A power-bounded operator T satisfying $\sup _n n\lVert T^n-T^{n+1}\rVert <\infty $ is a Ritt operator. For such operators, we study the generalized square function $Q_{\alpha ,s,r}^Tf=(\sum _n n^{\alpha } |T^n(I-T)^rf|^s)^{1/s}$. It is known that when T is a positive contraction and a Ritt operator on $L^p$, $1<p<\infty $, then for any integer $r\ge 1$, the square function $Q_{2r-1,2,r}^Tf$ defines a bounded operator [17] on $L^p$. In this work, we extend the theory to the endpoint case $p=1$. We show that if T is a Ritt operator on $L^1$, then the generalized square function $Q_{\alpha ,s,r}^Tf $ is bounded on $L^1$ whenever $\alpha +1<sr$. In the particular setting where T is a convolution operator of the form $T_{\mu }=\sum _k \mu (k) U^kf$, with $\mu $ a probability measure on $\mathbb Z$ and U the composition operator induced by an invertible measure-preserving transformation, we provide sufficient conditions on $\mu $ under which $Q_{2r-1,2,r}^{T_{\mu }}f$ is of weak type $(1,1)$, for $r>0$. We also establish bounds for variational and oscillation norms, $\lVert n^{\beta } T^n(1-T)^r\rVert _{v(s)}$ and $\lVert n^{\beta } T^n(1-T)^r\rVert _{o(s)}$, for Ritt operators, highlighting endpoint behavior.

We study Volterra-type operators on Bergman–Morrey spaces. First, we obtain sharp boundedness criteria between scales, identifying the exact Bloch-type regularity required of the symbol and showing that the companion operator is bounded precisely for bounded symbols. Next, we develop the holomorphic optimal domain, prove that it is a Banach space with bounded point evaluations and multiplier algebra $H^\infty $, and establish strict inclusions as the parameters vary. Then, we introduce the meromorphic optimal domain, characterize the constant-symbol case, and show that it is a Banach space with point evaluations bounded off the zero set of the derivative. Finally, we analyze the symbol classes $W_g$ and $V_g^{p}$, prove the structural identity $W_g=T_g(H^\infty )+\mathbb C$, and give equivalence and comparison criteria for optimal domains across symbols.

Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha $ is equal to $\pi /\sin ((2+\alpha )\pi /p)$ when $-1<\alpha <p-2$. In this article, we provide a proof of this conjecture for $0\leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}$, and this range of $\alpha $ improves the best known result when $\alpha>\frac {1}{47}$ and $\alpha \not =1$.

We consider an infinite-dimension SIS model introduced by Delmas, Dronnier, and Zitt, with a more general incidence rate, and study its equilibria. Unsurprisingly, there exists at least one endemic equilibrium if and only if the basic reproduction number is larger than 1. When the pathogen transmission exhibits one way propagation, it is possible to observe different possible endemic equilibria. We characterize in a general setting all the equilibria, using a decomposition of the space into atoms, given by the transmission operator. We also prove that the proportion of infected individuals converges to an equilibrium, which is uniquely determined by the support of the initial condition. We extend those results to infinite-dimensional SIS models with reservoir or with immigration.

In this article, we completely describe the numerical range of Toeplitz operators on weighted Bergman spaces with harmonic symbol. We also characterize the numerical range of weighted composition operators on weighted Bergman spaces and classify some sets which are the numerical range of composition operators. We investigate the inclusion of zero in the numerical range, and compute the radius of circular and elliptical disk contained in the numerical range of weighted composition operators on weighted Bergman spaces.

In this article, the authors establish an extension-restriction theorem between homogeneous weighted Besov spaces and weighted mixed-Riesz potential spaces. This general frame covers both the classical Besov spaces and their logarithmic analogs.

