at the origin, that is, at $\eta =0$ . Now we introduce a deterministic analog of the $\varepsilon $ -pseudospectrum. For this purpose we note that the map $w \mapsto \langle M(\zeta , w) \rangle $ is the Stieltjes transform of a probability measure on $\mathbb {R}$ . Indeed, [Reference Alt, Erdős and Krüger6, Proposition 2.1 and Definition 2.2] show that

(5.23) $$ \begin{align} \langle M(\zeta, w) \rangle = \int_{\mathbb{R}}\frac{\rho_\zeta(\mathrm{d} \tau )}{\tau-w} \end{align} $$

holds for all $w \in \mathbb {C}$ with $\operatorname {\mathrm {Im}} w>0$ and a unique probability measure $\rho _\zeta $ whose support satisfies $\operatorname {\mathrm {supp}} \rho _\zeta \subset \big ( \operatorname {\mathrm {Spec}}(D_{|a-\zeta |})\cup \operatorname {\mathrm {Spec}}(-D_{|a-\zeta |}) \big ) + [-2 \lVert S \rVert _\infty ^{1/2},2 \lVert S \rVert _\infty ^{1/2}]$ .

Definition 5.9. Let $\rho _\zeta $ be the unique probability measure on $\mathbb {R}$ for which (5.23) holds. Through $\rho _\zeta $ we define

(5.24) $$ \begin{align} { \mathbb S_\varepsilon:= \{\zeta \in \mathbb{C}: \mathrm{dist}(0, \operatorname{\mathrm{supp}} \rho_\zeta)\le \varepsilon\} } \end{align} $$

for any $\varepsilon \geq 0$ .

Remark 5.10. The sets $\mathbb S_\varepsilon $ defined in (5.24) are monotonously nondecreasing in $\varepsilon \geq 0$ , that is, $\mathbb S_{\varepsilon _1} \subset \mathbb S_{\varepsilon _2}$ if $\varepsilon _1 \le \varepsilon _2$ . Moreover, they are bounded, in fact, $\mathbb S_{\varepsilon } \subset \{\zeta \in \mathbb {C}: \lvert \zeta \rvert \le \varepsilon + \lVert a \rVert _\infty + 2 (\lVert S \rVert _{\infty })^{1/2}\}$ for all $\varepsilon \geq 0$ as a consequence of [Reference Alt, Erdős and Krüger6, Proposition 2.1]. Here, $\lVert S \rVert _\infty $ denotes the operator norm of S viewed as an operator from $\mathcal B$ to $\mathcal B$ .

If we stay away from $\mathbb S_\varepsilon $ , then the solution is extended to $\eta =0$ by setting $v_i=0$ by the following lemma.

Lemma 5.11. Let $\varepsilon>0$ . Let $\zeta \in (\mathbb {C} \setminus \mathbb S_\varepsilon )\cap \mathbb {D}_{1/\varepsilon }$ . Then $v_i(\zeta , \eta )\sim _\varepsilon \eta $ for all $\eta \in (0,1]$ and $i=1,2$ . In particular, $v_i$ is continuously extended to $ \zeta \in \mathbb {C}\setminus \mathbb S_0$ and $\eta =0$ by setting $v_i(\zeta , 0):=0$ .

Proof. From (4.4a), we conclude $v_1 (\lvert \zeta - a \rvert ^2 + (\eta + S v_2)(\eta + S^*v_1)) = \eta + S^* v_1 \geq \eta $ . Thus, $\lvert \zeta \rvert \leq \varepsilon ^{-1}$ , $a \in \mathcal B$ , $\eta \leq 1$ , (5.7) and Lemma 5.3 imply $v_1 \gtrsim _\varepsilon \eta $ for all $\eta \in (0,1]$ . Similarly, $v_2 \gtrsim _\varepsilon \eta $ for all $\eta \in (0,1]$ . On the other hand, as $\zeta \in \mathbb {C}\setminus \mathbb S_\varepsilon $ , the statement (v) of [Reference Alt, Erdős and Krüger6, Lemma D.1] holds for $\tau = 0$ . Hence, [Reference Alt, Erdős and Krüger6, Lemma D.1 (i)] implies $\max \{v_1(\zeta , \eta ), v_2(\zeta , \eta )\} \leq \lVert \mathrm {Im}\, M(\zeta ,\mathrm {i} \eta )) \rVert \lesssim \eta $ for all $\eta \in (0,c]$ for some sufficiently small $c \sim _\varepsilon 1$ . If $\eta \in (c,1]$ then the upper bound in Corollary 5.4 yields $v_i \lesssim \eta ^{-1} \sim \eta $ for all $\eta \in (c,1]$ . This completes the proof.

The next proposition states that if $\langle v_1 \rangle =\langle v_2 \rangle $ remains bounded away from zero as $\eta \downarrow 0$ , then the solution has an analytic extension to $\eta =0$ .

Proposition 5.12 (Analyticity in the bulk).

Let s satisfy A1 and $\zeta \in \mathbb {C}$ with $\limsup _{\eta \downarrow 0}\langle v_1(\zeta , \eta ) \rangle> 0$ . Then $v_1,v_2: \mathbb {C} \times (0,\infty ) \to (0,\infty )$ has an extension to a neighbourhood of $(\zeta ,0)$ in $\mathbb {C}\times \mathbb {R}$ which is real analytic in all variables.

