arXiv subjects

Georgios Tsikalas

Publications and source records attributed to Georgios Tsikalas.

11 recordsLinked to original sources

arXiv2026

Beurling Criteria for Reproducing Kernels

A classical theorem due to Beurling-Lax-Halmos characterizes the invariant subspaces of the unilateral shift as ranges of isometric multiplication operators acting on the Hardy space. The class of Beurling-Lax-Halmos (BLH) pairs of reproducing kernels is introduced, consisting of those pairs that admit an analogue of this theorem. The BLH class is, under varying hypotheses, characterized in several equivalent ways: dilation-theoretically, via a complete Leech interpolation property, and through a sums-of-squares-inspired Agler-style decomposition. These characterizations unify and extend a variety of related results in the literature. Examples are given to illustrate the theory, the hypotheses and compare and contrast with the recent developments in the study of complete Pick pairs. In particular, while it is anticipated, perhaps under some mild assumptions, that the class of complete Pick pairs of kernels is contained in the class of BLH pairs of kernels, it is shown that the reverse inclusion fails in a strong sense.

math.FA↗

arXiv2026

When do Kernels Admit Characteristic Functions?

A general framework for deriving characteristic functions for reproducing kernels that do not necessarily possess the complete Pick property was recently established by Bhattacharyya and Jindal. We show that, in this setting, the existence of a characteristic function is equivalent to a Beurling-type invariant subspace condition. Combined with recent results characterizing kernels satisfying this condition, our theorem implies that the existence of a characteristic function is equivalent to a concrete Agler-type decomposition of the underlying kernels.

math.FA↗

arXiv2026

Spectral Estimates over Multiply Connected Domains

We consider the quantum analogue of a disk with $n$ pairwise disjoint circular holes. We establish a uniform upper bound for the associated spectral constant that is independent of the geometry of the holes, together with a sharper asymptotic bound when the holes are well-separated. The latter generalizes recent estimates for the quantum annulus due to Crouzeix and Pascoe.

math.FA↗

arXiv2026

Operators with small Kreiss constants

We investigate matrices satisfying the Kreiss condition $$\|(zI-T)^{-1}\|\le\cfrac{K}{|z|-1}, \hspace{0.7 cm} |z|>1, $$ with $K$ lying arbitrarily close to $1.$ We provide lower bounds for the power growth of such matrices, which complement and refine related estimates due to Nikolski and Spijker-Tracogna-Welfert. We also study operators that satisfy a variant of the above Kreiss condition where $K$ is replaced by $1+ε(|z|)$, where the positive continuous function $ε(|z|)$ tends to $0$ as $|z|\to 1^+.$ We show that, if the spectrum of $T$ touches the unit circle only at a single point and the resolvent of $T$ satisfies a growth restriction along the unit circle, it is possible to choose $ε$ so that this Kreiss-type condition guarantees similarity to a contraction. At the core of our proof lies a positivity argument involving the double-layer potential operator. Counterexamples related to less restrictive choices of $ε$ are also provided.

math.FA↗

arXiv2025

The complete Pick property for pairs of kernels and Shimorin's factorization

Let $(\mathcal{H}_k, \mathcal{H}_{\ell})$ be a pair of Hilbert function spaces with kernels $k, \ell$. In a 2005 paper, Shimorin showed that a certain factorization condition on $(k, \ell)$ yields a commutant lifting theorem for multipliers $\mathcal{H}_k\to\mathcal{H}_{\ell}$, thus unifying and extending previous results due to Ball-Trent-Vinnikov and Volberg-Treil. Our main result is a strong converse to Shimorin's theorem for a large class of holomorphic pairs $(k, \ell),$ which leads to a full characterization of the complete Pick property for such pairs. We also present a short alternative proof of sufficiency for Shimorin's condition. Finally, we establish necessary conditions for abstract pairs $(k, \ell)$ to satisfy the complete Pick property, further generalizing Shimorin's work with proofs that are new even in the single-kernel case $k=\ell.$ Our approach differs from Shimorin's in that we do not work with the Nevanlinna-Pick problem directly; instead, we are able to extract vital information for $(k, \ell)$ through Carathéodory-Fejér interpolation.