Let $C_{\varphi }$ be a composition operator on the Bergman space $A^2$ of the unit disc. A well-known problem asks whether the condition $\int _D\big ({1-|z|^2\over 1-|\varphi (z)|^2}\big )^pd\lambda (z) < \infty $ is equivalent to the membership of $C_\varphi $ in the Schatten class ${\mathcal {C}}_p$, $1 < p < \infty $. This was settled in the negative for the case $2 < p < \infty $ in [3]. When $2 < p < \infty $, this condition is not sufficient for $C_\varphi \in {\mathcal {C}}_p$. In this article, we take up the case $1 < p < 2$. We show that when $1 < p < 2$, this condition is not necessary for $C_\varphi \in {\mathcal {C}}_p$.

In this paper, we study the ranges of the Schwartz space $\mathcal {S}$ and its dual $\mathcal {S}'$ (space of tempered distributions) under the Bargmann transform. The characterization of these two ranges leads to interesting reproducing kernel Hilbert spaces whose reproducing kernels can be expressed, respectively, in terms of the Touchard polynomials and the hypergeometric functions. We investigate the main properties of some associated operators and introduce two generalized Bargmann transforms in this framework. This can be considered as a continuation of an interesting research path that Neretin started earlier in his book on Gaussian integral operators.

In this paper, we will show the Carleson measure characterizations for the boundedness and compactness of the Cesàro-type operator

\begin{equation*}\mathcal{C}_{\mu}(f)(z)=\sum^{\infty}_{n=0}\left( \int_{[0,1)}t^nd\mu(t)\right) \left(\sum^{n}_{k=0}a_k \right)z^n, \quad z\in \mathbb{D},\end{equation*}

acting on a number of important analytic function spaces on $\mathbb{D}$, where µ is a positive finite Borel measure. The function spaces are some newly appeared analytic function spaces (e.g., Bergman–Morrey spaces $A^{p,\lambda}$ and Dirichlet–Morrey spaces $\mathcal{D}_p^{\lambda}$) . This work continues the lines of the previous characterizations by Blasco and Galanopoulos et al. for classical Hardy spaces and weighted Bergman spaces and so forth.

For commuting contractions $T_1,\dots,T_n$ acting on a Hilbert space $\mathscr{H}$ with $T=\prod_{i=1}^n T_i$, we find a necessary and sufficient condition such that $(T_1,\dots,T_n)$ dilates to a commuting tuple of isometries $(V_1,\dots,V_n)$ on the minimal isometric dilation space of T with $V=\prod_{i=1}^nV_i$ being the minimal isometric dilation of T. This isometric dilation provides a commutant lifting of $(T_1, \dots, T_n)$ on the minimal isometric dilation space of T. We construct both Schäffer and Sz. Nagy–Foias-type isometric dilations for $(T_1,\dots,T_n)$ on the minimal dilation spaces of T. Also, a different dilation is constructed when the product T is a $C._0$ contraction, that is, ${T^*}^n \rightarrow 0$ as $n \rightarrow \infty$. As a consequence of these dilation theorems, we obtain different functional models for $(T_1,\dots,T_n)$ in terms of multiplication operators on vectorial Hardy spaces. One notable fact about our models is that the multipliers are all analytic functions in one variable. The dilation when T is a $C._0$ contraction leads to a conditional factorization of T. Several examples have been constructed.

The article introduces and studies Hausdorff–Berezin operators on the unit ball in a complex space. These operators are a natural generalization of the Berezin transform. In addition, the class of such operators contains, for example, the invariant Green potential, and some other operators of complex analysis. Sufficient and necessary conditions for boundedness in the space of p – integrable functions with Haar measure (invariant with respect to involutive automorphisms of the unit ball) are given. We also provide results on compactness of Hausdorff–Berezin operators in Lebesgue spaces on the unit ball. Such operators have previously been introduced and studied in the context of the unit disc in the complex plane. Present work is a natural continuation of these studies.

This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity.