To prove this proposition, we show that the Dyson equation (4.4) is stable even for $\eta =0$ . However, the equation does not have a unique solution on $\mathcal {B}_+^2$ for $\eta =0$ without the additional constraint $\langle v_1 \rangle =\langle v_2 \rangle $ . Therefore, we have to reformulate the equation to incorporate this constraint. Proposition 5.12 is proved at the end of this subsection.

We recall that $\mathcal B_+:=\{w \in \mathcal B:w> 0\}$ and set

$$\begin{align*}e_- = (1, - 1) \in {\mathcal B}^2 := \mathcal B \oplus \mathcal B\,, \qquad e_-^\perp:= \{h=(h_1,h_2) \in\mathcal B^2: \langle h_1 \rangle = \langle h_2 \rangle \}. \end{align*}$$

For $\eta>0$ and $\zeta \in \mathbb {C}$ , we define $J\equiv J_{\zeta , \eta }\colon e_-^\perp \cap \mathcal B_+^2 \to e_-^\perp , (w_1,w_2) \mapsto (J_1(w_1,w_2),J_2(w_1,w_2))$ through

$$ \begin{align*} J_1(w_1,w_2)&:=(\eta + Sw_2)\bigg({w_1- \frac{\eta + S^*w_1}{(\eta + S^*w_1)(\eta + Sw_2)+ \lvert a-\zeta \rvert^2}}\bigg)\,, \\ J_2(w_1,w_2)&:=(\eta + S^*w_1)\bigg({w_2- \frac{\eta + Sw_2}{(\eta + S^*w_1)(\eta + Sw_2)+ \lvert a-\zeta \rvert^2}}\bigg)\,. \end{align*} $$

Then (4.4) takes the form $J(v)=0$ with $v=(v_1,v_2) \in \mathcal {B}_+^2$ .

On $\mathcal B^2$ , we introduce a tracial state and a scalar product defined through

(5.25) $$ \begin{align} { \bigg\langle \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \bigg\rangle := \frac{1}{2} \big( \langle x_1 \rangle + \langle x_2 \rangle \big), \qquad \qquad \bigg\langle{\begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} } \,\mspace{2mu},\, {\begin{pmatrix} y_1 \\ y_2 \end{pmatrix} }\bigg\rangle:= \frac{1}{2}\big({\langle \overline{x}_1y_1 \rangle+\langle \overline{x}_2y_2 \rangle}\big) } \end{align} $$

for $x_1$ , $x_2$ , $y_1$ , $y_2 \in \mathcal B$ . For $x \in \mathcal B^2$ , we write $\lVert x \rVert _2 := \sqrt {\langle {x} \mspace {2mu}, {x}\rangle }$ . We also interpret $\mathcal B^2$ as an algebra equipped with the componentwise multiplication $(x_1x_2)(y_1,y_2):=(x_1y_1, x_2y_2)$ .

For the rest of this section we will assume that s satisfies A1. Until the proof of Proposition 5.12, we fix $\zeta \in \mathbb {C}$ such that $\limsup _{\eta \downarrow 0}\langle v_1(\zeta , \eta ) \rangle \ge \delta $ for some $\delta>0$ . Under these conditions, J remains well defined on $e_-^\perp \cap \mathcal B_+^2$ even for $\eta =0$ and we set $J_0:= J_{\zeta ,\eta =0}$ . We now pick candidates for $v_1(\zeta ,0)$ and $v_2(\zeta ,0)$ by choosing weakly convergent subsequences in the limit $\eta \downarrow 0$ . By Lemma 5.3, there are $v_0 \in (L^2)^2 := L^2 \oplus L^2$ and a monotonically decreasing sequence $\eta _n \downarrow 0$ in $(0,1]$ such that $v^{(n)} = v(\zeta , \eta _n)$ is weakly convergent to $v_0$ in $(L^2)^2 $ , that is, for any $h \in (L^2)^2$ , $\langle {h} \mspace {2mu}, {v^{(n)} - v_0}\rangle \to 0$ in the limit $n \to \infty $ . We recall that $L^2 = L^2(\mathfrak X, \mathcal A, \mu )$ .

Lemma 5.13. Let $v_0=\lim _{n \to \infty }v^{(n)}$ be a weak limit as above. Then $v_0 \in \mathcal {B}_+^2 \cap e_-^\perp $ and $\delta \lesssim v_0 \lesssim \frac {1}{\delta }$ . Furthermore, $v_0$ satisfies (4.4) for $\eta =0$ , that is, $J_0(v_0)=0$ .

For the following arguments, we introduce the operators $S_o$ and $S_d$ on $\mathcal B^2$ defined through

(5.26) $$ \begin{align} S_o:= \left( \begin{array}{cc} 0 & S\\ S^* & 0 \end{array}\right)\,, \quad S_d := \left( \begin{array}{cc} S^* & 0\\ 0 & S\end{array} \right)\,. \end{align} $$

Owing to the upper bound in A1, $S_o$ and $S_d$ can be extended naturally to operators on $(L^2)^2$ .

Proof. Since $v^{(n)} \to v_0$ weakly and $\langle e_- v^{(n)} \rangle =0$ , we conclude $v_0 \perp e_-$ . Furthermore, for any $h \in \mathcal {B}_+^2$ we get $\langle h v_0 \rangle = \lim _{n \to \infty } \langle hv^{(n)} \rangle \gtrsim \delta \langle h \rangle $ because of Corollary 5.4 and $\limsup _{n \to \infty }\langle v^{(n)} \rangle \ge \delta $ . From this we conclude $ v_0 \gtrsim \delta $ . Similarly, Corollary 5.4 implies $v_0 \lesssim \frac {1}{\delta }$ and thus $v_0 \in \mathcal B_+^2$ .

