For fixed $0<r<1$, let $A_r=\{z \in \mathbb {C} : r<|z|<1\}$ be the annulus with boundary $\partial \overline {A}_r=\mathbb {T} \cup r\mathbb {T}$, where $\mathbb T$ is the unit circle in the complex plane $\mathbb C$. An operator having $\overline {A}_r$ as a spectral set is called an $A_r$-contraction. Also, a normal operator with its spectrum lying in the boundary $\partial \overline {A}_r$ is called an $A_r$-unitary. The $C_{1,r}$ class was introduced by Bello and Yakubovich in the following way:

$$\begin{align*}C_{1, r}=\{T: T \ \text{is invertible and} \ \|T\|, \|rT^{-1}\| \leq 1\}. \end{align*}$$

McCullough and Pascoe defined the quantum annulus $\mathbb Q \mathbb A_r$ by

$$\begin{align*}\mathbb Q\mathbb A_r = \{T \,:\, T \text{ is invertible and } \, \|rT\|, \|rT^{-1}\| \leq 1 \}. \end{align*}$$

If $\mathcal A_r$ denotes the set of all $A_r$-contractions, then $\mathcal A_r \subsetneq C_{1,r} \subsetneq \mathbb Q \mathbb A_r$. We first find a model for an operator in $C_{1,r}$ and also characterize the operators in $C_{1,r}$ in several different ways. We prove that the classes $C_{1,r}$ and $\mathbb Q\mathbb A_r$ are equivalent. Then, via this equivalence, we obtain analogous model and characterizations for an operator in $\mathbb Q \mathbb A_r$.

In Dong et al. (2022, Journal of Operator Theory 88, 365–406), the authors addressed the question of whether surjective maps preserving the norm of a symmetric Kubo-Ando mean can be extended to Jordan $\ast $-isomorphisms. The question was affirmatively answered for surjective maps between the positive definite cones of unital $C^{*}$-algebras for certain specific classes of symmetric Kubo-Ando means. Here, we give a comprehensive answer to this question for surjective maps between the positive definite cones of $AW^{*}$-algebras preserving the norm of any symmetric Kubo-Ando mean.

This article explores the notions of $\mathcal {F}$-transitivity and topological $\mathcal {F}$-recurrence for backward shift operators on weighted $\ell ^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal {F}$ represents a Furstenberg family of subsets of $\mathbb {N}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

McCullough and Trent generalize Beurling–Lax–Halmos invariant subspace theorem for the shift on Hardy space of the unit disk to the multi-shift on Drury–Arveson space of the unit ball by representing an invariant subspace of the multi-shift as the range of a multiplication operator that is a partial isometry. By using their method, we obtain similar representations for a class of invariant subspaces of the multi-shifts on Hardy and Bergman spaces of the unit ball or polydisk. Our results are surprisingly general and include several important classes of invariant subspaces on the unit ball or polydisk.

This article describes Hilbert spaces contractively contained in certain reproducing kernel Hilbert spaces of analytic functions on the open unit disc which are nearly invariant under division by an inner function. We extend Hitt’s theorem on nearly invariant subspaces of the backward shift operator on $H^2(\mathbb {D})$ as well as its many generalizations to the setting of de Branges spaces.

We establish a Central Limit Theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb {N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.

In the present paper, we characterize the Fredholmness of Toeplitz pairs on Hardy space over the bidisk with the bounded holomorphic symbols, and hence, we obtain the index formula for such Toeplitz pairs. The key to obtain the Fredholmness of such Toeplitz pairs is the $L^p$ solution of Corona Problem over $\mathbb {D}^2$.

Viruses present an amazing genetic variability. An ensemble of infecting viruses, also called a viral quasispecies, is a cloud of mutants centered around a specific genotype. The simplest model of evolution, whose equilibrium state is described by the quasispecies equation, is the Moran–Kingman model. For the sharp-peak landscape, we perform several exact computations and derive several exact formulas. We also obtain an exact formula for the quasispecies distribution, involving a series and the mean fitness. A very simple formula for the mean Hamming distance is derived, which is exact and does not require a specific asymptotic expansion (such as sending the length of the macromolecules to $\infty$ or the mutation probability to 0). With the help of these formulas, we present an original proof for the well-known phenomenon of the error threshold. We recover the limiting quasispecies distribution in the long-chain regime. We try also to extend these formulas to a general fitness landscape. We obtain an equation involving the covariance of the fitness and the Hamming class number in the quasispecies distribution. Going beyond the sharp-peak landscape, we consider fitness landscapes having finitely many peaks and a plateau-type landscape. Finally, within this framework, we prove rigorously the possible occurrence of the survival of the flattest, a phenomenon which was previously discovered by Wilke et al. (Nature 412, 2001) and which has been investigated in several works (see e.g. Codoñer et al. (PLOS Pathogens 2, 2006), Franklin et al. (Artificial Life 25, 2019), Sardanyés et al. (J. Theoret. Biol. 250, 2008), and Tejero et al. (BMC Evolutionary Biol. 11, 2011)).

