2.1 NC Function Theory

The $d-$ dimensional complex NC universe, $\mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , is the graded set of all row $d-$ tuples of square matrices of any fixed size, $n \in \mathbb {N}$ :

$$ \begin{align*}\mathbb{C}^{(\mathbb{N} \times \mathbb{N}) \cdot d} := \bigsqcup_{n=1}^\infty \mathbb{C}^{(n\times n)\cdot d}; \quad \quad \mathbb{C}^{(n\times n)\cdot d} := \mathbb{C}^{n\times n} \otimes \mathbb{C}^{1 \times d}.\end{align*} $$

Given any row $d-$ tuple $Z = (Z_1, \cdots , Z_d) \in \mathbb {C}^{(n\times n)\cdot d}$ , we will view Z as a linear map from d copies of $\mathbb {C}^n$ into one copy, and we define the row-norm of Z as the norm of this linear map:

$$ \begin{align*}\| Z \|_{\mathrm{row}} := \| Z \|_{\mathscr{B} (\mathbb{C}^n \otimes \mathbb{C}^d , \mathbb{C}^n)} = \sqrt{ \left\| \sum_{j=1}^d Z_j Z_j^{*} \right\|}.\end{align*} $$

When we write $Z^{*}$ , this will then denote the Hilbert space adjoint of the linear map $Z : \mathbb {C}^n \otimes \mathbb {C}^d \rightarrow \mathbb {C}^n$ . That is,

$$ \begin{align*}Z^{*} = \begin{pmatrix} Z_1^{*} \\ \vdots \\ Z_d^{*} \end{pmatrix} : \mathbb{C}^n \rightarrow \mathbb{C}^n \otimes \mathbb{C}^d.\end{align*} $$

We will also, on occasion, need to consider $\mathrm {col} (Z) := \left ( \begin {smallmatrix} Z_1 \\ \vdots \\ Z_d \end {smallmatrix} \right ) $ and $\mathrm {row} (Z^{*} ) := (Z_1^{*}, \cdots , Z_d^{*} )$ . In this case, given $Z \in \mathbb {C}^{(n\times n)\cdot d}$ ,

$$ \begin{align*}\| Z \|_{\mathrm{col}} := \| \mathrm{col} (Z) \|_{\mathscr{B} (\mathbb{C}^n , \mathbb{C}^n \otimes \mathbb{C}^{d})} = \sqrt{\left\| \sum_{j=1}^d Z_j^{*} Z_j \right\|}.\end{align*} $$

If the row-norm of $Z \in \mathbb {C}^{(n\times n)\cdot d}$ is less than or equal to $1$ , we say Z is a row contraction, and if $\| Z \|_{\mathrm {row}} <1$ , we say Z is a strict row contraction. The NC unit row-ball, $\mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , is the open unit ball of $\mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ with respect to this row-norm. The uniform topology on $\mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ is related to the topology generated by the row pseudo-metric. Namely, given $Z \in \mathbb {C}^{(n\times n)\cdot d}$ and $W \in \mathbb {C}^{(m\times m)\cdot d}$ ,

$$ \begin{align*}d_{\mathrm{row}} (Z,W) := \| Z^{\oplus m} - W^{\oplus n} \|_{\mathrm{row}},\end{align*} $$

and the NC unit row-ball, $\mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , then consists of all Z so that $d_{\mathrm {row}} (Z,0) <1$ . A sub-base for the row pseudo-metric topology is given by the row-balls centred at any $Y \in \mathbb {C}^{(m\times m)\cdot d}$ ,

$$ \begin{align*}r \cdot \mathbb{B}^{(\mathbb{N} \times \mathbb{N}) \cdot d} (Y ) := \bigsqcup_{m=1}^\infty \{ X \in \mathbb{C}^{(\mathbb{N} \times \mathbb{N}) \cdot d} | \ d_{\mathrm{row}} (X,Y) < r \}.\end{align*} $$

The uniform topology on $\mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ is the topology generated by the sub-base of sets, $r \cdot \mathbb {B}^d_{m\mathbb {N}} (Y) \subseteq r \cdot \mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d} (Y)$ , where if $Y \in \mathbb {C}^{(m\times m)\cdot d}$ , then $\mathbb {B}^d_{m\mathbb {N}} (Y)$ contains only the levels $mn$ , $n \in \mathbb {N}$ ,

$$ \begin{align*}r \cdot \mathbb{B}^d_{m\mathbb{N}} (Y) := \bigsqcup_{n=1}^\infty \{ X \in \mathbb{C}^{(mn \times mn) \cdot d} | \ d_{\mathrm{row}} (X,Y) <r \}.\end{align*} $$

As explained in [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov35, Proposition 7.18], there is subtle difference between the uniform topology and the topology generated by $d_{\mathrm {row}}$ , and the topology generated by the row pseudo-metric is strictly weaker than the uniform topology. The main difference is that these open sets, $\mathbb {B}^d_{m\mathbb {N}} (Y)$ , need not be closed under direct summands. This difference, however, will not play a significant role in this paper.

An NC set is any subset, $\Omega \subseteq \mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , that is closed under direct sums. Hence, any such $\Omega $ can be written as a graded set,

$$ \begin{align*}\Omega = \bigsqcup_{n=1}^\infty \Omega_n, \quad \quad \Omega_n := \Omega \cap \mathbb{C}^{(n\times n)\cdot d}.\end{align*} $$

A freely non-commutative function, or more simply, NC function on an NC set $\Omega \subseteq \mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , is any function $f : \Omega \rightarrow \mathbb {C}^{\mathbb {N} \times \mathbb {N}}$ , where $\mathbb {C}^{\mathbb {N} \times \mathbb {N}}$ denotes the one-dimensional NC universe, that obeys the three axioms:

1. 1. $f : \Omega _n \rightarrow \mathbb {C}^{n\times n}$ , i.e., f is graded, or f preserves matrix size.

2. 2. Given $X \in \Omega _n$ and $Y \in \Omega _m$ ,

   $$ \begin{align*}f \begin{pmatrix} X & 0 \\ 0 & Y \end{pmatrix} = \begin{pmatrix} f(X) & 0 \\ 0 & f(Y) \end{pmatrix},\end{align*} $$
   i.e., f preserves direct sums.

3. 3. If $S \in \mathrm {GL}_n$ is any invertible matrix, $X \in \Omega _n$ , and

   $$ \begin{align*}S^{-1} X S := (S^{-1} X_1 S , \cdots , S^{-1} X_d S ) \in \Omega_n,\end{align*} $$
   then $f (S^{-1} X S ) = S^{-1} f(X) S$ , i.e., f preserves joint similarities.

We say that an NC function, h, on a uniformly open NC domain, $\mathrm {Dom} \, h$ , is locally uniformly bounded if for any $X \in \mathrm {Dom} \, h$ , there is a uniformly open neighbourhood of X in $\mathrm {Dom} \, h$ , on which $h(Y)$ is uniformly bounded in matrix-norm. As proven in [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov35, Corollary 7.6, Corollary 7.28], the axioms defining NC functions and this local boundedness condition imply that any such h is holomorphic, in the sense that it is Fréchet, and hence Gâteaux differentiable at any point in its domain, and uniformly analytic in the sense that it has a Taylor-type power series expansion about any point, X, in its NC domain (i.e., a Taylor–Taylor series) with nonzero radius of convergence, $R_X>0$ [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov35, Corollary 7.26, Corollary 7.28, Theorem 8.11]. That is, its Taylor–Taylor series converges absolutely and uniformly on any (uniformly) open row-ball of radius $0<r <R_X$ , centred at any $X \in \mathrm {Dom} \, h$ . Here, if h is uniformly analytic in a uniformly open neighbourhood, $\mathscr {U}$ , of $Y \in \mathbb {C}^{(n\times n)\cdot d}$ , and $X \in \mathscr {U}_{mn}$ is sufficiently close to Y, then the Taylor–Taylor series of h, centred at Y and evaluated at X, is

$$ \begin{align*}h(X) = \sum_{j =0}^\infty \frac{\left( \partial_{X - Y^{\oplus m}}^j h\right) (Y^{\oplus m})}{j!},\end{align*} $$

where $\partial ^j_{X - Y^{\oplus m}} h (Y^{\oplus m})$ is the jth directional or Gâteaux derivative of h in the direction $X - Y^{\oplus m}$ . That is,

$$ \begin{align*}\partial_X h (Y) := \lim_{t \rightarrow 0} \frac{h(Y+tX) - h(Y)}{t},\end{align*} $$

for $X, Y$ of the same size. In particular, the Taylor–Taylor series of h at the origin, $0$ , of the NC universe can be written

$$ \begin{align*}h (X) = \sum_{j=0}^\infty \frac{1}{j!} \partial_X^j h (0),\end{align*} $$

where if $X \in \mathbb {C}^{(n\times n)\cdot d}$ , then $\partial ^j_X h (0) := \partial ^j_X h (0_n)$ is a homogeneous free polynomial of degree j in $X = (X_1, \cdots , X_d)$ . This Taylor–Taylor series at $0$ can be expanded further as

$$ \begin{align*}h(X) = \sum_{\omega \in \mathbb{F}^d_+} \Delta^{\omega^{\mathrm{t}}}_X f (0).\end{align*} $$

Here, $\Delta ^{\omega ^{\mathrm {t}} }_{(\cdot )} f (0)$ is a partial difference-differential operator of f at $0$ of order $|\omega |$ , where if $\omega = i_1 \cdots i_m$ , then $|\omega | =m$ is the length of $\omega $ and $\omega ^{\mathrm {t}} = i_m \cdots i_1$ . Given any $n \in \mathbb {N}$ , $\Delta ^{\omega ^{\mathrm {t}}}_{(\cdot ), \cdots , (\cdot )} f (0) = \Delta ^{\omega ^{\mathrm {t}}}_{(\cdot ), \cdots , (\cdot )} f (0_n) : \mathbb {C}^{(n\times n)\cdot d} \otimes \mathbb {C}^{|\omega |} \rightarrow \mathbb {C}^{n \times n}$ is a multi-linear map from $|\omega |$ copies of $\mathbb {C}^{(n\times n)\cdot d}$ into $\mathbb {C}^{n \times n}$ , and we have written $\Delta ^{\omega ^{\mathrm {t}} }_X f(0)$ as a short-form notation for $\Delta ^{\omega ^{\mathrm {t}} }_{X, \cdots , X} f (0, \cdots , 0)$ , where $X \in \mathbb {C}^{(n\times n)\cdot d}$ is repeated $|\omega |$ times and $0 =(0_n, \cdots , 0_n ) \in \mathbb {C}^{(n\times n)\cdot d}$ is repeated $|\omega | +1$ times. That is, $\Delta ^{\omega ^{\mathrm {t}}}_{H^{(1)}, \cdots , H^{(|\omega |)}} f(0)$ is multilinear in the ‘directional arguments’, $H^{(j)} \in \mathbb {C}^{(n\times n)\cdot d}$ , $1 \leq j \leq |\omega |$ . See [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov35, Section 3.5 and Corollary 4.4] for a precise definition of these higher-order partial difference-differential operators. Given any $\omega \in \mathbb {F}^d_+$ , one can compute that

$$ \begin{align*}\Delta^{\omega^{\mathrm{t}}}_X h (0) = c_\omega X^\omega, \quad c_\omega \in \mathbb{C},\end{align*} $$

so that this gives a free power series expansion of $h(X)$ , and any h that is uniformly analytic in a uniformly open neighbourhood of $0$ can be identified with a free FPS [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov35, Corollary 4.4].

2.2 Free formal power series

Recall that any $h \in \mathbb {C} \langle \! \langle \mathfrak {z} \rangle \! \rangle $ can be written as

$$ \begin{align*}h(\mathfrak{z} ) = \sum_{\omega \in \mathbb{F}^d_+} \hat{h}_\omega \mathfrak{z}^\omega; \quad \quad \hat{h}_\omega \in \mathbb{C}.\end{align*} $$

There is a natural ‘letter reversal’ involution on the free monoid, $\mathrm {t} : \mathbb {F}^d_+ \rightarrow \mathbb {F}^d_+$ defined by $\omega \mapsto \omega ^{\mathrm {t}}$ , where if $\omega = i_1 \cdots i_n$ , $\omega ^{\mathrm {t}} := i_n \cdots i_1$ and $\emptyset ^{\mathrm {t}} := \emptyset $ . This gives rise to a natural involutive anti-isomorphism of the ring of all complex free FPS, $\mathrm {t} : \mathbb {C} \langle \! \langle \mathfrak {z} \rangle \! \rangle \twoheadrightarrow \mathbb {C} \langle \! \langle \mathfrak {z} \rangle \! \rangle $ , via

$$ \begin{align*}h (\mathfrak{z} ) = \sum_{\omega \in \mathbb{F}^d_+} \hat{h}_\omega \mathfrak{z}^\omega \quad \mapsto \quad h^{\mathrm{t}} (\mathfrak{z} ):= \sum_{\omega \in \mathbb{F}^d_+} h_{\omega^{\mathrm{t}}} \mathfrak{z}^\omega = \sum h_{\omega } \mathfrak{z}^{\omega^{\mathrm{t}}},\end{align*} $$

and we call $h^{\mathrm {t}}$ the transpose of h.