The natural extensions of S and $S^*$ to operators on $L^2$ are Hilbert-Schmidt operators because ${s \in L^2(\mathfrak X \times \mathfrak X, \mu \otimes \mu )}$ due to the upper bound in A1. In particular, S and $S^*$ are compact operators on $L^2$ and, thus, $S_ov^{(n)} \to S_ov_0$ and $S_d v^{(n)} \to S_d v_0$ in $(L^2)^2$ . The bounds $\delta \lesssim v^{(n)} \lesssim \delta ^{-1}$ then imply that $J_{\zeta , \eta _n}(v^{(n)}) \to J_{0}(v_0)$ weakly in $(L^2)^2$ . Consequently, $J_{0}(v_0)=0$ .

For the formulation of the next lemma, we note that $\lVert T \rVert _\infty $ denotes the operator norm of an operator $T \colon \mathcal B^2 \to \mathcal B^2$ and, analogously, $\lVert T \rVert _2$ is the operator norm if $T \colon (L^2)^2 \to (L^2)^2$ .

Lemma 5.14. Let $v_0$ be a weak limit of a sequence $v^{(n)}=v(\zeta ,\eta _n)$ as above. Then

$$\begin{align*}\lVert (\nabla J_0|_{w =v_0})^{-1} \rVert_2+\lVert (\nabla J_0|_{w =v_0})^{-1} \rVert_\infty \lesssim_\delta 1. \end{align*}$$

Proof of Lemma 5.14.

Within this proof we will make use of the auxiliary Lemma 5.15 below whose proof relies on ideas from [Reference Alt, Erdős and Krüger5]. Therefore we introduce notations that match the ones from [Reference Alt, Erdős and Krüger5], namely

(5.27) $$ \begin{align} { \tau := (\lvert \zeta-a \rvert^2,\lvert \zeta-a \rvert^2)\, } \end{align} $$

and recall the definitions of $S_o$ and $S_d$ from (5.26). Using the notations (5.26) and (5.27), we write J in the form

$$\begin{align*}J(w) = (\eta + S_o w)\bigg({w-\frac{1}{\eta + S_o w + \frac{\tau}{\eta + S_d w}}}\bigg)\,. \end{align*}$$

Now we take the directional derivative $\nabla _h J$ of J in the direction $h \in \mathcal B^2$ with $h \perp e_-$ , that is, $\langle h e_- \rangle =0$ , and evaluate at the solution $v=(v_1,v_2)$ . Thus, we find

(5.28) $$ \begin{align} \nabla_h J |_{w=v} = (\eta + S_o v)\bigg({h +v^2S_o h - \frac{v^2\tau}{(\eta + S_d v)^2}S_d h}\bigg) =(\eta + S_o v)\mathscr L h\,, \end{align} $$

where we used $J(v)=0$ and introduced the linear operator $\mathscr L\equiv \mathscr L_{\zeta ,\eta }(v) \colon \mathcal B^2 \to \mathcal B^2$ as

(5.29) $$ \begin{align} { \mathscr Lh:=h +v^2S_o h - \frac{v^2\tau}{(\eta + S_d v)^2}S_d h } \end{align} $$

to guarantee the last equality. Here, $v^2 =(v_1^2, v_2^2)$ since the algebra $\mathcal B^2$ is naturally equipped with entrywise multiplication. We now restrict our analysis to $\eta =0$ and use the following lemma that provides a resolvent estimate for $\mathscr L_0=\mathscr L_{\zeta ,0}(v_0)$ , the operator evaluated on the weak limit $v_0$ .

Lemma 5.15. There is $\varepsilon _\ast \sim _\delta 1$ such that for any $\varepsilon \in (0,\varepsilon _\ast )$ we have the bound

(5.30) $$ \begin{align} { \sup\Big\{{\lVert (\mathscr L_0-\omega)^{-1} \rVert_\#: \omega \not \in \mathbb{D}_\varepsilon \cup ((1 + \mathbb{D}_{1+\varepsilon}) \setminus \mathbb{D}_{2\varepsilon}) }\Big\} \lesssim_{\delta, \varepsilon} 1 } \end{align} $$

for $\#=2, \infty $ . Here, $\mathbb {D}_\varepsilon $ contains the single isolated eigenvalue $0$ of $\mathscr L_0$ with corresponding right and left eigenvectors $v_-:=e_- v_0$ and $S_ov_-$ , that is,

$$\begin{align*}\mathbb{D}_\varepsilon\cap \operatorname{\mathrm{Spec}}(\mathscr L_0) =\{0\}\,, \qquad \ker \mathscr L_0^2 =\mathrm{Span}(v_-) \,, \qquad \mathscr L_0v_-=0\,, \qquad \mathscr L_0^* S_ov_- =0\,. \end{align*}$$

Here, $\mathscr L_0^*$ is the adjoint of $\mathscr L$ with respect to the $L^2$ -scalar product introduced in (5.25).

