arXiv subjects
Publications and source records attributed to Mishko Mitkovski.
In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.
In this paper, we give a harmonic analysis proof of the Neumann boundary observability inequality for the wave equation in an arbitrary space dimension. Our proof is elementary in nature and gives a simple, explicit constant. We also extend the method to prove the observability inequality of a visco-elastic wave equation.
We derive necessary conditions for localization of continuous frames in terms of generalized Beurling densities. As an important application we provide necessary density conditions for sampling and interpolation in a very large class of reproducing kernel Hilbert spaces.
Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we will focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and P{ó}lya sequences. Second, if the kernel is non-trivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.
We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers, Calderon-Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transform.
In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < \infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < \infty$ case.
We study the general moment problem for measures on the real line, with polynomials replaced by more general spaces of entire functions. As a particular case, we describe measures that are uniquely determined by a restriction of their Fourier transform to a finite interval. We apply our results to prove an extension of a theorem by Eremenko and Novikov on the frequency of oscillations of measures with a spectral gap (high-pass signals) near infinity.
In this paper we characterize the compact operators on the Bergman space $A^p(\mathbb{D}^n)$. The main result shows that an operator on $A^p(\mathbb{D}^n)$ is compact if and only if it belongs to the Toeplitz algebra $\mathcal{T}_{p}$ and its Berezin transform vanishes on the boundary.
It was known to von Neumann in the 1950's that the integer lattice $\mathbb{Z}^2$ forms a uniqueness set for the Bargmann-Fock space. It was later demonstrated by Seip and Wallstén that a sequence of points $Γ$ that is uniformly close to the integer lattice is still a uniqueness set. We show in this paper that the uniqueness sets for the Fock space are preserved under much more general perturbations.
In this paper we characterize the compact operators on $A^p_α(\mathbb{B}_n)$ when $1 -1$. The main result shows that an operator on $A^p_α(\mathbb{B}_n)$ is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.
We give a Herglotz-type representation of an arbitrary generalized spectral measure. As an application, a new proof of the classical Naimark's dilation theorem is given. The same approach is used to describe the spectrum of all unitary rank-one perturbations of a given partial isometry.
A separated sequence $Λ$ on the real line is called a Polya sequence if any entire function of zero exponential type bounded on $Λ$ is constant. In this paper we solve the problem by Polya and Levinson that asks for a description of Polya sets. We also show that the Polya-Levinson problem is equivalent to a version of the so-called Beurling gap problem on Fourier transforms of measures. The solution is obtained via a recently developed approach based on the use of Toeplitz kernels and de Branges spaces of entire functions.