Given a free formal power series, $h (\mathfrak {z} ) = \sum \hat {h}_\omega \mathfrak {z}^\omega $ , its radius of convergence, $R_h \in [0, +\infty ]$ , is the largest value so that the power series $h(Z)$ converges absolutely and uniformly on $r \cdot \mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ for any $0<r<R_h$ . This radius can be computed by Popescu’s generalization of the Hadamard formula [Reference Popescu44],

(2.1) $$ \begin{align} \frac{1}{R_h} = \limsup_{n\rightarrow \infty} \sqrt[2n]{\sum_{|\omega| =n} | \hat{h}_\omega |^2 }. \end{align} $$

Hence, any free FPS with radius of convergence $r>0$ defines a uniformly analytic NC function on $r\cdot \mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ .

Every free FPS that converges with radius $r> 0$ will be bounded on every row sub-ball of radius $r'$ , $0<r'<r$ . However, in [Reference Pascoe43], J.E. Pascoe has constructed an example of an NC function that is uniformly locally bounded everywhere in the $d-$ dimensional NC universe and yet is not bounded on the unit row-ball. In order to avoid such pathologies, we will define an NC function, h, to be uniformly entire if it is defined by a free FPS at $0$ with an infinite radius of convergence. (If a free FPS, $h \in \mathbb {C} \langle \! \langle \mathfrak {z} \rangle \! \rangle $ , has a positive radius of convergence, then this FPS is necessarily equal to the Taylor–Taylor series of h at $0$ , as described at the end of Subsection 2.1.) By [Reference Popescu44, Theorem 1.1 and Section 3], or by a special case of [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov35, Corollary 7.26 and Theorem 8.11], this is equivalent to h being defined and uniformly locally bounded (hence uniformly analytic) everywhere in the NC universe, and bounded on any row-ball of finite radius which is centered at the origin. In particular, the example of [Reference Pascoe43] is not uniformly entire, according to our definition.

We will denote the ring of uniformly entire NC functions in d NC variables by $\mathscr {O}^d$ .

2.3 The free Hardy space

Let $\mathbb {H}^2_d$ denote the set of all free FPS with square–summable coefficients:

$$ \begin{align*}\mathbb{H}^2_d := \left\{ h(\mathfrak{z}) = \sum \hat{h}_\omega \mathfrak{z}^\omega \left| \ \sum |\hat{h}_\omega|^2 < + \infty \right. \right\}.\end{align*} $$

Any $h \in \mathbb {H}^2_d$ has radius of convergence at least $1$ , so that $\mathbb {H}^2_d$ is a vector space of uniformly analytic NC functions on $\mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ . In fact, equipping $\mathbb {H}^2_d$ with the $\ell ^2-$ norm of the coefficients, this is a Hilbert space of NC functions that is clearly a natural multivariate and NC generalization of the classical Hardy space, $H^2$ , of square-summable Taylor series in the complex unit disk.

Left multiplication by each of the d independent variables, $L_k := M^L_{\mathfrak {z}_k}$ , defines a $d-$ tuple of isometries on $\mathbb {H}^2_d$ with pairwise orthogonal ranges which we call the left free shifts. Since these have pairwise orthogonal ranges, $L := (L_1 , \cdots , L_d ) : \mathbb {H}^2_d \otimes \mathbb {C}^d \rightarrow \mathbb {H}^2_d$ , is a row isometry, i.e., an isometry from several copies of a Hilbert space into one copy. The standard orthonormal basis for $\mathbb {H}^2_d \simeq \ell ^2 (\mathbb {F}^d_+ )$ is given by the free monomials $\{ \mathfrak {z}^\omega \}_{\omega \in \mathbb {F}^d_+}$ , and it follows that the letter reversal involution on free FPS defines an involutive unitary operator on $\mathbb {H}^2_d$ , the flip unitary, $U_{\mathrm {t}} = U_{\mathrm {t}}^{*}$ . One can check that $U_{\mathrm {t}} L_k U_{\mathrm {t}} = M^R_{\mathfrak {z}_k} =: R_k$ is the right free shift of right multiplication by $\mathfrak {z}_k$ . Hence, $R = U_{\mathrm {t}} L U_{\mathrm {t}}$ is also a row isometry. In the single-variable setting, $d=1$ , $\mathbb {H}^2_1 =H^2$ is the classical Hardy space of analytic Taylor series in the complex unit disk with square-summable coefficients. In this case, we use the standard notation, $S=M_z$ , for the isometric shift operator on $H^2$ .

We also consider the Banach algebra of uniformly bounded, and hence uniformly analytic, NC functions in the NC unit row-ball, $\mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , which we denote by $\mathbb {H}^\infty _d$ . That is, $h \in \mathbb {H}^\infty _d$ if

$$ \begin{align*}\| h \|_\infty := \sup_{Z \in \mathbb{B}^{(\mathbb{N} \times \mathbb{N}) \cdot d}} \| h (Z ) \| < +\infty.\end{align*} $$

Any $h \in \mathbb {H}^\infty _d$ is necessarily a uniformly bounded and hence analytic NC function in $\mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , and as proven in [Reference Popescu44, Theorem 3.1] and [Reference Salomon, Shalit and Shamovich50, Theorem 3.1], $h \in \mathbb {H}^\infty _d$ if and only if h belongs to the left multiplier algebra of $\mathbb {H}^2_d$ . Namely, an NC function, h, on $\mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ is a left multiplier of the free Hardy space if $h \cdot f \in \mathbb {H}^2_d$ for any $f \in \mathbb {H}^2_d$ . Moreover, the left multiplier algebra is equal to $\mathbb {H}^\infty _d$ , and if $M^L_h : \mathbb {H}^2_d \rightarrow \mathbb {H}^2_d$ denotes the linear map of left multiplication by $h \in \mathbb {H}^\infty _d$ , then we have equality of norms, $\| h \|_\infty = \| M^L_h \|$ [Reference Popescu44, Theorem 3.1], [Reference Salomon, Shalit and Shamovich50, Theorem 3.1]. Note that since $1 \in \mathbb {H}^2_d$ , $\mathbb {H}^\infty _d \subseteq \mathbb {H}^2_d$ .

If $h \in \mathbb {H}^2_d$ , then $h_r (Z) := h(rZ) \in \mathbb {H}^\infty _d$ ; see, for example, [Reference Jury, Martin and Shamovich32, Lemma 4.5]. Given any $h \in \mathbb {H}^\infty _d$ , the operator $h(L)$ , obtained by substituting the $L_j$ in for the variables $\mathfrak {z}_j$ in the free power series for h is the operator, $M^L_h$ , of left multiplication by h. However, one can check that $h(R) = M^R_{h^{\mathrm {t}}}$ .

Any $h \in \mathbb {H}^2_d$ is familiar, i.e., is given by an operator realization $(A,b,c)$ . Indeed, given any $h \in \mathbb {H}^2_d$ ,

$$ \begin{align*}h(\mathfrak{z} ) = \sum_{\omega \in \mathbb{F}^d_+} \hat{h}_\omega \mathfrak{z}^\omega, \quad \mbox{and} \quad (L_j^{*} h) (\mathfrak{z}) = \sum \hat{h}_{j\omega} \mathfrak{z}^\omega.\end{align*} $$

Since the monomials, $\{ \mathfrak {z}^\omega \}$ , are an ON basis of $\mathbb {H}^2_d$ , it follows that for any $\omega \in \mathbb {F}^d_+$ ,

$$ \begin{align*}1^{*} L^{*\omega} h^{\mathrm{t}} = \sum_\alpha \hat{h}_{\alpha} \langle { \mathfrak{z}^{\omega^{\mathrm{t}}} } , {\mathfrak{z}^{ \alpha^{\mathrm{t}}}} \rangle_{\mathbb{H}^2_d} = \hat{h}_\omega.\end{align*} $$

Hence, $h\sim ( L^{*} , 1, h^{\mathrm {t}})$ has an operator realization. Note that the constant function, $1$ , is $L-$ cyclic. Here, and throughout, all inner products are conjugate linear in their first argument.

If $h \in \mathbb {H}^2_d$ has the realization $(A,b,c)$ , then

$$ \begin{align*}h (\mathfrak{z}) = \sum_{\omega \in \mathbb{F}^d_+} \hat{h}_\omega \mathfrak{z}^\omega, \quad \mbox{where} \quad \hat{h}_\omega = b^{*} A^\omega c.\end{align*} $$

As one can readily calculate,

$$ \begin{align*}(L_j^{*} h) (\mathfrak{z}) = \sum_{\omega \in \mathbb{F}^d_+} \hat{h}_{j\omega} \mathfrak{z}^\omega, \quad \mbox{and} \quad (R_j^{*} h) (\mathfrak{z}) = \sum_{\omega \in \mathbb{F}^d_+} \hat{h}_{\omega j} \mathfrak{z}^\omega,\end{align*} $$

so that

(2.2) $$ \begin{align} L_j^{*} h \sim (A, A_j^{*} b, c), \quad \mbox{and} \quad R_j^{*} h \sim (A, b, A_j c). \end{align} $$

Moreover, given any $p \in \mathbb {C} \langle \mathfrak {z} \rangle $ , it then follows that

(2.3) $$ \begin{align} p(L)^{*} h \sim (A,p(A^{*})b, c), \quad \quad p(R)^{*} h \sim (A, b, \overline{p}(A) c), \end{align} $$

where $\overline {p} \in \mathbb {C} \langle \mathfrak {z} \rangle $ denotes the free polynomial obtained by complex conjugation of the coefficients of p.

FM realizations from descriptor realizations and vice versa.

Any NC function that admits an FM realization admits a descriptor realization and vice versa. Namely, if h has descriptor realization $(A,b,c)$ , then one can define

(3.6) $$ \begin{align} \mathcal{H}' := \bigvee_{\omega \neq \emptyset} A^\omega c, \quad A' := A |_{\mathcal{H}^{\prime\prime}}, \quad B_j := A_j c, \quad C := (P_{\mathcal{H} '} b)^{*}, \quad \mbox{and} \quad D := h(0). \end{align} $$

It follows that $(A',B,C,D)$ is an FM realization of h. Moreover, it is readily checked that if $(A,b,c)$ is a minimal descriptor realization, then $(A',B,C,D)$ is a minimal FM realization. Conversely, if $(A,B,C,D)$ is an FM realization of h on $\mathcal {H}$ , one can construct a descriptor realization, $(\hat {A}, b , c)$ on $\hat {\mathcal {H}} := \mathcal {H} \oplus \mathbb {C}$ by setting $\hat {A}_j |_{\mathcal {H} } := A_j $ and defining $\hat {A}_j 0 \oplus 1 := B_j \in \mathcal {H}$ . That is, with respect to the direct sum decomposition $\hat {\mathcal {H}} = \mathcal {H} \oplus \mathbb {C}$ ,

(3.7) $$ \begin{align} \hat{A}_j = \begin{pmatrix} A_j & B_j \langle {0 \oplus 1} , {\cdot} \rangle_{\hat{\mathcal{H}}} \\ 0 & 0 \end{pmatrix}. \end{align} $$

Next, one defines $b:= C^{*} \oplus \overline {D} 1$ and $c:= 0 \oplus 1$ . It is easy to verify that $(\hat {A} , b, c )$ is then a descriptor realization of h. If $(A,B,C,D)$ is a minimal FM realization, then $(\hat {A}, b, c )$ is controllable, but it may not be observable. Even it is not observable, however, its observable subspace has codimension at most one in $\hat {\mathcal {H}}$ .

3.1 Uniqueness of minimal realizations and domains

In this subsection, we show, as in the finite setting, that minimal realizations are unique up to a suitable notion of isomorphism.

Remark 3.3 (On closed and closeable operators).

Recall that a linear map $L : \mathrm {Dom} \, L \subseteq \mathcal {H} \rightarrow \mathcal {J}$ , with dense linear domain, $\mathrm {Dom} \, L$ , in a Hilbert space, $\mathcal {H}$ , and range in a Hilbert space, $\mathcal {J}$ , is said to be closed if its graph is a closed subspace of $\mathcal {H} \oplus \mathcal {J}$ . Such a linear map is closeable if it has a closed extension, $\hat {L}$ , i.e., $\mathrm {Dom} \, L \subseteq \mathrm {Dom} \, \hat {L} \subseteq \mathcal {H}$ and $\hat {L} |_{\mathrm {Dom} \, L} = L$ . In this case, we write $L \subseteq \hat {L}$ and say that L is a restriction of $\hat {L}$ . If L is closeable, its closure, $\overline {L}$ , is defined as its minimal closed extension, which can be obtained by taking the closure of the graph of L in $\mathcal {H} \oplus \mathcal {J}$ . In particular, L is closeable if and only if the closure of its graph in $\mathcal {H} \oplus \mathcal {J}$ does not contain any vectors of the form, $0 \oplus y$ for $y \neq 0$ . Equivalently, L is closeable if and only if whenever $x_n \in \mathrm {Dom} \, L$ , $x_n \rightarrow 0$ and $y_n = Lx_n \rightarrow y \in \mathcal {J}$ , then necessarily $y =0$ .