The proof of Lemma 5.15 follows a strategy similar to the one used to prove stability of the Dyson equation in [Reference Alt, Erdős and Krüger5], where the case $a=0$ was treated. For completeness we present the proof, adjusted to our setup with nontrivial a, in Appendix B below. Using Lemma 5.15 we now show that

(5.31) $$ \begin{align} { \lVert \mathscr L_0^{-1}|_{(S_o v_-)^\perp } \rVert_\#\lesssim_{\delta} 1\,, \qquad \#=2, \infty\,, } \end{align} $$

from which the claim of Lemma 5.14 immediately follows due to (5.28), A1 and $v_0 \gtrsim \delta $ . To see (5.31), we apply Lemma B.4 to $A:=C\mspace {1 mu}\mathscr L_0$ for some appropriately large positive constant $C\sim _\delta 1$ . We now check the assumptions of the lemma. Note that $\mathscr L_0$ maps $e_-^\perp $ to $(S_o v_-)^\perp $ . By Lemma 5.15 the right and left eigenvectors of $\mathscr L_0$ corresponding to the eigenvalue $0$ are $v_-$ and $S_ov_-$ , respectively. Moreover, $\langle {v_-} \mspace {2mu}, {e_-}\rangle \gtrsim _\delta 1$ as $v_0 \gtrsim \delta $ , $\lvert \langle {e_-} \mspace {2mu}, {w}\rangle \rvert \le \lVert w \rVert _\#$ and that $\lVert \mathscr L_0w \rVert _\# \gtrsim _\delta \lVert w \rVert _\#$ for any $w \perp S_ov_-$ due to Lemma 5.15. By Lemma B.4 with the choices $\alpha :=0$ , $x:= v_-$ and $y:= \frac {1}{2}e_-$ we get $\lVert \mathscr L_0w \rVert _\# \gtrsim _\delta \lVert w \rVert _\#$ for any $w \perp e_-$ . Thus, (5.31) is shown.

Now we use the stability at $\eta =0$ to finish the proof of the main result of this subsection.

Proof of Proposition 5.12.

Let $\zeta _0 \in \mathbb {C}$ be such that $\limsup _{\eta \downarrow 0}\langle v_1(\zeta _0, \eta ) \rangle> 0$ . Let $\eta _n \downarrow 0$ such that $v^{(n)}=v(\zeta _0, \eta _n)$ is weakly convergent in $(L^2)^2$ . This is possible, because the family $v(\zeta _0, \eta )$ with $\eta \in (0,1]$ is bounded in $(L^2)^2$ due to Lemma 5.3. By Lemma 5.13 the weak limit $v_0=\lim _{n \to \infty }v^{(n)} $ satisfies the Dyson equation, $J_{\zeta ,0}(v_0) = 0$ , and by Lemma 5.14 the Dyson equation is stable at $v=v_0$ and $\eta =0$ . By the implicit function theorem we find a real analytic function $\widetilde {v}$ , defined on a neighbourhood U of $(\zeta _0,0)$ in $\mathbb {C}\times \mathbb {R}$ , such that $\widetilde {v}(\zeta ,\eta )$ solves (4.4) and $\widetilde {v}(\zeta _0,0)=v_0$ . Since $v_0 \gtrsim \delta $ according to Lemma 5.13, $\widetilde {v}(\zeta ,\eta )>0$ on U if the neighbourhood U is chosen sufficiently small. By uniqueness of the solution to the Dyson equation we conclude $\widetilde {v}(\zeta ,\eta )= v(\zeta , \eta )$ for all $(\zeta , \eta ) \in U$ .

5.4 Characterisation of $\mathbb {S}$

Throughout this section we assume that $a \in \mathcal B$ and s satisfies A1 and A2. To generalize (2.8), (2.9) and (2.10) to the setup introduced in Section 5, we define an operator $B_\zeta \colon \mathcal {B} \to \mathcal {B}$ , a function $\beta \colon \mathbb {C} \to \mathbb {R}$ and a subset $\mathbb {S} \subset \mathbb {C}$ through

(5.32) $$ \begin{align} { \beta(\zeta) := \inf_{x \in \mathcal B_+} \sup_{y \in \mathcal B_+} \frac{\langle{x} \mspace{2mu}, {B_\zeta \mspace{2 mu}y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle}\,, \qquad B_\zeta:= D_{\lvert a-\zeta \rvert^2}-S\,, \qquad \mathbb{S}:=\{\zeta \in \mathbb{C}: \beta(\zeta) <0\}\,. } \end{align} $$

We also set

(5.33) $$ \begin{align} \rho_\zeta(0):=\frac{1}{\pi} \lim_{\eta \downarrow 0}\langle v_1(\zeta, \eta) \rangle\,. \end{align} $$

This limit exists, because either $\limsup _{\eta \to 0}\langle v_1(\zeta , \eta ) \rangle>0$ , in which case $v_1$ can be analytically extended to $\eta =0$ by Proposition 5.12, or $\limsup _{\eta \to 0}\langle v_1(\zeta , \eta ) \rangle =0$ in which case the limit equals zero as well. The definition in (5.33) is motivated by the fact that the measure $\rho _\zeta $ from Definition 5.9 can be shown to have a density on $\mathbb {R}$ and the value of this density at zero would be given by the right hand side in (5.33).

In the following we will denote by $\lambda _{\mathrm {PF}}(T)$ the spectral radius of a compact and positivity preserving operator T. By the Krein-Rutman theorem $\lambda _{\mathrm {PF}}(T)$ is an eigenvalue of T with a positive eigenvector. We also refer to this eigenvalue as the Perron-Frobenius eigenvalue of T. In particular the operators S and $S^*$ are compact as mentioned in the proof of Lemma 5.13 and therefore so are $D_x S D_y$ and $D_x S^* D_y$ for $x,y \in \mathcal {B}$ . We use this fact in the statement of the following proposition.