We introduce and study Dirichlet-type spaces $\mathcal D(\mu _1, \mu _2)$ of the unit bidisc $\mathbb D^2,$ where $\mu _1, \mu _2$ are finite positive Borel measures on the unit circle. We show that the coordinate functions $z_1$ and $z_2$ are multipliers for $\mathcal D(\mu _1, \mu _2)$ and the complex polynomials are dense in $\mathcal D(\mu _1, \mu _2).$ Further, we obtain the division property and solve Gleason’s problem for $\mathcal D(\mu _1, \mu _2)$ over a bidisc centered at the origin. In particular, we show that the commuting pair $\mathscr M_z$ of the multiplication operators $\mathscr M_{z_1}, \mathscr M_{z_2}$ on $\mathcal D(\mu _1, \mu _2)$ defines a cyclic toral $2$-isometry and $\mathscr M^*_z$ belongs to the Cowen–Douglas class $\mathbf {B}_1(\mathbb D^2_r)$ for some $r>0.$ Moreover, we formulate a notion of wandering subspace for commuting tuples and use it to obtain a bidisc analog of Richter’s representation theorem for cyclic analytic $2$-isometries. In particular, we show that a cyclic analytic toral $2$-isometric pair T with cyclic vector $f_0$ is unitarily equivalent to $\mathscr M_z$ on $\mathcal D(\mu _1, \mu _2)$ for some $\mu _1,\mu _2$ if and only if $\ker T^*,$ spanned by $f_0,$ is a wandering subspace for $T.$

Let $\sigma \in (0,\,2)$, $\chi ^{(\sigma )}(y):={\mathbf 1}_{\sigma \in (1,2)}+{\mathbf 1}_{\sigma =1} {\mathbf 1}_{y\in B(\mathbf {0},\,1)}$, where $\mathbf {0}$ denotes the origin of $\mathbb {R}^n$, and $a$ be a non-negative and bounded measurable function on $\mathbb {R}^n$. In this paper, we obtain the boundedness of the non-local elliptic operator

\[ Lu(x):=\int_{\mathbb{R}^n}\left[u(x+y)-u(x)-\chi^{(\sigma)}(y)y\cdot\nabla u(x)\right]a(y)\,\frac{{\rm d}y}{|y|^{n+\sigma}} \]

$\mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$\lambda \in (0,\,\infty )$

$Lu-\lambda u=f$

$\mathbb {R}^n$

$f\in \mathrm {BMO}(\mathbb {R}^n)\cap (\bigcup _{p\in (1,\infty )}L^p(\mathbb {R}^n))$

$H^1(\mathbb {R}^n)$

$\mathrm {BMO}(\mathbb {R}^n)$

$H^1(\mathbb {R}^n)$

$L^p(\mathbb {R}^n)$

$p\in (1,\,\infty )$

$p=1$

$p=\infty$

In this paper, by the introduction of several parameters, we construct a new kernel function which is defined in the whole plane and includes some classical kernel functions. Estimating the weight functions with the techniques of real analysis, we establish a new Hilbert-type inequality in the whole plane, and the constant factor of the newly obtained inequality is proved to be the best possible. Additionally, by means of the partial fraction expansion of the tangent function, some special and interesting inequalities are presented at the end of the paper.

In this note, we mainly study operator-theoretic properties on the Besov space $B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.

In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$, as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$. The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

For a nondecreasing function $K: [0, \infty)\rightarrow [0, \infty)$ and $0<s<\infty $, we introduce a Morrey type space of functions analytic in the unit disk $\mathbb {D}$, denoted by $\mathcal {D}^s_K$. Some characterizations of $\mathcal {D}^s_K$ are obtained in terms of K-Carleson measures. A relationship between two spaces $\mathcal {D}^{s_1}_K$ and $\mathcal {D}^{s_2}_K$ is given by fractional order derivatives. As an extension of some known results, for a positive Borel measure $\mu $ on $\mathbb {D}$, we find sufficient or necessary condition for the embedding map $I: \mathcal {D}^{s}_{K}\mapsto \mathcal {T}^s_{K}(\mu)$ to be bounded.

Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by

\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]