Given a densely-defined linear map, $L : \mathrm {Dom} \, L \subseteq \mathcal {H} \rightarrow \mathcal {J}$ , as above, its Hilbert space adjoint, $L^{*}$ , is a linear map defined on the linear domain, $\mathscr {D}_*$ , consisting of all $y \in \mathcal {J}$ for which there exists a $y_* \in \mathcal {H}$ , so that

$$ \begin{align*}\langle {Lx} , {y} \rangle_{\mathcal{J}} = \langle {x} , {y_*} \rangle_{\mathcal{H}}, \quad \quad \forall x \in \mathrm{Dom} \, L.\end{align*} $$

One then defines $\mathrm {Dom} \, L^{*} := \mathscr {D}_*$ and $L^{*} y := y_*$ . The adjoint of a densely-defined linear map is always a closed linear map. However, it may not be densely-defined. One can prove that $L^{*}$ is densely-defined if and only if L is closeable, in which case $\overline {L} = L^{**}$ , i.e., the bi-adjoint is the minimal closed extension of L. Finally, a subset $\mathscr {D} \subseteq \mathrm {Dom} \, L$ of the domain of a closed linear operator, L, is called a core for L if $(\mathscr {D} , L \mathscr {D} )$ is dense in the graph of L. In particular, if L is closeable with closure $\overline {L}$ , then $\mathrm {Dom} \, L$ is a core for $\overline {L}$ . Standard references for the theory of closed linear operators include the functional analysis texts, [Reference Reed and Simon48, Reference Akhiezer and Glazman2, Reference Conway17].

In particular, the above theorem implies that $Sc =c'$ , $S^{-*} b = b'$ and $S^{-1} A' S = A$ , suitably interpreted. An analogous result holds for FM realizations. This theorem has been previously established, essentially, in one-variable in [Reference Helton29, Theorem 3b.1] and [Reference Ball and Cohen10, Theorem 3.2], and it is a natural extension of a well-known result on uniqueness of finite-dimensional minimal realizations from systems theory [Reference Chen14, Theorem 5-21].

Proof. By assumption, we have that for any $\omega \in \mathbb {F}^d_+$ ,

$$ \begin{align*}b^{*} A^\omega c = b^{\prime *} A^{'\omega} c'.\end{align*} $$

We define a linear map, $S_0$ , with dense domain $\mathbb {C} \langle A \rangle c \subseteq \mathcal {H}$ by $S_0 p(A) c = p(A') c'$ . First, in order that this be well defined, if $p(A)c = \widetilde {p} (A) c$ for $p,\widetilde {p} \in \mathbb {C} \langle \mathfrak {z} \rangle $ , then we need to check that $p(A') c' = \widetilde {p} (A') c'$ . This follows from minimality. Indeed, if $p(A)c = \widetilde {p} (A)c$ in $\mathcal {H}$ , then for any $q \in \mathbb {C} \langle \mathfrak {z} \rangle $ ,

$$ \begin{align*}b^{*} q(A) p(A) c = b^{*} q(A) \widetilde{p} (A) c,\end{align*} $$

so that

$$ \begin{align*}b^{\prime *} q(A') p(A') c' = b^{\prime *} q(A') \widetilde{p} (A') c'.\end{align*} $$

By minimality of the realization $(A',b',c')$ , it follows that $p(A') c' = \widetilde {p} (A' ) c' \in \mathcal {H} '$ .

The linear map, $S_0$ , has dense range in $\mathcal {H}'$ , by construction. We next prove that $S_0$ is closeable. If $p_n (A) c \in \mathrm {Dom} \, S_0$ , $p_n (A) c \rightarrow 0$ and $S_0 p_n (A) c = p_n (A' ) c' \rightarrow y' \in \mathcal {H} '$ , we need to show that $y' =0$ . For any $q \in \mathbb {C} \langle \mathfrak {z} \rangle $ , consider

$$ \begin{align*} \langle { q(A^{\prime *} ) b'} , {y'} \rangle_{\mathcal{H}'} & = \lim_n \langle {b'} , {\overline{q} (A') p_n (A') c'} \rangle_{\mathcal{H} '} \\ & = \lim b^{*} \overline{q} (A) p_n (A) c =0, \end{align*} $$

since $p_n (A) c \rightarrow 0$ in $\mathcal {H}$ and $\overline {q} (A)$ is a bounded linear operator for any $q \in \mathbb {C} \langle \mathfrak {z} \rangle $ . It follows that $y' \in \mathcal {H}' $ is orthogonal to $\mathscr {O}_{A^{\prime *}, b'} = \mathcal {H} '$ and hence $y' =0$ . This proves that $S_0$ is closeable, and we set $S := \overline {S_0}$ , the minimal closed extension of $S_0$ .

The closed operator, S, has dense range since $S_0$ does, and we claim that S is also injective. Indeed, if S were not injective, then $Sh =0$ for some $h \in \mathrm {Dom} \, S \subseteq \mathcal {H}$ . Since $S = \overline {S_0}$ , $\mathrm {Dom} \, S_0 = \mathbb {C} \langle A \rangle c$ is a core for S. In particular, there is a sequence $p_n (A) c \in \mathrm {Dom} \, S_0$ so that $p_n (A) c \rightarrow h$ and $S p_n (A) c \rightarrow Sh =0$ . By symmetry, if we defined instead $S^{\prime }_0 : \mathbb {C} \langle A' \rangle \subseteq \mathcal {H} ' \rightarrow \mathcal {H}$ by $S' p(A' ) c' = p(A) c$ , $S'$ is closeable which implies that $h =0$ . (It is also easy to argue directly that S is injective.) Since S is injective, closed and has dense range, $S^{-1}$ is also closed and densely-defined on $\mathrm {Ran} \, S$ . It is also injective and has dense range.

Finally, we claim that $\mathbb {C} \langle A^{\prime *} \rangle b' \subseteq \mathrm {Dom} \, S^{*}$ and that $S^{*} q(A^{\prime *} ) b' = q(A^{*}) b$ for any $q \in \mathbb {C} \langle \mathfrak {z} \rangle $ . Indeed, for any $p \in \mathbb {C} \langle \mathfrak {z} \rangle $ ,

$$ \begin{align*} \langle {q(A^{\prime *} ) b'} , {Sp(A)c} \rangle_{\mathcal{H} ' } & = \langle {b'} , {\overline{q} (A') p(A') c'} \rangle_{\mathcal{H}'} \\ & = \langle {b} , {\overline{q} (A) p(A) c} \rangle_{\mathcal{H}} = \langle {q(A^{*})b} , {p(A)c} \rangle_{\mathcal{H}}. \end{align*} $$

By the definition of the Hilbert space adjoint of a densely-defined linear map, $q(A^{\prime *})b' \in \mathrm {Dom} \, S^{*}$ and $S^{*} q(A^{\prime *})b' = q(A^{*} ) b$ . The fact that $\mathbb {C} \langle A^{\prime *} \rangle b'$ is a core for $S^{*}$ also follows from a symmetric argument.

Definition 3.7. Let $A \in \mathscr {B} (\mathcal {H} )^d$ and $A' \in \mathscr {B} (\mathcal {H} ' )^d$ be two column $d-$ tuples of bounded linear operators. We say that A and B are analytically equivalent if there exist $b,c \in \mathcal {H}$ and $b',c' \in \mathcal {H}'$ so that the realizations $(A,b,c)$ and $(A',b',c')$ are both minimal and define the same uniformly analytic NC function in a uniformly open neighbourhood of $0$ . That is, A and $A'$ are analytically equivalent if there exists an $r>0$ so that

$$ \begin{align*}b^{*} L_A (Z)^{-1} c = b^{\prime *} L_{A'} (Z)^{-1} c'; \quad \quad \forall Z \in r \cdot \mathbb{B}^{(\mathbb{N} \times \mathbb{N}) \cdot d},\end{align*} $$

or equivalently,

$$ \begin{align*}b^{*} A^\omega c = b^{\prime *} A^{'\omega}c' \quad \quad \forall \omega \in \mathbb{F}^d_+.\end{align*} $$

Similarly, we say that two realizations $(A,b,c)$ and $(A',b',c')$ are analytically equivalent and write $(A,b,c) \sim (A',b',c')$ if they define the same uniformly analytic NC function in a uniformly open neighbourhood of $0$ . (We do not require that analytically equivalent realizations be minimal in this second definition.)

Theorem 3.1 implies, in particular, that any two analytically equivalent $A \in \mathscr {B} (\mathcal {H})^d$ and $A' \in \mathscr {B} (\mathcal {H} ' )^d$ are pseudo-similar.

Note that if $h \sim (A,b,c)$ is familiar, then h is a uniformly analytic NC function on $\mathscr {D} (A)$ , $\mathscr {D} (A)$ is uniformly open by operator-norm continuity of the inversion map, $r \cdot \mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d} \subseteq \mathscr {D} (A)$ for $r := \| A \|^{-1}_{\mathrm {col}}>0$ , and $\mathscr {D} (A)$ is joint similarity invariant.

Example 3.10. Consider the entire function, $f(z) = \frac {e^z - 1}{z}$ . By a result of Douglas, Shapiro and Shields [Reference Douglas, Shapiro and Shields21, Theorem 2.2.4], any function that is analytic in a disk of radius $r>1$ (centred at $0$ ) is either cyclic for the backward shift on the Hardy space, $H^2 (\mathbb {D})$ , or rational. Hence, this $f \in H^2$ is cyclic for the backward shift. (It would be interesting to see whether there is an analogue of this result for the free Hardy space.) In other words, $H^2 = \bigvee _{n=0}^{\infty } S^{*n} f$ , where $S:= M_z$ denotes the shift on $H^2$ . Moreover, $1$ is cyclic for S, so we have a minimal realization for f, i.e., $f \sim (S^{*}, 1, f)$ .

To construct another realization, we consider the Volterra operator on $L^2[0,1]$ given by $(T g)(x) = \int _0^x g(t) dt$ . It is well known that T is a quasinilpotent Hilbert–Schmidt operator [Reference Gohberg and Krein25]. Moreover, the vector $1 \in L^2 [0,1]$ is $T-$ cyclic, since $T^n 1 = \frac {x^n}{n!}$ . Since $(T^{*} g)(x) = \int _x^1 g(t)dt$ , it follows that $1$ is also $T^{*}-$ cyclic. The power series generated by the realization $(T, 1, 1)$ is

$$\begin{align*}\sum_{n=0}^{\infty} \langle 1, T^n 1 \rangle z^n = \sum_{n=0}^{\infty} \left( \int_0^1 \frac{x^n}{n!}dx \right) z^n = \sum_{n=0}^{\infty} \frac{z^n}{(n+1)!} = f(z), \end{align*}$$

so that $(T,1,1)$ is also a minimal realization of f. Theorem 3.6 now implies that $S^{*}$ and T are analytically equivalent, and hence pseudo-similar.

Domain of a familiar NC function

Recall that NC rational functions are defined as certain evaluation equivalence classes of NC rational expressions [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov34]. Namely, an NC rational expression, $\mathrm {r}$ , is any syntactically valid expression obtained by applying the arithmetic operations of inversion, addition and multiplication to the free algebra. The domain, $\mathrm {Dom} \, \mathrm {r}$ , of such an expression is the set of all $X \in \mathbb {C}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ for which the expression can be evaluated at X, and $\mathrm {r}$ is said to be valid, if this domain is not empty. Two valid NC rational expressions, $\mathrm {r}_1$ and $\mathrm {r}_2$ , are said to be equivalent if $\mathrm {r}_1 (X) = \mathrm {r}_2 (X)$ for all $X \in \mathrm {Dom} \, \mathrm {r}_1 \bigcap \mathrm {Dom} \, \mathrm {r}_2$ . One can prove that for any two valid NC rational expressions, there is a finite $N \in \mathbb {N}$ so that $\mathrm {Dom}_n \, \mathrm {r}_1 \bigcap \mathrm {Dom}_n \, \mathrm {r}_2$ is nonempty, and in fact is Zariski open and dense in $\mathbb {C}^{(n\times n)\cdot d}$ for any $n \geq N$ ; see [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov34, Footnote 1, Section 2]. It follows that this evaluation relation is transitive, and an equivalence relation on valid NC rational expressions. One then defines an NC rational function, $\mathfrak {r}$ , as an evaluation equivalence class of valid NC rational expressions with domain $\mathrm {Dom} \, \mathfrak {r} := \bigcup _{\mathrm {r} \in \mathfrak {r}} \mathrm {Dom} \, \mathrm {r}$ . One views $\mathfrak {r}$ as an NC function on $\mathrm {Dom} \, \mathfrak {r}$ , and we simply write $\mathfrak {r} (X) := \mathrm {r} (X)$ if $X \in \mathrm {Dom} \, \mathrm {r}$ and $\mathrm {r} \in \mathfrak {r}$ . Any NC rational function, , is uniformly analytic on its domain, and its domain is a joint similarity-invariant and uniformly open NC set. This follows from the fact that the arithmetic operations $+, \cdot , (\cdot )^{-1}$ are all operator-norm continuous and free polynomials are uniformly entire.

The evaluation equivalence relation, $\sim $ , described above, applied to arbitrary uniformly analytic NC functions need not be well defined. First, given uniformly analytic $f,g$ , there is no reason for $\mathrm {Dom} \, f \cap \mathrm {Dom} \, g$ to be nonempty. Also, this relation need not be transitive when applied to such arbitrary functions. If $f\sim g$ and $g \sim h$ , and $\mathrm {Dom} \, f \cap \mathrm {Dom} \, g \neq \emptyset $ , $\mathrm {Dom} \, g \cap \mathrm {Dom} \, h \neq \emptyset $ , it could still be that $\mathrm {Dom} \, f \cap \mathrm {Dom} \, h = \emptyset $ , or worse, it could be that this intersection is nonempty but $f(X) \neq h (X)$ for some $X \in \mathrm {Dom} \, f \cap \mathrm {Dom} \, h$ .