Proposition 5.16. Let $a \in \mathcal B$ and s satisfy A1 and A2. The following relations between $\beta $ , $\mathbb {S}$ , $\mathbb {S}_\varepsilon $ and $\rho _\zeta $ apply.

1. (i) The function $\mathbb {C} \ni \zeta \mapsto \beta (\zeta )$ is continuous and satisfies $\lim _{\zeta \to \infty } \beta (\zeta ) = + \infty $ . In particular, $\mathbb {S}$ is bounded.

2. (ii) The spectrum of $D_a$ lies inside $\mathbb {S}$ , that is,

   (5.34) $$ \begin{align} { \operatorname{\mathrm{Spec}} (D_a) \subset {\mathbb{S}}\,. } \end{align} $$

3. (iii) The sign of $\beta $ satisfies

   (5.35) $$ \begin{align} { \operatorname{\mathrm{sign}}{\beta(\zeta)} = \mathrm{sign} \Big(1-\lambda_{\mathrm{PF}} \Big(SD_{\lvert a-\zeta \rvert}^{-2}\Big)\Big) \,, \qquad \zeta \in \mathbb{C}\,. } \end{align} $$

4. (iv) For any $\zeta \in \mathbb {C}$ with $\beta (\zeta )>0$ the operator $B_\zeta $ is invertible. Furthermore, all such $\zeta $ are characterised by

   (5.36) $$ \begin{align} { \{\zeta \in \mathbb{C}: \beta(\zeta)>0\} = \{\zeta \in \mathbb{C}: \mathrm{dist}(0, \operatorname{\mathrm{supp}} \rho_\zeta)>0\}=\mathbb{C} \setminus \mathbb S_0\,. } \end{align} $$

5. (v) The set $\mathbb {S}$ is characterised by having a positive singular value density at the origin, that is,

   (5.37) $$ \begin{align} { \mathbb{S}= \{\zeta \in \mathbb{C} : \rho_\zeta(0)>0\}\,. } \end{align} $$

Before proving Proposition 5.16, we state a corollary, which follows directly from Proposition 5.12 and Proposition 5.16 (v).

Corollary 5.17 (Existence and uniqueness for (1.1)).

Let $a \in \mathcal B$ and s satisfy A1 and A2. Then for each $\zeta \in \mathbb S$ , the relations (1.1) have a unique solution $(v_1, v_2) \in \mathcal B_+ \times \mathcal B_+$ .

Proof. Proof of (i): The continuity of $\zeta \mapsto \beta (\zeta )=\beta $ with $B = B_\zeta $ is a consequence of the bound

$$\begin{align*}\bigg\lvert \inf_{x \in \mathcal B_+} \sup_{y \in \mathcal B_+}\frac{\langle{x} \mspace{2mu}, {(B+D_w)y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} - \beta \bigg\rvert \le \sup_{x \in \mathcal B_+} \sup_{y \in \mathcal B_+}\frac{\langle{x} \mspace{2mu}, {D_{\lvert w \rvert}y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} \le \lVert w \rVert_\infty \end{align*}$$

for any real valued $w \in \mathcal B$ since $B-B_{\zeta + \omega } = D_w$ with $w = a-\zeta |^2-|a-\zeta + \omega |^2$ and $\lVert w \rVert _\infty \to 0$ for $\omega \in \mathbb {C}$ with $|\omega | \to 0$ . This bound follows from

$$\begin{align*}\frac{\langle{x} \mspace{2mu}, {(B+D_w)y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} \le \frac{\langle{x} \mspace{2mu}, {By}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} +\sup_{\widetilde x \in \mathcal B_+} \sup_{\widetilde y \in \mathcal B_+}\frac{\langle{\widetilde x} \mspace{2mu}, {D_{|w|}\widetilde y}\rangle}{\langle{\widetilde x} \mspace{2mu}, {\widetilde y}\rangle} \le \frac{\langle{x} \mspace{2mu}, {By}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} + \lVert w \rVert_\infty \end{align*}$$

for every $x,y \in \mathcal B_+$ , as well as

$$\begin{align*}\frac{\langle{x} \mspace{2mu}, {(B+D_w)y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} \ge \frac{\langle{x} \mspace{2mu}, {By}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} - \sup_{\widetilde x \in \mathcal B_+} \sup_{\widetilde y \in \mathcal B_+}\frac{\langle{\widetilde x} \mspace{2mu}, {D_{|w|}\widetilde y}\rangle}{\langle{\widetilde x} \mspace{2mu}, {\widetilde y}\rangle} \ge \frac{\langle{x} \mspace{2mu}, {By}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} - \lVert w \rVert_\infty, \end{align*}$$

and then taking the supremum over $y \in \mathcal B_+$ and the infimum over $x \in \mathcal B_+$ in both inequalities. The statement $\beta (\zeta ) \to + \infty $ as $\zeta \to \infty $ is obvious.

Before we start with the proof of other individual statements of the proposition, we show that $\mathbb {S}$ can be classified in terms of the Perron-Frobenius eigenvalue of $SD_{\lvert a-\zeta \rvert }^{-2}$ in the sense that

(5.38) $$ \begin{align} { \mathbb{S} = \{\zeta \in \mathbb{C} :\lambda(\zeta)>1\} \,, } \end{align} $$

where we introduced $\lambda : \mathbb {C} \to [0,\infty ]$ as the limit of a strictly increasing sequence via

$$\begin{align*}\lambda(\zeta):= \lim_{\varepsilon\downarrow 0} \lambda_\varepsilon(\zeta)\,, \qquad \lambda_\varepsilon(\zeta):= \lambda_{\mathrm{PF}} \big({S(\varepsilon+D_{ \lvert a-\zeta \rvert^2})^{-1}}\big). \end{align*}$$

We recall that $\lambda _{\mathrm {PF}}(T)$ denotes the Perron-Frobenius eigenvalue of T and refer to the comments before the statement of the proposition for its definition.

