### A Berezin-type map and a class of weighted composition operators

In this paper we consider the map L defined on the Bergman space [...] of the right half plane [...] .

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In this paper we consider the map L defined on the Bergman space [...] of the right half plane [...] .

This article provided some sufficient or necessary conditions for a class of integral operators to be bounded on mixed norm spaces in the unit ball.

Let μ be a finite positive Borel measure on [0,1). Let ${\mathscr{H}}_{\mu}={\left({\mu}_{n,k}\right)}_{n,k\ge 0}$ be the Hankel matrix with entries ${\mu}_{n,k}={\int}_{[0,1)}{t}^{n+k}d\mu \left(t\right)$. The matrix ${}_{\mu}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula ${\mathscr{H}}_{\mu}\left(f\right)\left(z\right)={\sum}_{n=0}^{\infty}i\left({\sum}_{k=0}^{\infty}{\mu}_{n,k}{a}_{k}\right)z\u207f$, z ∈ , where $f\left(z\right)={\sum}_{n=0}^{\infty}a\u2099z\u207f$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that ${\mathscr{H}}_{\mu}\left(f\right)\left(z\right)={\int}_{[0,1)}f\left(t\right)/(1-tz)d\mu \left(t\right)$ for all f in the Hardy space H¹, and among them we describe those for which ${\mathscr{H}}_{\mu}$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A².

We present a change of variable method and use it to prove the equivalence to bundle shifts for certain analytic Toeplitz operators on the Banach spaces ${H}^{p}\left(G\right)(1\le p<\infty )$. In Section 2 we see this approach applied in the analysis of essential spectra. Some partial results were obtained in [9] in the Hilbert space case.

We define and analyze Toeplitz operators whose symbols are the elements of the complex quantum plane, a non-commutative, infinite dimensional algebra. In particular, the symbols do not come from an algebra of functions. The process of forming operators from non-commuting symbols can be considered as a second quantization. To do this we construct a reproducing kernel associated with the quantum plane. We also discuss the commutation relations of creation and annihilation operators which are defined...

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1-\Phi \left(z\right)$ inside the unit disk, where $\Phi $ is the generating function of $X$, under some mild conditions

This paper is devoted to Banach algebras generated by Toeplitz operators with strongly oscillating symbols, that is, with symbols of the form b[eia(x)] where b belongs to some algebra of functions on the unit circle and a is a fixed orientation-preserving homeomorphism of the real line onto itself. We prove the existence of certain interesting homomorphisms and establish conditions for the normal solvability, Fredholmness, and invertibility of operators in these algebras.