Mubariz T. Garayev

Hardy–Hilbert type inequality in reproducing kernel Hilbert space: its applications and related results

ABSTRACT A reproducing kernel Hilbert space is a Hilbert space of complex-valued functions on a (non-empty) set Ω, which has the property that point evaluation is continuous on for all . Then the Riesz representation theorem guarantees that for every there is a unique element such that for all . The function is called the reproducing kernel of and the function is the normalized reproducing kernel in . The Berezin symbol of an operator A on a reproducing kernel Hilbert space is defined by The Berezin number of an operator A on is defined by The so-called Crawford number is defined by We also introduce the number defined by It is clear that By using the Hardy–Hilbert type inequality in reproducing kernel Hilbert space, we prove Berezin number inequalities for the convex functions in Reproducing Kernel Hilbert Spaces. We also prove some new inequalities between these numerical characteristics. Some other related results are also obtained.

Berezin symbols and Borel summability

Abstract We prove in terms of so-called Berezin symbols some theorems for Borel summability method for sequences and series of complex numbers. Namely, we characterize the Borel convergent sequences and series; prove regularity of Borel summability method, and prove a new Tauberian type theorem for Borel summability.

Two Applications of the Nordgren-Rosenthal Theorem

In this article, a concept of (Ber)-convergence for the multiple sequences (a k 1,…, k N ) of complex numbers is introduced and a Tauberian theorem for such a summability method is proved. We also solve an open question for a bounded single sequence (a n ) n≥0 posed by Zorboska in [10] in connection with the compactness problem for so-called radial operators on the classical Bergman space over the unit disc 𝔻 = {z ∈ ℂ: |z| <1}. Our proofs essentially use the Berezin symbols technique of operator theory in the reproducing kernel Hilbert spaces. Namely, we apply the Nordgren-Rosenthal theorem regarding compact operators on a reproducing kernels Hilbert space.

The Berezin Number, Norm of a Hankel Operator and Related Topics

We give, in terms of the Berezin number, necessary and sufficient conditions providing unitarity of an invertible operator. Also we obtain in terms of the Berezin number a new inequality for the norm of the Hankel operator \(H_{\varphi}\) which is better than the classical inequality \(\|H_{\varphi}\|\leq\|\varphi\|_{\infty}\) The Berezin number is also used to generalize the Douglas lemma on zero Toeplitz products.

Basisity problem and weighted shift operators

We investigate a basisity problem in the space ℒAp(D) and in its invariant subspaces. Namely, let W denote a unilateral weighted shift operator acting in the space ℒAp(D), 1≤ p∞, by Wzn=λnzn+1,n≥0, with respect to the standard basis {zn}n≥0. Applying the so-called “discrete Duhamel product” technique, it is proven that for any integer k≥1 the sequence {(wi+nk)−1(W|Ei)knf}n≥0 is a basic sequence in Ei:=span {zi+n:n≥0} equivalent to the basis {zi+n}n≥0 if and only if fˆ(i)≠0. We also investigate a Banach algebra structure for the subspaces Ei,i≥0.