To show (5.38) let $\varepsilon \in (0,1)$ , $D:=D_{\lvert \zeta -a \rvert ^2 } $ , $\lambda _\varepsilon =\lambda _\varepsilon (\zeta )$ and $C>0$ such that $1+\lvert \zeta -a \rvert ^2 \le C$ . For $\zeta \in \mathbb {C}$ with $\beta (\zeta ) \ge 0$ we get

$$ \begin{align*} \beta + \varepsilon \le C\inf_{x\in \mathcal B_+}\sup_{y\in \mathcal B_+} \frac{\langle{x} \mspace{2mu}, {(\varepsilon+B) y}\rangle}{\langle{x} \mspace{2mu}, {(\varepsilon+D) y}\rangle} = C\bigg({1- \sup_{x\in \mathcal B_+}\inf_{y\in \mathcal B_+}\frac{\langle{x} \mspace{2mu}, {S(\varepsilon+D)^{-1}y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle}}\bigg)=C\big({1-\lambda_\varepsilon}\big) , \end{align*} $$

where we used $\varepsilon + \lvert \zeta -a \rvert ^2 \le C$ in the first inequality, and conclude

(5.39) $$ \begin{align} { \beta \le C(1-\lambda) \qquad \text{ if } \beta \ge 0\,. } \end{align} $$

For $\zeta \in \mathbb {C}$ with $\beta (\zeta ) < 0$ we use

$$\begin{align*}-\beta-\varepsilon = \sup_{x\in \mathcal B_+}\inf_{y\in \mathcal B_+} \frac{-\langle{x} \mspace{2mu}, {(\varepsilon +B) y}\rangle}{\langle{x} \mspace{2mu}, { y}\rangle} \end{align*}$$

for sufficiently small $\varepsilon>0$ and find analogously that

(5.40) $$ \begin{align} { \beta\ge C\big({1-\lambda}\big) \qquad \text{ if } \beta < 0\,. } \end{align} $$

From (5.39) and (5.40) we conclude (5.38).

We also show that

(5.41) $$ \begin{align} { \operatorname{\mathrm{Spec}} (D_a) \subset \{\zeta \in \mathbb{C}: \beta(\zeta) \le 0\}. } \end{align} $$

We will improve this to (5.34) below. Let $\zeta \in \operatorname {\mathrm {Spec}} (D_a)$ . Then $\operatorname {\mbox {ess inf}} \lvert \zeta -a \rvert =0$ . Thus, for any $\varepsilon>0$ we find $x \in \mathcal B\setminus \{ 0\}$ with $x \geq 0$ such that $ \lvert \zeta -a \rvert ^2 x \le \varepsilon x$ . In the definition of $\beta $ from (5.32) we can take the supremum over all $x \in \overline {\mathcal B}_+$ , Thus, we get

$$\begin{align*}\beta \le \varepsilon- \inf_{y\in \mathcal B_+} \frac{\langle{x} \mspace{2mu}, {Sy}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} \le \varepsilon\,. \end{align*}$$

Since $\varepsilon>0$ was arbitrarily small, we conclude $\beta \le 0$ .

Proof of (iv): Let $\zeta \in \mathbb {C}$ such that $\beta (\zeta )=\beta>0$ . Then (5.39) implies $\lambda _{\mathrm {PF}} (SD^{-1}) <1$ with $D=D_{\lvert \zeta -a \rvert ^2 }$ . Here, D is invertible because $\operatorname {\mbox {ess inf}} \lvert \zeta -a \rvert>0$ due to (5.41). In particular, $B= (1- SD^{-1})D$ is invertible.

Now we show $\mathrm{dist} (0, \operatorname {\mathrm {supp}} \rho _\zeta )>0$ to see one inclusion in the characterisation (5.36). The Dyson equation in the matrix representation, (4.3) is solved by

(5.42) $$ \begin{align} { M_0:= \left( \begin{array}{cc} 0 & (\overline{a - \zeta})^{-1}\\ (a-\zeta)^{-1} &0 \end{array}\right) } \end{align} $$

at $w=0$ . Furthermore, the associated stability operator (cf. (5.20))

(5.43) $$ \begin{align} { \mathcal{L}_0\colon \mathcal B^{2\times 2} \to \mathcal B^{2\times 2}, \quad R \mapsto M_0^{-1}RM_0^{-1} - \Sigma R = \left( \begin{array}{cc} \lvert a-\zeta \rvert^2r_{22}- Sr_{22} & (a-\zeta)^2r_{21}\\ (\overline{a-\zeta})^{2}r_{12} & \lvert a-\zeta \rvert^2r_{11}- S^{ *}r_{11} \end{array} \right) } \end{align} $$

is invertible because $B_\zeta $ is invertible and $\operatorname {\mbox {ess inf}} \lvert a-\zeta \rvert>0$ . Therefore (4.3) can be uniquely solved for sufficiently small w as an analytic function $w \mapsto M(\zeta , w)$ with $M(\zeta ,0)=M_0$ and we get

$$\begin{align*}\mathcal{L}_0[\partial_w M(\zeta,w)|_{w=0} ]= 1\quad \text{and} \quad \partial_\eta M(\zeta, \mathrm{i} \eta) |_{\eta =0} = \mathrm{i}\mspace{1 mu}\mathcal{L}_0^{-1} [1] \end{align*}$$