The domains of familiar (and hence) uniformly analytic NC functions exhibit better behaviour. If $f\sim (A,b,c)$ and $g \sim (A' ,b' ,c')$ , then $f,g$ are uniformly analytic on $\mathscr {D} (A)$ and $\mathscr {D} (A')$ , respectively, and $\mathscr {D} (A) \cap \mathscr {D} (A') \supseteq r \cdot \mathbb {B}^{(\mathbb {N} \times \mathbb {N}) \cdot d}$ , where $r := \min \{ \| A \|_{\mathrm {col}}^{-1} , \| A ' \|_{\mathrm {col}}^{-1} \}$ . It follows, by the identity theorem in several complex variables, that we can define an evaluation relation on such familiar NC functions by $f\sim g$ if $f(X) = g(X)$ for all X that belong to the (level-wise) connected component of $0$ in $\mathscr {D} (A) \cap \mathscr {D} (A')$ . If $\mathscr {D}^{(0)} (A)$ is this level-wise connected component of $0$ in $\mathscr {D} (A) \cap \mathscr {D} (A')$ , we view any familiar NC function as an evaluation equivalence class, $[f]$ , defined on the domain

(3.8) $$ \begin{align} \mathrm{Dom} \, [f] := \bigcup_{(A,b,c) \sim g \in [f]} \mathscr{D}^{(0)} (A), \end{align} $$

and we write $[f](X) := g(X)$ if $g\in [f]$ , and $X \in \mathscr {D}^{(0)} (A)$ . We will typically write f in place of $[f]$ .

3.2 Kalman decomposition

Given any operator realization for a fixed NC function, one can construct a minimal realization by compressing the realization to a certain semi-invariant subspace. This extends what is called the Kalman decomposition from finite-dimensional realization theory [Reference Kalman, Falb and Arbib36, Reference Bart, Gohberg and Kaashoek12].

Let $(A,b,c) \sim h$ be any descriptor realization of h on $\mathcal {H}$ . Define the minimal space of $(A,b,c)$ as the subspace

(3.9) $$ \begin{align} \mathscr{M}_{A,b,c} := \mathscr{C}_{A,c} \ominus \mathscr{O}_{A^{*},b}^\perp \cap \mathscr{C}_{A,c}. \end{align} $$

Then, $\mathscr {M}_{A,b,c}$ is the direct difference of the nested, $A-$ invariant subspaces, $\mathscr {C}_{A,c}$ and $\mathscr {O}_{A^{*},b}^\perp \cap \mathscr {C}_{A,c}$ , and is hence semi-invariant for A [Reference Sarason52]. That is, if $Q_0 : \mathcal {H} \rightarrow \mathscr {M}_{A,b,c}$ denotes the orthogonal projection, then

$$ \begin{align*}Q_0 A^\omega Q_0 = (Q_0 A Q_0 )^\omega, \quad \omega \in \mathbb{F}^d_+,\end{align*} $$

so that compression to $\mathscr {M}_{A,b,c}$ is a surjective, unital homomorphism of the unital algebra generated by A, $\mathrm {Alg} \{ I, A_1, \cdots , A_d \}$ , onto $\mathrm {Alg} \{ I, A^{(0)}_1, \cdots , A^{(0)}_d \}$ where $A^{(0)}:= Q_0 A |_{\mathscr {M}_{A,b,c}}$ .

One can construct a minimal FM realization from any FM realization similarly.

Proof. Given $(A,b,c)$ , first let $A'$ be the restriction of A to the $A-$ invariant controllable subspace, $\mathscr {C}_{A,c}$ . If $P'$ is the orthogonal projection onto $\mathscr {C}_{A,c}$ , let $b' = P' b$ and $c' = c \in \mathscr {C}_{A,c}$ . Then, $(A',b',c')$ is a controllable realization of h on $\mathscr {C}_{A,c}$ since

$$ \begin{align*}b^{\prime *} A^{'w} c' = (P'b)^{*} (P' A P')^\omega P' c = b^{*} A^\omega P' c = b^{*} A^\omega c\end{align*} $$

for any word $\omega \in \mathbb {F}^d_+$ .

Define the subspace $\mathscr {M} := \bigvee A^{\prime \omega *} b' \subseteq \mathscr {C}_{A,c}$ with orthogonal projection $Q_0 \leq P'$ . First observe that $x \in \mathscr {C}_{A,c} \ominus \mathscr {M}$ if and only if, for all $\omega \in \mathbb {F}^d_+$ ,

$$ \begin{align*} 0 & = \langle {A^{\prime\omega *} b'} , {x} \rangle = \langle {(P' A P')^{\omega *} P'b} , {x} \rangle \\ & = \langle {P' A^{\omega *} b} , {x} \rangle, \quad \quad \mbox{since Ran }\, P' \text{ is co-invariant for } A^{*}, \\ & = \langle {A^{\omega *}b} , {x} \rangle, \end{align*} $$

since $P' x = x$ . It follows that $x \in \mathscr {C}_{A,c} \ominus \mathscr {M}$ if and only if $x \in \mathscr {O}_{A^{*},b}^\perp \cap \mathscr {C}_{A,c}$ , so that

$$ \begin{align*}\mathscr{M} = \mathscr{C}_{A,c} \ominus \mathscr{C}_{A,c} \cap \mathscr{O}_{A^{*},b}^\perp =\mathscr{M}_{A,b,c}.\end{align*} $$

Set $A^{(0)} = Q_0 A |_{\mathscr {M}_{A,b,c}}$ , $b_0 = b'$ and $c_0 = Q_0 c$ . We claim that $(A^{(0)}, b_0 , c_0)$ is a minimal realization of h on $\mathscr {M}_{A,b,c}$ . Indeed, for any word $\omega \in \mathbb {F}^d_+$ ,

$$ \begin{align*} b_0^{*} A^{(0) \omega} c_0 & = b^{\prime *} (Q_0 A' Q_0 )^\omega Q_0 c' \\ & = b^{\prime *} Q_0 A^{\prime\omega} c \quad \quad \quad (Q_0 \mbox{ is co-invariant for } A' \text{ and } c'=c)\\ & = (P' b)^{*} A^{\prime\omega} P'c \quad \quad \quad (b' \in \mathrm{Ran} \, Q_0 = \mathscr{M}_{A,b,c}) \\ & = b^{*} A^\omega c = \hat{h}_\omega, \end{align*} $$

since $c = c' \in \mathrm {Ran} \, P'$ and $\mathrm {Ran} \, P' = \mathscr {C}_{A,c}$ is $A-$ invariant. This proves that $(A^{(0)}, b_0, c_0)$ is a realization for h. Moreover,

$$ \begin{align*}\bigvee A^{(0) * \omega} b_0 = \bigvee A^{\prime *\omega} \underbrace{Q_0 b'}_{=b '} = \mathscr{M}_{A,b,c}, \quad \quad \mbox{and} \end{align*} $$

$$ \begin{align*} \bigvee A^{(0) \omega} c_0 & = \bigvee Q_0 A^{\prime\omega} Q_0 c \\ & = Q_0 \bigvee A^{\prime\omega} c, \quad \quad \mbox{since } Q_0 \text{ is } A'\text{-co-invariant,} \\ &= Q_0 \bigvee A^\omega c, \quad \quad \mbox{since } A' = A|_{\mathscr{C}_{A,c}}, \\ &= Q_0 \mathscr{C}_{A,c} = \mathscr{M}_{A,b,c}, \end{align*} $$

since $\mathscr {M}_{A,b,c} \subseteq \mathscr {C}_{A,c}$ . This proves that the realization $(A^{(0)}, b_0, c_0)$ on $\mathscr {M}_{A,b,c}$ is minimal.

If $(A,b,c)$ is a finite-dimensional realization of an NC rational function, the classical Kalman decomposition theorem further asserts that the invertibility domain of $A^{(0)}$ contains that of A, $\mathscr {D} (A) \subseteq \mathscr {D} (A^{(0)})$ . Although we are unable to prove this in general, we can extend this to the case where $A \in \mathscr {C} = \mathscr {C} (\mathcal {H})^d$ is a $d-$ tuple of compact operators.

This corollary allows one to extend $h \sim (A,b,c)$ , $A \in \mathscr {C}$ , to an NC function on a potentially larger NC domain. (Later, we will see that if $A \in \mathscr {C}$ , then $\mathscr {D} (A)$ is level-wise open and connected in matrix-norm, so that $(A,b,c)$ and $(A^{(0)}, b_0, c_0)$ define the same uniformly analytic NC function on $\mathscr {D} (A)$ .)

Lemma 3.16. Let $T \in \mathscr {B} (\mathcal {H})$ have a closed, invariant subspace, $\mathcal {J}$ , so that with respect to the orthogonal direct sum decomposition, $\mathcal {H} = \mathcal {J} \oplus \mathcal {J}^\perp $ ,

$$ \begin{align*}T \simeq \begin{pmatrix} T_1 & T_2 \\ 0 & T_3 \end{pmatrix}.\end{align*} $$

If T is invertible, then $T_1$ is injective, $T_1^{*}$ is surjective on $\mathcal {J}$ , and $T_3 \simeq P_{\mathcal {J}}^\perp T |_{\mathcal {J}^\perp }$ is surjective on $\mathcal {J}^\perp $ . Moreover, T and $T_1$ are both invertible if and only if T and $T_3$ are both invertible, in which case

$$ \begin{align*}T^{-1} = \begin{pmatrix} T_1^{-1} & -T_1^{-1} T_2 T_3^{-1} \\ 0 & T_3^{-1} \end{pmatrix},\end{align*} $$

so that $\mathcal {J}$ is also $T^{-1}-$ invariant.

Proof. If T is invertible and $0 \neq x \in \mathcal {J}$ , then

$$ \begin{align*}\begin{pmatrix} 0 \\ 0 \end{pmatrix} \neq T \begin{pmatrix} x \\ 0 \end{pmatrix} = \begin{pmatrix} T_1 x \\ 0 \end{pmatrix},\end{align*} $$

so that $T_1 \simeq T |_{\mathcal {J}}$ must be injective on $\mathcal {J}$ . Similarly, since T is surjective, given any $y \in \mathcal {J}^\perp $ , there exists $x \in \mathcal {J}$ and $x' \in \mathcal {J}^\perp $ so that

$$ \begin{align*}\begin{pmatrix} 0 \\ y \end{pmatrix} = T \begin{pmatrix} x \\ x' \end{pmatrix} = \begin{pmatrix} T_1 x + T_2 x' \\ T_3 x' \end{pmatrix}.\end{align*} $$

Hence, $T_3 x' = y$ and $T_3$ is surjective. Since T is invertible so is $T^{*} \simeq \left ( \begin {smallmatrix} T_1^{*} & 0 \\ T_2^{*} & T_3^{*} \end {smallmatrix} \right ) $ , and a similar analysis shows that $T_1^{*} \in \mathscr {B} (\mathcal {J} )$ is surjective on $\mathcal {J}$ .

If $T_3$ and T are both invertible, suppose that $y \in \mathcal {J} \ominus \mathrm {Ran} \, T_1$ . Then since T is bijective, there exists $x_1 \in \mathcal {J}$ and $x_2 \in \mathcal {J}^\perp $ so that

$$ \begin{align*}\begin{pmatrix} y \\ 0 \end{pmatrix} = T \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} T_1 x_1 + T_2 x_2 \\ T_3 x_2 \end{pmatrix}.\end{align*} $$

Hence, $T_3 x_2 =0$ so that $x_2 =0$ by injectivity of $T_3$ , and $y = T_1 x_1$ . However, $y \in \mathcal {J} \ominus \mathrm {Ran} \, T_1$ , so that $T_1 x_1 = y=0$ . This proves that $T_1$ is surjective, and since T is invertible, $T_1$ is also injective, and hence invertible.

Similarly, if $T_1$ and T are both invertible, then so are $T^{*}$ and $T_1^{*}$ , and $T_3^{*}$ is injective. An analogous argument now proves that $T_3^{*}$ is invertible so that $T_3$ is invertible as well. If $T_1$ and $T_3$ are both invertible, then it is easily verified that the block upper triangular formula in the lemma statement is the inverse of T, so that T is invertible and $T^{-1} \mathcal {J} \subseteq \mathcal {J}$ .

Example 3.17. If $T \in \mathscr {B} (\mathcal {H})$ has a closed, invariant subspace $\mathcal {J}$ , it is generally not true that $\mathcal {J}$ is invariant for $T^{-1}$ , or that the ‘diagonal blocks’ of T are both invertible. A counterexample is given by the bilateral shift, T, on $\ell ^2 (\mathbb {Z} )$ , with $\mathcal {J} = \ell ^2 (\mathbb {N}_0 )$ . In this case, $T_1 = P_{\mathcal {J}} T |_{\mathcal {J}}$ is isomorphic to the unilateral shift, which is an isometry, hence injective, but it is not surjective, $T_3 = P^\perp _{\mathcal {J}} T |_{\mathcal {J}^\perp }$ is isomorphic to the backward shift, which is a surjective co-isometry, but not injective, and if $(e_n )_{n\in \mathbb {Z}}$ denotes the standard orthonormal basis of $\ell ^2 (\mathbb {Z})$ , then $T_2 = \langle {e_{-1}} , {\cdot } \rangle e_0$ . Moreover, T is unitary so that, with respect to the direct sum decomposition, $\ell ^2 (\mathbb {Z} ) = \ell ^2 (\mathbb {N}_0) \oplus \ell ^2 (-\mathbb {N} )$ ,

$$ \begin{align*}T^{-1} = T^{*} \simeq \begin{pmatrix} T_1^{*} & 0 \\ T_2^{*} & T_3^{*} \end{pmatrix},\end{align*} $$

so that $\ell ^2 (\mathbb {N}_0) = \mathcal {J}$ is not $T^{-1}-$ invariant.