math.FA↗

arXiv2023

Positivity conditions on the annulus via the double-layer potential kernel

We introduce and study a scale of operator classes on the annulus that is motivated by the $\mathcal{C}_ρ$ classes of $ρ$-contractions of Nagy and Foiaş. In particular, our classes are defined in terms of the contractivity of the double-layer potential integral operator over the annulus. We prove that if, in addition, complete contractivity is assumed, then one obtains a complete characterization involving certain variants of the $\mathcal{C}_ρ$ classes. Recent work of Crouzeix-Greenbaum and Schwenninger-de Vries allows us to also obtain relevant K-spectral estimates, generalizing existing results from the literature on the annulus. Finally, we exhibit a special case where these estimates can be significantly strengthened.

math.FA↗

arXiv2023

Denjoy-Wolff points on the bidisk via models

Let $F=(ϕ, ψ):\mathbb{D}^2\to\mathbb{D}^2$ denote a holomorphic self-map of the bidisk without interior fixed points. It is well-known that, unlike the case with self-maps of the disk, the sequence of iterates $$\{F^n:=F\circ F\circ \cdots \circ F\}$$ needn't converge. The cluster set of $\{F^n\}$ was described in a classical 1954 paper of Hervé. Motivated by Hervé's work and the Hilbert space perspective of Agler, McCarthy and Young on boundary regularity, we propose a new approach to boundary points of Denjoy-Wolff type for the coordinate maps $ϕ, ψ.$ We establish several equivalent descriptions of our Denjoy-Wolff points, some of which only involve checking specific directional derivatives and are particularly convenient for applications. Using these tools, we are able to refine Hervé's theorem and show that, under the extra assumption of $ϕ$ and $ψ$ possessing Denjoy-Wolff points with certain regularity properties, one can draw much stronger conclusions regarding the behavior of $\{F^n\}.$

math.FA↗

arXiv2022

Interpolating sequences for pairs of spaces

We characterize interpolating sequences for pairs of reproducing kernels $(s, \ell)$, where $s$ is a complete Pick factor of $\ell.$ This answers a question of Aleman, Hartz, McCarthy and Richter.

math.FA↗

arXiv2022

Subinner-free outer factorizations on an annulus

Recent work of Aleman, Hartz, McCarthy and Richter generalizes the classical inner-outer factorization of Hardy space functions to the complete Pick space setting, establishing an essentially unique "subinner-free outer" factorization. In this note, we investigate certain special examples of such factorizations in the setting of the function space induced on the annulus $A_r=\{r<|z|<1\}$ by the complete Pick kernel $$k_{r}(λ,μ):=\frac{1-r^2}{(1-λ\barμ)(1-r^2/λ\barμ)}.$$

math.FA↗

arXiv2021

A von Neumann type inequality for an annulus

Let $A_r=\{r<|z|<1\}$ be an annulus. We consider the class of operators $\mathcal{F}_r:=\{T\in\mathcal{B}(H): r^2T^{-1}(T^{-1})^*+TT^*\le r^2+1,\hspace{0.08 cm}σ(T)\subset A_r\}$ and show that for every bounded holomorphic function $ϕ$ on $A_r:$ $$\sup_{T\in\mathcal{F}_r}||ϕ(T)||\le\sqrt{2}||ϕ||_{\infty},$$ where the constant $\sqrt{2}$ is the best possible. We do this by characterizing the calcular norm induced on $H^{\infty}(A_r)$ by $\mathcal{F}_r$ as the multiplier norm of a suitable holomorphic function space on $A_r$.

math.FA↗

arXiv2021

A note on a spectral constant associated with an annulus

Fix $R>1$ and let $A_R=\{1/R\le |z|\le R \}$ be an annulus. Also, let $K(R)$ denote the smallest constant such that $A_R$ is a $K(R)$-spectral set for the bounded linear operator $T\in \mathcal{B}(H)$ whenever $||T||\le R$ and $||T^{-1}||\le R.$ We show that $K(R)\ge 2, \text{ for all } R>1. $ This improves on previous results by Badea, Beckermann and Crouzeix.

math.FA↗