In particular, $\operatorname {\mathrm {Im}} M(\zeta , \mathrm {i} \eta )$ is positive definite for sufficiently small $\eta>0$ because $\mathcal {L}_0^{-1}$ is positivity preserving, as can be seen from a Neumann series expansion using that $\lambda _{\mathrm {PF}}(SD^{-1}) < 1$ . Therefore $M(\zeta , \mathrm {i} \eta )$ is the unique solution of (4.3) with $m_{11}(\zeta ,\mathrm {i} \eta ) =\mathrm {i} v_1(\zeta , \eta )$ and $m_{22}(\zeta ,\mathrm {i} \eta )=\mathrm {i} v_2(\zeta , \eta )$ . Since $m_{11}|_{\eta =0} =m_{22}|_{\eta =0}=0$ we conclude that $\rho _\zeta (0) = \langle v_1(\zeta ,0) \rangle =0$ . The invertibility of $\mathcal {L}_0$ also implies analyticity of $M(\zeta , w)$ in w in a small neighbourhood of zero. Thus, $\rho _\zeta ( [-\varepsilon ,\varepsilon ])=0$ for $\varepsilon>0$ sufficiently small and $\mathrm{dist} (0, \operatorname {\mathrm {supp}} \rho _\zeta )>0$ .

To see the other inclusion in (5.36), let $\zeta \in \mathbb {C}$ be such that $\mathrm{dist} (0, \operatorname {\mathrm {supp}} \rho _\zeta )\ge \delta $ for some $\delta>0$ . From [Reference Alt, Erdős and Krüger6, Lemma D.1 (iv)] we know that $M=M(\zeta , \mathrm {i} \eta )$ is locally a real analytic function of $\eta $ with an expansion $M= M_0 + \mathrm {i}\mspace {1 mu}\eta \mspace {2 mu}M_1 + O(\eta ^2)$ , where $M_0=M_0^*$ . Taking the imaginary part of (4.3) at $w=\mathrm {i} \eta $ , dividing both sides by $\eta $ shows that

(5.44) $$ \begin{align} { (M^*)^{-1}K_\eta M^{-1} = 1 + \Sigma[K_\eta]\,, } \end{align} $$

where $K_\eta = \frac {1}{\eta } \operatorname {\mathrm {Im}} M= \operatorname {\mathrm {Re}} M_1 + O(\eta )$ . In particular, $K_0:= \lim _{\eta \downarrow 0}K_\eta $ exists and since $\operatorname {\mathrm {Im}} M(\zeta , \mathrm {i} \eta ) \sim _\delta \eta $ by Lemma 5.11 we get $K_0 \sim _\delta 1$ . Evaluating (5.44) at $\eta =0$ yields

(5.45) $$ \begin{align} { M_0(\Sigma K_0) M_0 = K_0 -1\,. } \end{align} $$

Taking the scalar product of (5.45) with the left Perron-Frobenius eigenvector of $R \mapsto M_0 (\Sigma R ) M_0$ and using $K_0 \sim _\delta 1$ we see that $\lambda _{\mathrm {PF}}(R \mapsto M_0 (\Sigma R ) M_0) <1$ . This is equivalent to $\lambda :=\lambda _{\mathrm {PF}}(SD^{-1})<1$ with $D:=D_{\lvert a-\zeta \rvert ^2}$ . Now let $u \in \mathcal B_+$ be the Perron-Frobenius eigenvector of $SD^{-1}$ . Since $\varepsilon :=\operatorname {\mbox {ess inf}} \lvert a-\zeta \rvert>0$ we get with $y_0:=D^{-1}u$ that

$$\begin{align*}(D -S)y_0=(1-\lambda)Dy_0>0\,. \end{align*}$$

Thus,

$$\begin{align*}\beta = \inf_{x>0} \sup_{y >0} \frac{\langle{x} \mspace{2mu}, {B y}\rangle}{\langle{x} \mspace{2mu}, {y}\rangle} \ge \inf_{x >0} \frac{\langle{x} \mspace{2mu}, {B y_0}\rangle}{\langle{x} \mspace{2mu}, {y_0}\rangle} \ge (1-\lambda)\mspace{1 mu}\varepsilon^2>0\,. \end{align*}$$

This finishes the proof of (5.36), that is, of (iv).

Proof of (iii): We have now collected enough information to improve (5.38) to (5.35). Indeed, by (5.39) and (5.40) it remains to show that $\lambda <1$ implies $\beta>0$ . Due to (5.38) we already know $\beta \ge 0$ in case $\lambda <1$ . Now let $\beta =0$ and $\lambda \le 1$ . Then we show that $\lambda =1$ . Indeed by the characterisation (5.36) we have $0 \in \operatorname {\mathrm {supp}} \rho _\zeta $ . Now we consider the identity

$$\begin{align*}B_\zeta v_2 = \eta-v_2(\eta + S^*v_1)(\eta +Sv_2) \end{align*}$$

which follows from (4.4b). For some $\varepsilon>0$ we add $\varepsilon \mspace {1 mu} v_2$ to both sides and apply the inverse of $\varepsilon +D$ with $D=D_{\lvert a-\zeta \rvert ^2}$ . Then we take the scalar product with the right Perron-Frobenius eigenvector $x_\varepsilon \in \mathcal B_+$ of $S^*(\varepsilon +D)^{-1}$ corresponding to its Perron-Frobenius eigenvalue $\lambda _\varepsilon>0$ . Note that the Perron-Frobenius eigenvalues of $S^*(\varepsilon +D)^{-1}$ , $(\varepsilon +D)^{-1}S$ and $S(\varepsilon +D)^{-1}$ all coincide. Thus we get