Proof of Corollary 3.15.

With respect to the orthogonal decomposition,

$$ \begin{align*}\mathscr{H} = \mathscr{M}_{A,b,c} \oplus ( \mathscr{C}_{A,c} \ominus \mathscr{M}_{A,b,c} ) \oplus \mathscr{C}_{A,c}^\perp,\end{align*} $$

we have that

Hence, the entire operator, $A_j$ , is block upper triangular with respect to the orthogonal decompostion $\mathcal {H} = \mathscr {C}_{A,c} \oplus \mathscr {C}_{A,c}^\perp $ , while the upper left block corresponding to the decomposition of $\mathscr {C}_{A,b}$ is block lower triangular. Let, $A^{\prime }_j$ denote this upper left block so that

$$ \begin{align*}A_j = \begin{pmatrix} A^{\prime}_j & * \\ 0 & A^{(2)}_j \end{pmatrix}.\end{align*} $$

By the previous lemma, if $X \in \mathscr {D}_m (A)$ , so that $L_A (X)$ is invertible, it follows that $L_{A'} (X)$ is injective and $L_{A^{(2)}} (X)$ is surjective. However, if $L_{A'} (X) = I_m \otimes I - X \otimes A'$ is not invertible, it is not surjective, so that $\lambda =1$ belongs to the spectrum of $X \otimes A'$ . Since each $A_j$ is compact, it follows that $X \otimes A'$ is compact, so that any nonzero point in its spectrum is an eigenvalue. This contradicts injectivity of $L_{A'} (X)$ , and we conclude that $L_{A'} (X)$ is invertible. Similarly, since $L_{A'} (X)$ is invertible, we must have that $L_{A^{(0)}} (X)$ is surjective and $L_{A^{(1)}} (X)$ is injective by the previous lemma. Again, both $X \otimes A^{(0)}$ and $X \otimes A^{(1)}$ are compact, so that both $L_{A^{(0)}} (X)$ and $L_{A^{(1)}} (X)$ must be invertible. Hence, $X \in \mathscr {D}_m (A^{(0)})$ and $\mathscr {D} (A) \subseteq \mathscr {D} ( A^{(0)} )$ .

3.3 The Fornasini–Marchesini and descriptor algorithms

Let $(A,B,C,D) \sim g$ and $(A', B' , C' , D') \sim h$ be FM operator realizations on separable, complex Hilbert spaces, $\mathcal {H}$ and $\mathcal {H} '$ . An FM operator realization for $g +h$ is given by $(A^+, B^+, C^+, D^+)$ , where

(3.10) $$ \begin{align} A^+_j := A_j \oplus A^{\prime}_j, \quad B^+_j := \begin{pmatrix} B_j \\ B^{\prime}_j \end{pmatrix}, \quad C^+_j := ( C_j , C^{\prime}_j ), \quad \mbox{and} \quad D^+ := D + D'. \end{align} $$

An FM realization for $g \cdot h$ is given by $(A^\times , B^\times , C^\times , D^\times )$ , where

(3.11) $$ \begin{align} A^\times_j := \begin{pmatrix} A_j & B_j C' \\ 0 & A^{\prime}_j \end{pmatrix}, \quad B^\times_j := \begin{pmatrix} B_j D' \\ B^{\prime}_j \end{pmatrix}, \quad C^\times = ( C , D C'), \quad \mbox{and} \quad D^\times := D \cdot D'. \end{align} $$

And an FM operator realization for $g^{-1}$ , assuming that $g(0) = D \neq 0$ , is given by $( A^{(-1)}, B^{(-1)}, C^{(-1)}, D^{(-1)} )$ , where

(3.12) $$ \begin{align} A^{(-1)}_j := A_j - \frac{1}{D} B_j C, \quad B^{(-1)}_j := -\frac{1}{D} B_j, \quad C^{(-1)} := \frac{1}{D} C, \quad \mbox{and} \quad D^{(-1)} := \frac{1}{D}. \end{align} $$

It is well known that these formulas give realizations for sums, products and inverses in the case of finite-dimensional realizations. It is straightforward to verify, by direct computation, that these formulas still hold for operator-realizations, so we omit the proof. (The arguments are independent of dimension.) The fact that Equation (3.12) above gives an FM realization for $g^{-1}$ , for example, can be readily checked using the Woodbury identity [Reference Woodbury64].

Remark 3.18. If $(A,B,C,D)$ is a minimal FM realization so that $D \neq 0$ , then it is easy to check that $( A^{(-1)}, B^{(-1)}, C^{(-1)}, D^{(-1)} )$ is also minimal, as in [Reference Helton, Klep and Volčič30, Section 5.2]. Indeed, assuming $(A,B,C,D)$ is minimal,

$$ \begin{align*}\bigvee_{\substack{\omega \in \mathbb{F}^d_+ \\ 1 \leq j \leq d}} A^{(-1) \omega } B^{(-1)}_j \supseteq \bigvee A^\omega B_j = \mathcal{H}, \quad \mbox{and} \quad \bigvee_{\omega \in \mathbb{F}^d_+} A^{(-1) * \omega} C^{(-1)*} \supseteq \bigvee A^{*\omega} C^{*} = \mathcal{H}.\end{align*} $$

Similarly, there is an algorithm for constructing descriptor realizations of sums, products and inverses as follows. Let $(A,b,c) \sim g$ and $(A',b',c') \sim h$ be descriptor realizations on complex, separable Hilbert spaces $\mathcal {H}$ and $\mathcal {H} '$ . A descriptor realization for $g+h$ is given by $(A^+ , b^+ , c^+)$ , where

(3.13) $$ \begin{align} A_j^+ := A_j \oplus A_j ', \quad b^+ := \begin{pmatrix} b \\ b' \end{pmatrix}, \quad \mbox{and} \quad c^+ := \begin{pmatrix} c \\ c' \end{pmatrix}. \end{align} $$

A descriptor realization for $g \cdot h$ is given by $(A^\times , b^\times , c^\times )$ , where

(3.14) $$ \begin{align} A_j^\times := \begin{pmatrix} A_j & A_j c b^{\prime *} \\ 0 & A' \end{pmatrix}, \quad b^\times := \begin{pmatrix} b \\ c^{*} b \, b' \end{pmatrix}, \quad \mbox{and} \quad c^\times := \begin{pmatrix} 0 \\ c' \end{pmatrix}. \end{align} $$

And a descriptor realization for $g^{-1}$ , assuming that $g(0) = b^{*} c \neq 0$ , on the Hilbert space $\hat {\mathcal {H}} := \mathcal {H} \oplus \mathbb {C}$ , is given by $(A^{(-1)}, b^{(-1)}, c^{(-1)} )$ , where $A^{(-1)}_j \in \mathscr {B} (\hat {\mathcal {H}} )$ , $1 \leq j \leq d$ , $b^{(-1)}, c^{(-1)} \in \hat {\mathcal {H}}$ and

(3.15) $$ \begin{align} A^{(-1)}_j := \begin{pmatrix} A_j - \frac{1}{b^{*}c} A_j c b^{*} & \frac{1}{b^{*} c} A_j c \langle {0 \oplus 1} , {\cdot} \rangle_{\hat{\mathcal{H}}} \\ 0 & 0 \end{pmatrix},\, b^{(-1)} := \frac{1}{c^{*}b} \begin{pmatrix} -b \\ 1 \end{pmatrix}, \mbox{ and } c^{(-1)} := \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \end{align} $$

3.4 Operator realizations around a matrix centre

In [Reference Porat and Vinnikov46, Reference Porat and Vinnikov47], Porat and Vinnikov show that any NC rational function, , has a ‘realization around a matrix–centre’, $Y \in \mathrm {Dom} \, \mathfrak {r}$ . Namely, suppose that $Y \in \mathrm {Dom}_m \, \mathfrak {r}$ . Then there is a quadruple, $(\boldsymbol {A} , \boldsymbol {B} , \boldsymbol {C}, \boldsymbol {D})$ , where each $\boldsymbol {A}_j : \mathbb {C}^{m \times m} \rightarrow \mathbb {C}^{M \times M}$ is a linear map, each $\boldsymbol {B}_j : \mathbb {C}^{m \times m} \rightarrow \mathbb {C}^{M \times m}$ is a linear map, $\boldsymbol {C} \in \mathbb {C}^{m \times M}$ , and $\boldsymbol {D} = \mathfrak {r} (Y) \in \mathbb {C}^{m \times m}$ , so that for any $X \in \mathrm {Dom}_m \, \mathfrak {r}$ that is sufficiently close to Y,

$$ \begin{align*}\mathfrak{r} (X) = \boldsymbol{D} + \boldsymbol{C} \left( I_M - \sum_j \boldsymbol{A}_j ( X_j - Y_j ) \right)^{-1} \sum_j \boldsymbol{B}_j (X_j - Y_j ).\end{align*} $$

This can be extended to points $X \in \mathbb {C}^{(mn \times mn) \cdot d}$ , $n \in \mathbb {N}$ by viewing each $X_j$ as an $n \times n$ block matrix with blocks of size $m \times m$ ,

$$ \begin{align*}X_j = \left( X_{j}^{(k,\ell)} \right)_{ 1 \leq k , \ell \leq n} = \sum_{k,\ell =1}^n E_{k, \ell} \otimes X^{(k,\ell)}_j; \quad \quad X_j^{(k, \ell)} \in \mathbb{C}^{m \times m},\end{align*} $$

where $E_{k,\ell }$ are the standard matrix units of $\mathbb {C}^{n \times n}$ . We then define

$$ \begin{align*}\boldsymbol{A}_j (X_j ) := \left( \boldsymbol{A}_j \left( X_j^{(k, \ell )} \right) \right) \in \mathbb{C}^{n \times n} \otimes \mathbb{C}^{M \times M},\end{align*} $$

and then the action of $\boldsymbol {B}_j$ on $X_j$ is defined similarly. That is, we are identifying $\mathbb {C}^{mn \times mn}$ with $\mathbb {C}^{n\times n} \otimes \mathbb {C}^{m \times m}$ and $\boldsymbol {A}_j$ with its ampliation, $\mathrm {id}_n \otimes \boldsymbol {A}_j$ , where $\mathrm {id}_n$ is the identity map on $\mathbb {C}^{n \times n}$ . In this case,

$$ \begin{align*}\mathfrak{r} (X) = I_n \otimes \boldsymbol{D} + I_n \otimes \boldsymbol{C} \left( I_n \otimes I_M - \sum \boldsymbol{A}_j (X_j - I_n \otimes Y_j) \right)^{-1} \sum \boldsymbol{B}_j (X_j - I_n \otimes Y_j).\end{align*} $$

As for regular realizations, we will write $\boldsymbol {A} (X), \boldsymbol {B} (X)$ in place of $\sum \boldsymbol {A}_j (X_j)$ and $\sum \boldsymbol {B}_j (X_j)$ , respectively, and we define the linear pencil of $\boldsymbol {A}$ as

$$ \begin{align*}L_{\boldsymbol{A}} (X) := I_n \otimes I_M - \boldsymbol{A} (X); \quad \quad X \in \mathbb{C}^{(mn \times mn) \cdot d}.\end{align*} $$

Let $(A,B,C,D)$ , $A \in \mathscr {B} (\mathcal {H})^d$ , be a minimal FM realization of a familiar NC function, f, and suppose that $Y \in \mathscr {D} (A) \subseteq \mathrm {Dom} \, f$ , $Y \in \mathbb {C}^{(m\times m)\cdot d}$ . Then f also has an FM ‘matrix–realization’ centred at Y. To see this, consider the Taylor–Taylor series of f centred at Y.

$$ \begin{align*}f(X) = \sum_{\omega \in \mathbb{F}^d_+} \Delta^{\omega^{\mathrm{t}}}_{X-Y} f (Y).\end{align*} $$

Given any $H \in \mathbb {C}^{(m\times m)\cdot d}$ , let $\vec {H}_j := (0_m, \cdots , H_j , 0_m, \cdots , 0_m ) \in \mathbb {C}^{(m\times m)\cdot d}$ , with $H_j$ in the jth slot. For any $1 \leq j \leq d$ , $\Delta ^{(j)}_H f (Y)$ can be computed as the upper right corner of

$$ \begin{align*} f \begin{pmatrix} Y & \vec{H}_j \\ & Y \end{pmatrix} &= D I_m + I_m \otimes C \left( I_m \otimes I - \left ( \begin{smallmatrix} Y \otimes A & H_j \otimes A_j \\ & Y \otimes A \end{smallmatrix} \right) \right)^{-1} \left ( \begin{smallmatrix} Y \otimes B & H_j \otimes B_j \\ & Y \otimes B\end{smallmatrix} \right) \\ & = DI_{2m} + I_{2m} \otimes C \begin{pmatrix} L_A (Y)^{-1} & L_A (Y)^{-1} H_j \otimes A_j L_A (Y)^{-1} \\ & L_A (Y)^{-1} \end{pmatrix} \begin{pmatrix} Y \otimes B & H_j \otimes B_j \\ & Y \otimes B\end{pmatrix} \\ & = \begin{pmatrix} * & I_m \otimes C \left( L_A (Y)^{-1} H_j \otimes B_j + L_A (Y)^{-1} H_j \otimes A_j L_A (Y)^{-1} Y \otimes B \right) \\ & * \end{pmatrix}. \end{align*} $$