We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szegö positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions), but for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria, and Sarkar (Isometric dilations and von Neumann inequality for a class of tuples in the polydisc. Trans. Amer. Math. Soc. 372 (2019), 1429–1450). Our dilation is explicitly constructed, and we present some applications.

The goal of this paper is to show that the theory of curvature invariant, as introduced by Arveson, admits a natural extension to the framework of ${\mathcal U}$-twisted polyballs $B^{\mathcal U}({\mathcal H})$ which consist of k-tuples $(A_1,\ldots, A_k)$ of row contractions $A_i=(A_{i,1},\ldots, A_{i,n_i})$ satisfying certain ${\mathcal U}$-commutation relations with respect to a set ${\mathcal U}$ of unitary commuting operators on a Hilbert space ${\mathcal H}$. Throughout this paper, we will be concerned with the curvature of the elements $A\in B^{\mathcal U}({\mathcal H})$ with positive trace class defect operator $\Delta_A(I)$. We prove the existence of the curvature invariant and present some of its basic properties. A distinguished role as a universal model among the pure elements in ${\mathcal U}$-twisted polyballs is played by the standard $I\otimes{\mathcal U}$-twisted multi-shift S acting on $\ell^2({\mathbb F}_{n_1}^+\times\cdots\times {\mathbb F}_{n_k}^+)\otimes {\mathcal H}$. The curvature invariant $\mathrm{curv} (A)$ can be any non-negative real number and measures the amount by which A deviates from the universal model S. Special attention is given to the $I\otimes {\mathcal U}$-twisted multi-shift S and the invariant subspaces (co-invariant) under S and $I\otimes {\mathcal U}$, due to the fact that any pure element $A\in B^{\mathcal U}({\mathcal H})$ with $\Delta_A(I)\geq 0$ is the compression of S to such a co-invariant subspace.

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.

We study the quasi-ergodicity of compact strong Feller semigroups $U_t$, $t> 0$, on $L^2(M,\mu )$; we assume that M is a locally compact Polish space equipped with a locally finite Borel measue $\mu $. The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $\mu $ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,\mu )$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of t) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty $; the propagation rate is determined by the decay of . We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schrödinger operators with confining potentials.

Given a probability space $(X,\mu )$, a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means $N^{-1}\sum _{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$. We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.

In this work, we study early warning signs for stochastic partial differential equations (SPDEs), where the linearisation around a steady state is characterised by continuous spectrum. The studied warning sign takes the form of qualitative changes in the variance as a deterministic bifurcation threshold is approached via parameter variation. Specifically, we focus on the scaling law of the variance near the transition. Since we are dealing here, in contrast to previous studies, with the case of continuous spectrum and quantitative scaling laws, it is natural to start with linearisations of the drift operator that are multiplication operators defined by analytic functions. For a one-dimensional spatial domain, we obtain precise rates of divergence. In the case of the two- and three-dimensional domains, an upper bound to the rate of the early warning sign is proven. These results are cross-validated by numerical simulations. Our theory can be generically useful for several applications, where stochastic and spatial aspects are important in combination with continuous spectrum bifurcations.

The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$’ or linear constraints on Borel measures. Such problems can be viewed as specific instances of the generalized moment problem (GMP), whose list of important applications in various domains of science and engineering is almost endless. We describe this methodology in optimization and also in two other applications for illustration. Finally we also introduce the Christoffel function and reveal its links with the Moment-SOS hierarchy and positive polynomials.

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

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 $T.$

We consider the range inclusion and the diagonalization in the Jordan algebra $\mathcal {S}_C$ of C-symmetric operators, that are, bounded linear operators T satisfying $CTC =T^{*}$, where C is a conjugation on a separable complex Hilbert space $\mathcal H$. For $T\in \mathcal {S}_C$, we aim to describe the set $C_{\mathcal {R}(T)}$ of those operators $A\in \mathcal {S}_C$ satisfying the range inclusion $\mathcal {R}(A)\subset \mathcal {R}(T)$. It is proved that (i) $C_{\mathcal {R}(T)}=T\mathcal {S}_C T$ if and only if $\mathcal {R}(T)$ is closed, (ii) $\overline {C_{\mathcal {R}(T)}}=\overline {T\mathcal {S}_C T}$, and (iii) $C_{\overline {\mathcal {R}(T)}}$ is the closure of $C_{\mathcal {R}(T)}$ in the strong operator topology. Also, we extend the classical Weyl–von Neumann Theorem to $\mathcal {S}_C$, showing that every self-adjoint operator in $\mathcal {S}_C$ is the sum of a diagonal operator in $\mathcal {S}_C$ and a compact operator with arbitrarily small Schatten p-norm for $p\in (1,\infty )$.

Consider the multiplication operator MB in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of MB in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.