(5.46) $$ \begin{align} { (1-\lambda_\varepsilon) \langle x_\varepsilon v_2 \rangle = \eta \mspace{1 mu} \langle (\varepsilon + D)^{-1}x_\varepsilon \rangle -\langle x_\varepsilon (\varepsilon+D)^{-1}(v_2(\eta + S^*v_1)(\eta +Sv_2)-\varepsilon\mspace{1 mu} v_2) \rangle\,. } \end{align} $$

From [Reference Alt, Erdős and Krüger6, Corollary D.2] and $\langle v_i \rangle \sim v_i$ by Lemma 5.7 (i) we see that $\eta /\langle v_i \rangle \to 0$ for $\eta \downarrow 0$ . Thus, dividing (5.46) by $\langle v_2 \rangle $ , taking the limit $\eta \downarrow 0$ and using (5.7) reveals

$$\begin{align*}( 1-\lambda_\varepsilon)\langle x_\varepsilon \rangle\sim ( 1-\lambda_\varepsilon)\langle x_\varepsilon k \rangle = \varepsilon \langle x_\varepsilon(\varepsilon+D)^{-1}k \rangle\sim \varepsilon \langle x_\varepsilon(\varepsilon+D)^{-1}S1 \rangle= \varepsilon \langle S^*(\varepsilon+D)^{-1}x_\varepsilon \rangle=\varepsilon \mspace{1 mu} \lambda_\varepsilon\langle x_\varepsilon \rangle\,, \end{align*}$$

where $k:= \limsup _{\eta \downarrow 0} \frac {v_2}{\langle v_2 \rangle }\sim 1$ . Letting $\varepsilon \downarrow 0$ shows $\lambda =1$ . Thus, (5.35) is proven.

Proof of (v): By (5.36) we know that $\rho _\zeta (0)>0$ implies $\beta (\zeta ) \le 0 $ . Thus, it suffices to show that for $\zeta \in \mathbb {C}$ with $\beta (\zeta ) \le 0$ we get $\beta (\zeta ) =0$ if and only if $ \rho _\zeta (0)=0$ . Now let $\beta =\beta (\zeta ) \le 0$ . As above, we consider the identity (5.46). First, suppose $\rho _\zeta (0)>0$ , that is, we can analytically extend v to $\eta =0$ by Proposition 5.12 and have $v|_{\eta =0}>0$ . Then in the limit $\eta \downarrow 0$ we find

$$\begin{align*}(\lambda_\varepsilon-1) \langle x_\varepsilon v_2 \rangle = \langle x_\varepsilon (\varepsilon+D)^{-1}(v_2 (S^*v_1Sv_2-\varepsilon)) \rangle \end{align*}$$

Using $v_2 \sim \langle v_2 \rangle \sim \rho _\zeta (0)$ for small enough $\varepsilon>0$ the right hand side satisfies

$$\begin{align*}\langle x_\varepsilon (\varepsilon+D)^{-1}(v_2 (S^*v_1Sv_2-\varepsilon)) \rangle\sim \rho_\zeta(0)^3 \langle x_\varepsilon (\varepsilon+D)^{-1}S1 \rangle \sim \lambda_\varepsilon\rho_\zeta(0)^3 \langle x_\varepsilon \rangle. \end{align*}$$

Since $\langle x_\varepsilon v_2 \rangle \sim \rho _\zeta (0) \langle x_\varepsilon \rangle $ we infer $\lambda _\varepsilon -1 \sim \lambda _\varepsilon \mspace {2 mu}\rho _\zeta (0)^2$ . Thus, $\lambda>1$ and by (5.35) therefore $\beta < 0$ .

Conversely, let $v|_{\eta =0}=0$ . Then we know from Lemma 5.7 (ii) that $\delta :=\operatorname {\mbox {ess inf}} \lvert a-\zeta \rvert>0$ . Since $\beta (\zeta )\le 0$ the characterisation (5.36) implies $0 \in \operatorname {\mathrm {supp}} \rho _\zeta $ and by (5.35) we have $\lambda \ge 1$ . Since $\eta /\langle v \rangle \to 0$ for $\eta \downarrow 0$ by [Reference Alt, Erdős and Krüger6, Corollary D.2] we get, dividing (5.46) by $\langle v_2 \rangle $ and taking the limit $\eta \downarrow 0$ , the scaling behaviour

$$\begin{align*}1-\lambda_\varepsilon \sim \varepsilon \mspace{1 mu} \lambda_\varepsilon\,. \end{align*}$$

This implies $\lambda _\varepsilon \le 1 $ , thus $\lambda =1$ , and completes the proof of (v).

Proof of (ii): Let $\zeta \in \operatorname {\mathrm {Spec}}(D_a)$ . By (5.41) we know $\beta (\zeta ) \le 0$ . Suppose $\beta (\zeta )=0$ . Then (5.37) would imply $\rho _\zeta (0)=0$ . However, this contradicts $\zeta \in \operatorname {\mathrm {Spec}}(D_a)$ because of Lemma 5.7 (ii) and the definition of $\rho _\zeta (0)$ in (5.33). This finishes the proof of the proposition.