Hence,

(3.16) $$ \begin{align} \Delta^{(j)}_H f (Y) = I_m \otimes C \left( L_A (Y)^{-1} H_j \otimes B_j + L_A (Y)^{-1} H_j \otimes A_j L_A (Y)^{-1} Y \otimes B \right). \end{align} $$

Similarly, $\Delta ^{(kj)}_H f (Y)$ can be calculated as the upper right corner of f applied to

$$ \begin{align*}\begin{pmatrix} Y & \vec{H}_j & 0 \\ & Y & \vec{H}_k \\ & & Y \end{pmatrix},\end{align*} $$

and this yields

$$ \begin{align*} & {\scriptstyle D I_{3m} + I_{3m} \otimes C} \left( {\scriptstyle I_{3m} \otimes I -} \left(\begin{smallmatrix} Y \otimes A & H_j \otimes A_j & 0 \\ & Y \otimes A & H_k \otimes A_k \\ & & Y \otimes A \end{smallmatrix}\right)\right)^{-1} \left(\begin{smallmatrix} Y \otimes B & H_j \otimes B_j & 0 \\ & Y \otimes B & H_k \otimes B_k \\ & & Y \otimes B \end{smallmatrix}\right) \\ & = {\scriptstyle D I_{3m} + I_{3m} \otimes C } \left(\begin{smallmatrix} L_A (Y)^{-1} & L_A (Y)^{-1} H_j \otimes A_j L_A (Y)^{-1} & L_A (Y)^{-1} H_j \otimes A_j L_A (Y)^{-1} H_k \otimes A_k L_A (Y)^{-1} \\ & * & * \\ & & * \end{smallmatrix} \right) \left(\begin{smallmatrix} Y\otimes B & H_j \otimes B_j & 0 \\ & * & H_k \otimes B_k \\ & & Y \otimes B \end{smallmatrix} \right). \end{align*} $$

That is,

(3.17) $$ \begin{align} \Delta^{(kj)}_H f (Y) = I_m \otimes C \, L_A (Y)^{-1} H_j \otimes A_j \left( L_A (Y)^{-1} H_k \otimes B_k + L_A(Y)^{-1} H_k \otimes A_k L_A (Y)^{-1} Y \otimes B \right). \end{align} $$

Hence, for any fixed $G \in \mathbb {C}^{m \times m}$ and any $1\leq j \leq d$ , define

$$ \begin{align*}\boldsymbol{A}_j (G) := L_A (Y)^{-1} G \otimes A_j, \quad \quad \boldsymbol{B}_j (G) := L_A (Y)^{-1} G \otimes B_j + L_A (Y)^{-1} G \otimes A_j L_A (Y)^{-1} Y \otimes B,\end{align*} $$

$\boldsymbol {C} := I_m \otimes C$ , and $\boldsymbol {D} := f(Y)$ . Then each $\boldsymbol {A}_j : \mathbb {C}^{m \times m} \rightarrow \mathscr {B} ( \mathbb {C}^m \otimes \mathcal {H} )$ and each $\boldsymbol {B}_j : \mathbb {C}^{m \times m} \rightarrow \mathbb {C}^{m \times m} \otimes \mathcal {H}$ are linear maps, , $\boldsymbol {D} =f(Y) \in \mathbb {C}^{m \times m}$ , and f is given by the matrix-realization $(\boldsymbol {A}, \boldsymbol {B}, \boldsymbol {C}, \boldsymbol {D} )$ in the sense of the following theorem. We will write $f \sim _Y (\boldsymbol {A}, \boldsymbol {B} , \boldsymbol {C} , \boldsymbol {D} )$ to denote that $(\boldsymbol {A} , \boldsymbol {B} , \boldsymbol {C} , \boldsymbol {D} )$ is a matrix-centre realization of f about Y, and we define the invertibility domain of the matrix-centre realization $(\boldsymbol {A} , \boldsymbol {B} , \boldsymbol {C} , \boldsymbol {D} )$ , $\mathscr {D}^Y (\boldsymbol {A} )$ , $Y \in \mathbb {C}^{(m\times m)\cdot d}$ , as the set of all $X \in \mathbb {C}^{(sm \times sm ) \cdot d}$ , $s\in \mathbb {N}$ , so that $L_{\boldsymbol {A}} (X - I_s \otimes Y)$ is invertible.

Theorem 3.19. Let $f \sim (A,B,C,D)$ be a familiar NC function and suppose that $Y \in \mathscr {D}_m (A) = \mathscr {D} (A) \cap \mathbb {C}^{(m\times m)\cdot d}$ . Then f has a matrix-centre realization about Y, $f \sim _Y (\boldsymbol {A},\boldsymbol {B}, \boldsymbol {C}, \boldsymbol {D})$ , where $\boldsymbol {A}_j : \mathbb {C}^{m\times m} \rightarrow \mathscr {B} (\mathbb {C}^m \otimes \mathcal {H})$ , $\boldsymbol {B}_j : \mathbb {C}^{m\times m} \rightarrow \mathbb {C}^m \otimes \mathcal {H}$ , $1\leq j \leq d$ , , and $\boldsymbol {D} \in \mathbb {C}^{m\times m}$ are given by

(3.18) $$ \begin{align} \boldsymbol{A}_j (G) & = L_A (Y)^{-1} G \otimes A_j, \quad \quad \boldsymbol{B}_j (G) = L_A (Y)^{-1} G \otimes B_j + L_A (Y)^{-1} G \otimes A_j L_A (Y)^{-1} Y \otimes B, \nonumber \\ & \boldsymbol{C} = I_m \otimes C, \quad \quad \mbox{and} \quad \quad \boldsymbol{D} = f (Y). \end{align} $$

That is, for any $X \in \mathscr {D}^Y_{sm} (\boldsymbol {A} )$ ,

(3.19) $$ \begin{align} f(X) = I_s \otimes \boldsymbol{D} + \boldsymbol{C} \underbrace{\left( I_{sm} \otimes I_{\mathcal{H}} - \sum \boldsymbol{A}_j (X_j - I_s \otimes Y_j ) \right)^{-1}}_{= L_{\boldsymbol{A}} (X-I_s \otimes Y)^{-1}} \sum \boldsymbol{B}_j (X_j - I_s \otimes Y_j). \end{align} $$

Note that each linear map,

$$ \begin{align*}\boldsymbol{A}_j : \mathbb{C}^{sm \times sm} \rightarrow \mathbb{C}^{sm \times sm} \otimes \mathscr{B} (\mathcal{H} ) = \mathscr{B} (\mathbb{C}^{sm} \otimes \mathcal{H} ),\end{align*} $$

in the above theorem statement, is completely bounded so that $\mathscr {D}^Y (\boldsymbol {A})$ is uniformly open, joint-similarity invariant, and contains a uniformly open row-ball centred at Y of radius $r = \| \boldsymbol {A} \|_{\mathrm {col}}^{-1}$ , where $\| \boldsymbol {A} \|_{\mathrm {col}}$ is the completely bounded norm of the ‘column’ $\boldsymbol {A} : \mathbb {C}^{m \times m} \rightarrow \mathscr {B} (\mathbb {C}^m \otimes \mathcal {H} ) \otimes \mathbb {C}^d$ .

As in the theory of NC rational functions, if $f \sim _Y (\boldsymbol {A}, \boldsymbol {B}, \boldsymbol {C}, \boldsymbol {D} )$ is given by the matrix-centre realization formula (3.19), then one can compute the Taylor–Taylor series of f expanded about the point Y in terms of the matrix-centre realization [Reference Porat and Vinnikov46, Lemma 2.12].

Lemma 3.20. Let $Y \in \mathrm {Dom}_m \, f$ , where f is a familiar NC function. If $f \sim _Y (\boldsymbol {A} , \boldsymbol {B} , \boldsymbol {C} , \boldsymbol {D} )$ , then the terms of the Taylor–Taylor series of f at Y are given by the multilinear partial difference-differential maps, $\Delta ^{\omega ^{\mathrm {t}}}_{(\cdot )} f (Y)$ , where if $\emptyset \neq \omega = i_1 \cdots i_k \in \mathbb {F}^d_+$ and $H \in \mathbb {C}^{(m\times m)\cdot d}$ , then

(3.20) $$ \begin{align} \Delta^{\omega^{\mathrm{t}}}_{H} f (Y) = \boldsymbol{C} \boldsymbol{A}_{i_1} (H_{i_1}) \boldsymbol{A}_{i_2} (H_{i_2}) \cdots \boldsymbol{A}_{i_{k-1}} (H_{i_{k-1}}) \boldsymbol{B}_{i_k} (H_{i_k}). \end{align} $$

That is, if the Taylor–Taylor series of $f \sim _Y (\boldsymbol {A}, \boldsymbol {B}, \boldsymbol {C}, \boldsymbol {D})$ centred at $Y \in \mathrm {Dom}_m \, f$ converges at $X \in \mathbb {C}^{(sm \times sm) \cdot d}$ , then

$$ \begin{align*}f(X) = \sum_{\omega \in \mathbb{F}^d_+} \Delta^{\omega^{\mathrm{t}}}_{X - I_s \otimes Y} f (Y),\end{align*} $$

where the terms of this series are given by Equation (3.20). This lemma can be readily established with a proof identical to that of [Reference Porat and Vinnikov46, Lemma 2.12]. Namely, these terms of the Taylor–Taylor series of f, centred at Y, can be computed by evaluating f at points in the ‘jointly nilpotent ball centred at Y’, which consists of all $X \in \mathbb {C}^{(sm \times sm) \cdot d}$ for which $X - I_s \otimes Y$ is jointly nilpotent. We also omit the inductive proof of Theorem 3.19, which can be accomplished by showing that the Taylor–Taylor series of the familiar NC function $f \sim (A,B,C,D)$ , at the point $Y \in \mathscr {D}_m (A)$ , coincides with the Taylor–Taylor series at Y of $\widetilde {f} \sim _Y (\boldsymbol {A}, \boldsymbol {B}, \boldsymbol {C}, \boldsymbol {D})$ . (We verified this for all words of length $\leq 2$ in Equations (3.16–3.17).) Here, $\widetilde {f}$ is the NC function given by the matrix-centre realization formula of Equation (3.19) and $(\boldsymbol {A}, \boldsymbol {B}, \boldsymbol {C}, \boldsymbol {D})$ are defined by the FM realization, $(A,B,C,D)$ , of f, as in the statement of Theorem 3.19.

Also as in [Reference Porat and Vinnikov46, Theorem 2.4], if $f \sim _Y (\boldsymbol {A}, \boldsymbol {B}, \boldsymbol {C} , \boldsymbol {D})$ and $g \sim _Y (\boldsymbol {A} ', \boldsymbol {B}', \boldsymbol {C}', \boldsymbol {D}')$ , then $f+g$ , $f\cdot g$ , and, assuming $f(Y)$ is invertible, $f^{-1}$ , all have matrix-centre realizations about Y given by a natural extension of the Fornasini–Marchesini algorithm described in Subsection 3.3. Namely, a matrix-centre realization for $f+g$ is given by $(\boldsymbol {A}^+ , \boldsymbol {B}^+ , \boldsymbol {C}^+, \boldsymbol {D}^+)$ , where

(3.21) $$ \begin{align} \boldsymbol{A}_j^+ := \boldsymbol{A}_j \oplus \boldsymbol{A}^{\prime}_j, \quad \boldsymbol{B}_j^+ := \begin{pmatrix} \boldsymbol{B}_j \\ \boldsymbol{B} ^{\prime}_j \end{pmatrix}, \quad \boldsymbol{C}^+ := (\boldsymbol{C} , \boldsymbol{C} ' ), \quad \mbox{and} \quad \boldsymbol{D}^+ = \boldsymbol{D} + \boldsymbol{D}'. \end{align} $$

A matrix-centre realization for $f\cdot g$ is given by $(\boldsymbol {A}^\times , \boldsymbol {B}^\times , \boldsymbol {C}^\times , \boldsymbol {D}^\times )$ , where

(3.22) $$ \begin{align} \boldsymbol{A}^\times_j := \begin{pmatrix} \boldsymbol{A}_j & \boldsymbol{B}_j (\cdot ) \boldsymbol{C} ' \\ & \boldsymbol{A}_j ' \end{pmatrix}, \quad \boldsymbol{B}^\times = \begin{pmatrix} \boldsymbol{B}_j (\cdot ) \boldsymbol{D} ' \\ \boldsymbol{B}_j ' \end{pmatrix}, \quad \boldsymbol{C}^\times = ( \boldsymbol{C} , \boldsymbol{D} \boldsymbol{C} ' ), \quad \mbox{and} \quad \boldsymbol{D}^\times = \boldsymbol{D} \boldsymbol{D} '. \end{align} $$

Finally, a matrix-centre realization for $f^{-1}$ , assuming that $\boldsymbol {D} = f (Y)$ is invertible, is given by $(\boldsymbol {A}^{(-1)}, \boldsymbol {B}^{(-1)}, \boldsymbol {C}^{(-1)}, \boldsymbol {D}^{(-1)} )$ , where

(3.23) $$ \begin{align} \boldsymbol{A}^{(-1)}_j := \boldsymbol{A}_j - \boldsymbol{B}_j (\cdot ) \boldsymbol{D}^{-1} \boldsymbol{C}, \quad \boldsymbol{B}_j^{(-1)} := -\boldsymbol{B}_j (\cdot ) \boldsymbol{D}^{-1}, \quad \boldsymbol{C}^{(-1)} := \boldsymbol{D}^{-1} \boldsymbol{C}, \quad \mbox{and} \quad \boldsymbol{D}^{(-1)} := \boldsymbol{D}^{-1}. \end{align} $$

Verification of these formulas is a straightforward computation, and we refer the reader to [Reference Porat and Vinnikov46, Theorem 2.4] for their proofs (in the NC rational/ finite-dimensional setting).

