arXiv subjects

M. Garayev

Publications and source records attributed to M. Garayev.

6 recordsLinked to original sources

arXiv2024

Convexity of the Berezin range of finite rank operators

For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over a nonempty set $\Omega$, the Berezin range of $T$ is defined by \[ \mathrm{Ber}(T)=\left\{\langle T\hat{k}_{\lambda},\hat{k}_{\lambda}\rangle_{\mathcal{H}} : \lambda \in \Omega \right\} \] and the Berezin radius is given by \[ \mathrm{ber}(T)=\sup\left\{ |\gamma| : \gamma \in \mathrm{Ber}(T) \right\}, \] where $\hat{k}_{\lambda}$ denotes the normalized reproducing kernel at $\lambda \in \Omega$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull of its Berezin range is also discussed.

math.FA↗

arXiv2024

On the convexity of the Berezin range of composition operators and related questions

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $B(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\mathbb{C}$ and the Dirichlet space of the unit disc $\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\mathcal{H}$.

math.FA↗

arXiv2023

Composition operators, convexity of their Berezin range and related questions

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $\text{Ber}(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators $A$ on the reproducing kernel Hilbert space $\mathcal{H}(\Omega)$, namely we prove that $f(\mathrm{Ber}(\Phi(A)))=\mathrm{Ber}(\Phi(f(A)))$, where $\Phi:\mathcal{B}%\left( \mathcal{H}\left( \Omega\right) \right) \mathcal{\rightarrow}\mathcal{B}\left( \mathcal{K(}Q\mathcal{)}\right) $ is a normalized positive linear map.

math.FA↗

arXiv2019

Fractional supersymmetry algebra and lacunary Hermite polynomials

We consider a realization of fractional supersymmetric of quantum mechanics of order $r$, where the Hamiltonian and supercharges involve reflection operators. It is shown that the Hamiltonian has $r$-fold degenerate spectrum and the eigenvalues of hermitian supercharges are zeros of the associated Hermite polynomials of Askey and Wimp. Also it is shown that the associated eigenfunctions involve lacunary Hermite polynomials.

hep-th↗

arXiv2018

Some aspects of number theory related to phase operators

We first extend the multiplicativity property of arithmetic functions to the setting of operators on the Fock space. Secondly, we use phase operators to get representation of some extended arithmetic functions by operators on the Hardy space. Finally, we show that radial limits to the boundary of the unit disc in the Hardy space is useful in order to go back to the classical arithmetic functions. Our approach can be understudied as a transition from the classical number theory to quantum setting.

math.FA↗

arXiv2018

Supersymmetric Quantum mechanics on the radial lines

We investigate a type of Hermite orthogonal polynomials on $r$ lines in the plane which have a common point at the origin and endpoints at the $r$ roots of unity and we show that their related Hermite functions are eigenfunctions of a differential-difference operator. A supersymmetric harmonic oscillator on $r$ radial lines is presented and analyzed. Its eigenfunctions are given in terms of these polynomials.

math.CA↗