4.1 Entire functions of a single complex variable

Let $h \in \mathscr {O} (\mathbb {C} )$ be an entire function (in one variable), with Taylor series at zero,

$$ \begin{align*}h (z) = \sum_{j=0}^\infty a_j z^j; \quad \quad a_j \in \mathbb{C}.\end{align*} $$

This Taylor series has infinite radius of convergence so that the coefficients obey:

$$ \begin{align*}\lim_{n\rightarrow \infty} \sqrt[n]{|a_n|} =0.\end{align*} $$

We can construct a realization of h as follows. Let $\mathcal {H} := \bigoplus _{n=0}^\infty \mathbb {C}^{n+1}$ . For $n=0$ , let $A_0 := 0$ , $b_0 = 1$ and $c_0 = a_0 1$ . For $n \in \mathbb {N}$ fix any complex nth root, $\sqrt [\uproot {3} n]{a_n}$ of $a_n \in \mathbb {C}$ , and define

$$ \begin{align*}A_n := \begin{pmatrix} 0 & \sqrt[\uproot{3} n]{a_n} & & & & \\ & 0 & \sqrt[\uproot{3} n]{a_n} & & & \\ & & \ddots & \ddots & & \\ & & & & & \\ & & & & & \sqrt[\uproot{3} n]{a_n} \\ & & & & & 0 \end{pmatrix} \in \mathbb{C}^{(n+1)\times (n+1)}, \quad b_n := e_1, \quad \mbox{and} \quad c_n := e_{n+1},\end{align*} $$

where $(e_{j} )_{j=1}^n$ is the standard orthonormal basis of $\mathbb {C}^n$ .

Theorem 4.3. Let $h (z) = \sum _{n=0}^\infty a_n z^n$ be an entire function. Then h has a compact and quasinilpotent realization $(A,b,c)$ , on $\mathcal {H} = \bigoplus _{n=0}^\infty \mathbb {C}^n$ , given by

$$ \begin{align*}A := A_0 \oplus \bigoplus_{n=1}^\infty \sqrt[n]{n^2} A_n, \quad b:= 1 \oplus \bigoplus_{n=1}^\infty \frac{1}{n} e_1, \quad \mbox{and} \quad c:= a_0 1 \oplus \bigoplus_{n=1}^\infty \frac{1}{n} e_{n+1}.\end{align*} $$

Proof. Note that if $E_{j,k}$ denote the standard matrix units of $\mathbb {C}^{(n+1) \times (n+1)}$ , and

$$ \begin{align*}S_n := \sum_{j=1}^{n} E_{j+1, j}, \quad \mbox{then,} \quad S_n^{*} = \begin{pmatrix} 0 & 1 & & \\ & \ddots & \ddots & \\ & & & 1 \\ & & & 0 \end{pmatrix} \in \mathbb{C}^{(n+1) \times (n+1)}.\end{align*} $$

Hence, $A_n = \sqrt [\uproot {3} n]{a_n} S_n^{*}$ . This ‘truncated backward shift’ matrix, $S_n^{*}$ , is a partial isometry with $\mathrm {Ker} \, S_n^{*} = \bigvee e_1$ and $\mathrm {Ran} \, S_n^{* \perp } = \bigvee e_{n+1}$ . Also notice that each $S_n^{*}$ is nilpotent as $S_n^{*(n+1)} = 0_n$ . It follows that $S_n S_n^{*} = P_1^\perp $ is the orthogonal projection onto $\bigvee \{ e_1 \}^\perp $ , and

$$ \begin{align*}A_n^{*} A_n = |a_n|^{\frac{2}{n}} (I_{n+1} - E_{1,1}).\end{align*} $$

Hence,

$$ \begin{align*}\| A \| = \sqrt{\| A A^{*} \| } = \sup \sqrt[n]{n^2} \sqrt[n]{|a_n|} < +\infty,\end{align*} $$

since $(a_n)$ is the sequence of Taylor coefficients of an entire function. This proves that $A \in \mathscr {B} (\mathcal {H})$ . Moreover, if $A^{(N)}$ denotes the $N+1$ st partial direct sum of A, $A^{(N)} := A_0 \oplus \bigoplus _{n=1}^N \sqrt [n]{n^2} A_n$ , then each $A^{(N)}$ is clearly finite-rank and nilpotent. Moreover,

$$ \begin{align*}\| A - A^{(N)} \| = \sup_{n>N} \sqrt[n]{n^2} \sqrt[n]{|a_n|},\end{align*} $$

which converges to $0$ as $N\rightarrow \infty $ , by Hadamard’s radius of convergence formula. This proves that A is the operator-norm limit of a sequence of finite-rank operators, so that A is compact. Setting $\hat {A}_n := \sqrt [n]{n^2} A_n$ , similar calculations show that

$$ \begin{align*}\| \hat{A}_n^k \| = (n^2 |a_n| )^{\frac{k}{n}},\end{align*} $$

which implies

$$ \begin{align*}\| A^k \| = \sup_{n> k} ( n^2 |a_n| )^{\frac{k}{n}}.\end{align*} $$

Given any $\epsilon>0$ , since $\sqrt [n]{n^2 |a_n|} \rightarrow 0$ , there is an $N_\epsilon \in \mathbb {N}$ so that $n> N_\epsilon $ implies that $\sqrt [n]{n^2 |a_n|} < \epsilon $ . Hence, for $n> k > N_\epsilon $ , $\| A^k \| < \epsilon ^{k}$ and by the Gelfand–Beurling spectral radius formula,

$$ \begin{align*}\rho (A) = \lim_{k \rightarrow \infty} \sqrt[k]{\| A^k \| } \leq \epsilon .\end{align*} $$

Since $\epsilon>0$ was arbitrary, we conclude that $\sigma (A) = \{ 0 \}$ and A is both compact and quasinilpotent.

Finally, it is easily checked that $(A,b,c)$ is a realization for h as

$$ \begin{align*}A_n^n = \begin{pmatrix} 0 & \cdots & 0 & a_n \\ \vdots & \ddots & & 0 \\ & & & \vdots \\ 0 & \cdots & & 0 \end{pmatrix}, \quad \mbox{and} \quad b_n^{*} A_n^j c_n = \delta_{j,n} a_n.\end{align*} $$

It follows that for all $n \in \mathbb {N} \cup \{ 0 \}$ , $ b^{*} A^n c = a_n, $ and $h \sim (A,b,c)$ .

Although the realization of an entire function, h, just constructed above need not be minimal, this is no matter.

Corollary 4.4. A formal power series, $h \in \mathbb {C} [ \! [ \mathfrak {z} ] \! ]$ in one variable, $\mathfrak {z}$ , is the Taylor series of an entire function in $\mathbb {C}$ if and only if it has a compact and quasinilpotent realization, that is, if and only if

$$ \begin{align*}h (\mathfrak{z}) = \sum_{n=0}^\infty \hat{h}_n \, \mathfrak{z}^n; \quad \mbox{where} \quad \hat{h}_n = b^{*} A^n c; \quad (A,b,c) \in \mathscr{B} (\mathcal{H}) \times \mathcal{H} \times \mathcal{H},\end{align*} $$

with A compact and quasinilpotent. Equivalently, $h \in \mathbb {C} [ \! [ \mathfrak {z} ] \! ]$ is the Taylor series of an entire function if and only if it has a minimal compact and quasinilpotent realization.

4.2 Realizations of uniformly entire NC functions

In order to establish this theorem, we will show that the construction of the realization in Theorem 4.3 above can be extended to several variables. It will be convenient to first prove several lemmas and preliminary results needed in this construction.

Fix any $\alpha = i_1 \cdots i_n \in \mathbb {F}^d_+$ , $1\leq i_k \leq d$ , and consider $\mathbb {C}^{n+1}$ with the standard basis $(e_j)_{j=1}^{n+1}$ . Also set $E_{j,k} := e_j e_k^{*} \in \mathbb {C}^{(n+1) \times (n+1)}$ to be the standard matrix units, and $E_j := E_{j,j}$ . We define the compressed shift matrix, $S_n := \sum _{j=1}^n E_{j+1 , j} \in \mathbb {C}^{(n+1) \times (n+1)}$ , so that $S_n^{*} = \sum _{j=1}^n E_{j, j+1}$ , and we call $S_n^{*}$ the truncated backward shift matrix of size $n+1$ . Then set

(4.1) $$ \begin{align} T (\alpha )_k := S_n^{*} \sum_{j=1}^n \delta_{i_j, k} E_{j+1} = \sum_{j=1}^n \delta_{i_j, k} E_j S_n^{*}; \quad 1\leq k \leq d. \end{align} $$

Let $T (\alpha ) := \left ( T (\alpha )_1 , \cdots , T (\alpha )_d \right ) \in \mathbb {C}^{\left ( (n+1) \times (n+1) \right ) \cdot d}$ , and also consider $C(\alpha )_k := T (\alpha )_k^{*} $ , for $1\leq k \leq d$ , $C(\alpha ) := \mathrm {row} ( T (\alpha )^{*} )$ . Observe that $T (\alpha )$ is jointly nilpotent of order n, i.e., $T (\alpha )^\beta \equiv 0_{n+1}$ for $|\beta |> |\alpha | = n$ . In particular, each $T (\alpha )_k \in \mathbb {C}^{(n+1) \times (n+1)}$ is nilpotent of order n. Similarly, $ C(\alpha ) := \left ( C(\alpha )_1 , \cdots , C (\alpha )_d \right )$ is jointly nilpotent of order n.

Lemma 4.6. Each $C_k := C (\alpha )_k $ , $1 \leq k \leq d$ is a partial isometry with range projection,

$$ \begin{align*}P_k := \sum_{\ell =1 }^n \delta_{i_\ell , k} E_{\ell +1},\end{align*} $$

and source projection,

$$ \begin{align*}Q_k := \sum_{\ell =1 }^n \delta_{i_\ell , k} E_{\ell }.\end{align*} $$

The partial isometries $C_k$ have pairwise orthogonal ranges and sources,

$$ \begin{align*}\sum_{k=1}^d P_k = I_{n+1} - E_1, \quad \quad \sum_{k=1}^d Q_k = I_{n+1} - E_{n+1},\end{align*} $$

and both $C, T:= T(\alpha )$ are jointly nilpotent row partial isometries of order n.

Proof. First,

$$ \begin{align*} C_j^{*}C_k &=\sum_{\ell =1}^n\delta_{i_\ell , j}E_{\ell}S_n^{*} S_n \sum_{p =1}^n \delta_{i_p , k} E_{p } \\ &=\sum_{\ell, p} \delta_{i_\ell, j} \delta_{i_p , k} \underbrace{ E_{\ell} (I_{n+1} - E_{n+1} ) E_p }_{= \delta_{\ell , p } E_{\ell} } \\ &=\delta_{j,k} \sum_{\ell} \delta_{i_{\ell} , j} E_{\ell } \\ &=\delta_{j,k} Q_j. \end{align*} $$

Similarly,

$$ \begin{align*} C_j C_k^{*} & = \sum_{p=1}^n \delta_{i_p , j } E_{p+1} S_n S_n^{*} \sum_{\ell = 1 }^n \delta_{i_\ell , k} E_{\ell +1 } \\ & = \sum_{p, \ell} \delta_{i_p, j} \delta_{i_\ell , k} \underbrace{E_{p+1} S_n S_n^{*} E_{\ell +1 }}_{=\delta_{p, \ell} E_{p+1}} \\ & = \delta_{k , j} \sum_{p=1}^n \delta_{i_p, j} E_{p +1} \\ & = \delta_{k , j} P_j. \end{align*} $$

It follows that

$$ \begin{align*} C C^{*} & = \sum_{k=1}^d P_k = \sum_{k=1}^d \sum_{p=1}^n \delta_{i_p, k} E_{p +1} \\ & = \sum_{p=1}^n E_{p+1} \\ & = I_{n+1} - E_1, \end{align*} $$

while a similar calculation shows that $TT^{*} = \sum _{k=1}^d Q_k = I_{n+1} - E_{n+1}$ .

Lemma 4.7. Let $C := C (\alpha )$ and $T:= T(\alpha )$ as above and set $\beta = j_1 \cdots j_m$ . Then

$$ \begin{align*}C^\beta e_k = \widetilde{e}_{k + m} = \left\{ \begin{array}{cc} e_{k+m} & \beta = i_{k+ m} \cdots i_k \\ 0 & \mbox{else} \end{array}. \right.\end{align*} $$

In particular,

$$ \begin{align*}T^{\beta *} e_1 = C^{\beta^{\mathrm{t}}} e_1 = \widetilde{e}_{1 + m} = \left\{ \begin{array}{cc} e_{1+m} & \beta = i_{1+ m} \cdots i_1 \\ 0 & \mbox{else} \end{array} \right. ,\end{align*} $$

so that this is zero unless $\alpha = \beta \gamma $ , or equivalently, $\alpha ^{\mathrm {t}} = \gamma ^{\mathrm {t}} \beta ^{\mathrm {t}}$ .

Proof.

$$ \begin{align*}C_j e_k = S_n \sum_{\ell =1}^n \delta_{i_\ell , j} \underbrace{E_\ell e_k}_{= e_k \delta_{\ell , k}} = \delta_{i_k , j} e_{k+1}.\end{align*} $$

Similarly,

$$ \begin{align*}C_{j_{m-1} } C_{j_m} e_k = \delta_{i_k , j_m } \delta_{i_{k+1} , j_{m-1} } e_{k+2},\end{align*} $$

and so on.

Similarly, one can prove the following.

Lemma 4.8. Let $T:= T (\alpha )$ as above and set $\beta = j_1 \cdots j_m$ . Then

$$ \begin{align*}T^\beta e_k = \widetilde{e}_{k -m} = \left\{ \begin{array}{cc} e_{k-m} & \beta = i_{k- m} \cdots i_{k-1} \\ 0 & \mbox{else} \end{array}. \right.\end{align*} $$

In particular,

$$ \begin{align*}T^\beta e_{n+1} = \widetilde{e}_{n+1 - m} = \left\{ \begin{array}{cc} e_{n+1 - m} & \beta = i_{n+1 - m} \cdots i_n \\ 0 & \mbox{else} \end{array} \right. ,\end{align*} $$

so that this is zero unless $\beta \leq \alpha $ .

In the above statement, we define a partial order on $\mathbb {F}^d_+$ by $\alpha \leq \omega $ if $\omega = \beta \alpha $ for some $\beta \in \mathbb {F}^d_+$ .

Lemma 4.9. Fix any $\alpha = i_1 \cdots i_n \in \mathbb {F}^d_+$ and $\beta = j_1 \cdots j_m \in \mathbb {F}^d_+$ as above. Then

$$ \begin{align*}\left( C (\alpha)^{\beta^{\mathrm{t}}} e_1 , e_{n+1} \right)_{\mathbb{C}^{n+1} } = \delta_{\alpha , \beta} = \left( e_1 , T(\alpha )^\beta e_{n+1} \right)_{\mathbb{C}^{n+1}}.\end{align*} $$

Proof.

$$ \begin{align*} \left( e_1 , T(\alpha )^\beta e_{n+1} \right)_{\mathbb{C}^{n+1}} & = \delta_{i_n ,j_m} \delta_{i_{n-1} , j_{m-1} } \cdots \delta_{i_{n+1 -m} , j_1} \left( e_1 , e_{n+1 -m} \right)_{\mathbb{C}^{n+1}} \\ & = \delta_{\alpha , \beta}.\\[-38pt] \end{align*} $$

Lemma 4.10. Given any $1 \leq m \leq n=|\alpha |$ ,

$$ \begin{align*}\sum_{|\omega | = m} T(\alpha )^\omega T(\alpha)^{\omega *} = \sum_{j=1}^{n-m+1} E_{j}.\end{align*} $$

Proof. If $m=1$ , this follows from Lemma 4.6. If $m>1$ , then again by Lemma 4.6,

$$ \begin{align*} \sum_{|\omega | = m} T(\alpha )^\omega T(\alpha)^{\omega *} & = \sum_{|\beta | = m-1} T(\alpha )^{\beta} \sum_{j=1}^d T(\alpha )_j T(\alpha )_j^{*} T(\alpha )^{\beta *} \\ & = \sum_{|\beta | = m-1} T(\alpha )^{\beta} ( I_{n+1} - e_{n+1} e_{n+1}^{*} ) T(\alpha )^{\beta *} \\ & = \sum_{|\beta | = m-1} T(\alpha )^{\beta} T(\alpha )^{\beta *} - E_{n-m+2}, \end{align*} $$

where the last equality follows from Lemma 4.8. Iterating this argument yields

$$ \begin{align*} \sum_{|\omega | = m} T(\alpha )^\omega T(\alpha)^{\omega *} & = I_{n+1} - \sum_{j =1}^m E_{n+2-j} \\ & = \sum_{j=1}^{n-m+1} E_j. \\[-47pt]\end{align*} $$

Proof of Theorem 4.5.

If $h \in \mathscr {O}^d$ is uniformly entire, then h has a Taylor–Taylor series at $0$ ,

$$ \begin{align*}h(\mathfrak{z} ) = \sum_{\omega \in \mathbb{F}^d_+} a_\omega \mathfrak{z}^\omega,\end{align*} $$

with infinite radius of convergence, $R_h$ . By Popescu’s Cauchy–Hadamard radius of convergence formula,

(4.2) $$ \begin{align} \frac{1}{R_h} =\lim_{n \rightarrow \infty} \sqrt[2n]{\sum_{|\omega| =n} |a_\omega |^2} = 0. \end{align} $$

Let $\mathcal {H} := \bigoplus _{n=0}^\infty \bigoplus _{ |\omega | = n} \mathbb {C}^{n+1}$ and set

$$ \begin{align*}A_j := 0 \oplus \bigoplus_{n=1}^\infty \bigoplus_{|\omega | = n} \underbrace{d \sqrt[n]{n^2} \sqrt[\uproot{3} n]{a_\omega} \cdot T (\omega)_j}_{=: A(\omega)_j} = 0 \oplus \bigoplus_{n=1}^\infty \bigoplus_{|\omega | = n} A(\omega)_j,\end{align*} $$

where, as before, $\sqrt [\uproot {3} n]{a_\omega }$ is any fixed nth root of $a_\omega \in \mathbb {C}$ . Then define

$$ \begin{align*}b:= 1 \oplus \bigoplus_{n=1}^\infty \bigoplus_{|\omega| =n } \frac{1}{n \sqrt{d}^n} e_1,\end{align*} $$

and

$$ \begin{align*}c:=a_\emptyset \oplus \bigoplus_{n=1}^\infty \bigoplus_{|\omega| =n } \frac{1}{n \sqrt{d}^n} e_{n+1}.\end{align*} $$

Here, $d^n$ is the number of words in $\mathbb {F}^d_+$ of length n so that $b,c \in \mathcal {H}$ .

As before, A is a compact and jointly quasinilpotent realization of h. Indeed,

$$ \begin{align*} \| A \|_{\mathrm{row}} & = \sqrt{\| A A^{*} \|} = \sup_{n \geq 0} \max_{|\omega | = n} \sqrt{\| A(\omega ) A (\omega )^{*} \|} \\ & = \sup_{n \geq 1} \max_{|\omega | = n} d \sqrt[n]{n^2|a_\omega|} = d \, \sup_{n \geq 1} \sqrt[n]{n^2} \, \max_{|\omega | =n} \sqrt[2n]{|a_\omega |^2} \\ & \leq d \, \sup_n \sqrt[n]{n^2} \, \sqrt[2n]{ \sum_{| \omega | = n } |a_\omega |^2}. \end{align*} $$

This is bounded, by the radius of convergence formula, Equation (4.2), so that $\mathrm {row} (A) \in \mathscr {B} (\mathcal {H} )^{1 \times d}$ , and hence, $A \in \mathscr {B} (\mathcal {H} )^d$ . Similarly, if $A^{(N)}$ denotes the Nth partial direct sum of A, then for each $1 \leq j \leq d$ ,

$$ \begin{align*}\| A_j - A^{(N)}_j \| \leq \| A - A^{(N)} \|_{\mathrm{row}} \leq \sup_{n> N } d \sqrt[n]{n^2} \sqrt[2n]{\sum_{|\omega | =n} |a_\omega|^2},\end{align*} $$

which converges to $0$ as $N \rightarrow \infty $ by the Popescu–Hadamard radius of convergence formula. Since each $A^{(N)}_j$ is of finite rank, it follows that $A \in \mathscr {C} (\mathcal {H} )^d$ is a $d-$ tuple of compact operators.

Moreover, and also as before, for any $m \in \mathbb {N}$ so that $n = |\omega | \geq m$ , Lemma 4.10 implies that

$$ \begin{align*} \left\| \sum_{|\alpha | =m } A^\alpha A^{\alpha *} \right\| & = \sup_{n \geq m} \max_{|\omega | = n} \left\| \sum_{j=1}^{n-m+1} E_j \right\| \, d^{2m} n^{\frac{4m}{n}} |a_\omega |^{\frac{2m}{n}}. \end{align*} $$

Hence,

$$ \begin{align*} \sqrt[2m]{ \left\| \sum_{|\alpha | = m } A^\alpha A^{\alpha^{*}} \right\|} & = d \, \sup_{n \geq m} \sqrt[n]{n^2} \, \max_{|\omega| = n} \sqrt[n]{|a_\omega|} \\ & \leq d \, \sup_{n\geq m} \sqrt[n]{n^2} \, \sqrt[2n]{\sum_{|\omega | =n} |a_\omega |^2}. \end{align*} $$

Again, this converges to $0$ as $m \rightarrow \infty $ by the Popescu–Hadamard radius of convergence formula. That is,

$$ \begin{align*}\rho (A) = \limsup_{m \rightarrow \infty} \sqrt[2m]{\left\| \sum_{|\alpha | = m } A^\alpha A^{\alpha^{*}} \right\|} = 0,\end{align*} $$

so that $A \in \mathscr {C} (\mathcal {H} )^d$ is a jointly compact and quasinilpotent tuple of operators.

If, for each $\omega \in \mathbb {F}^d_+$ , $|\omega | = n \in \mathbb {N}$ , we define $b_\omega := \frac {1}{n \sqrt {d}^n} e_1$ and $c_\omega := \frac {1}{n \sqrt {d}^n} e_{n+1}$ , it is easily checked that $(A(\omega ), b_\omega , c_\omega )$ is a minimal realization of $a_\omega \mathfrak {z}^\omega $ , and it follows that $(A,b,c)$ is a compact and quasinilpotent realization of h. Indeed, $b^{*} A^\emptyset c = b^{*} c = a_\emptyset $ , and by Lemma 4.9, for any $\alpha \neq \emptyset $ ,

$$ \begin{align*} b^{*} A^\alpha c & = \sum_{n=0}^\infty \sum_{|\omega | =n} \frac{1}{n^2 d^n} e_1^{*} A (\omega)^\alpha e_{n+1} \\ & = \sum_{n=0}^\infty \sum_{|\omega | =n} \frac{(n^2 )^{\frac{|\alpha|}{n}} d^{|\alpha|}}{n^2 d^n} \, \underbrace{e_1^{*} T (\omega)^\alpha e_{n+1}}_{\delta_{\alpha, \omega}} \, a_\omega^{\frac{|\alpha|}{n}} \\ & = a_\alpha. \end{align*} $$

As before, compressions of compact operators are compact, and the compression of a quasinilpotent tuple of operators to a jointly semi-invariant subspace is quasinilpotent, so that h has a minimal compact and quasinilpotent realization by the Kalman decomposition theorem, Theorem 3.12. The operator $d-$ tuples $A^{(N)}$ are all of finite rank, they are jointly nilpotent and the $A^{(N)}$ converge to A in the row operator norm. If $P_N$ denotes the orthogonal projection of $\mathcal {H}$ onto the finite dimensional subspace $\mathcal {H}_N$ on which $A^{(N)}$ acts, then observe that

$$ \begin{align*}P_N \mathscr{C}_{A,c} = P_N \bigvee A^\omega c = \bigvee A^{(N); \omega} c_N = \mathscr{C}_{A^{(N)}, c_N},\end{align*} $$

where $c_N = P_N c$ . Indeed, each finite-dimensional subspace, $\mathcal {H}_N$ is clearly jointly $A-$ reducing so that this formula follows readily. Similarly, it easy to check that

$$ \begin{align*}P_N \mathscr{O}_{A^{*}, b}^\perp = \mathscr{O}_{A^{(N)*}, b_N}^\perp, \quad b_N := P_N b,\end{align*} $$

and it follows that if $(\check {A}, \check {b} , \check {c})$ is the minimal realization obtained from the Kalman decomposition of $(A,b,c)$ , that compression of $(\check {A}, \check {b}, \check {c})$ to the minimal subspace of $(A^{(N)}, b_N , c_N )$ yields the minimal Kalman realization of $(A^{(N)}, b_N , c_N)$ . It follows that the minimal Kalman realizations of $(A^{(N)}, b_N , c_N)$ , which are all finite-rank and jointly nilpotent, converge to the minimal Kalman realization of $(A,b,c)$ in the sense that $A^{(N)} \rightarrow A$ in operator row-norm, $b_N \rightarrow b$ and $c_N \rightarrow c$ . In particular, we see that the minimal, jointly quasinilpotent and compact $d-$ tuple, $\check {A} \in \mathscr {B} (\mathcal {H} )^d$ , obtained from the Kalman decomposition of $(A,b,c)$ is the operator row-norm limit of the sequence of minimal, finite-rank and jointly nilpotent $d-$ tuples, $\check {A}^{(N)}$ , obtained from the Kalman decomposition of $(A^{(N)}, b_N, c_N)$ .