Academia.edu no longer supports Internet Explorer.
To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to upgrade your browser.
1994
Let ${\cal H}$ be a functional Hilbert space of analytic functions on a complex domain $\Omega,$ with the normalized reproducing kernel function $k\sb{z},\ z\in\Omega.$ If A is a linear map of ${\cal H}$ into itself, the Berezin symbol, A, of A is defined on $\Omega$ by $\tilde{A}(z) = \langle Ak\sb{z},\ k\ sb{z}\rangle.$ The purpose of this research is to study how the properties of an operator are reflected in the properties of its Berezin symbol. In summary, I have (1) studied the properties of the Berezin symbol as a complex-valued function; (2) characterized multiplication operators, induced by a multiplier of ${\cal H},$ using the multiplicative property of the map $A\mapsto\tilde{A};$ (3) expressed the boundedness and compactness properties of certain operators in terms of their Berezin symbols; and (4) characterized the set of multipliers as the set of analytic Berezin symbols of bounded operators.
Proceedings of the American Mathematical Society
The Berezin symbol and multipliers of functional Hilbert spaces1995 •
Indiana University Mathematics Journal
Compact Operators via the Berezin Transform1998 •
In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on the boundary of the disk. This result is new even when S equals a single Toeplitz operator. Our
We answer a question raised by Nordgren and Rosen-thal about the Schatten-von Neumann class membership of oper-ators in standard reproducing kernel Hilbert H spaces in terms of their Berezin symbols.
Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application
Berezin Transform of Invertible Positive OperatorsIn this paper we introduce a class A ⊂ L∞(D) such that if φ ∈ Aand satisfies certain positive-definite condition, then there exists aψ ∈ A such that φ(z) ≤ αeψ(z), for some constant α > 0. Further,if φ(z) = hAkz, kzi, for some bounded positive, invertible operator Afrom the Bergman space L2a(D) into itself then ψ(z) = h(log A)kz, kzi.Here kz, z ∈ D are the normalized reproducing kernel of L2a(D). Ap-plications of these results are also discusseonly a non-standard growth condition. We show that our problem admits at least one weak solution. In order to do this, the main tool is the Berkovits degree theory for abstract Hammerstein type mappings.
2011 •
Let D = {z ∈ C : |z| < 1} and L 2 a (D) be the Bergman space of the disk. In this paper we characterize the class A ⊂ L ∞ (D) such that if φ, ψ ∈ A, α ≥ 0 and 0 ≤ φ ≤ αψ then there exist positive operators S, T ∈ L(L 2 a (D)) such that φ(z) = S(z) ≤ α T (z) = αψ(z) for all z ∈ D. Further, we have shown that if S and T are two positive operators in L(L 2 a (D)) and T is invertible then there exists a constant α ≥ 0 such that S(z) ≤ α T (z) for all z ∈ D and S, T ∈ A. Here L(L 2 a (D)) is the space of all bounded linear operators from L 2 a (D) into L 2 a (D) and A(z) = Akz, kz is the Berezin transform of A ∈ L(L 2 a (D)) and kz is the normalized reproducing kernel of L 2 a (D). Applications of these results are also obtained.
2016 •
Using techniques from the theory of reproducing kernels and Berezin symbols, we investigate some problems related to classes of linear operators acting on reproducing kernel Hilbert spaces (RKHS’s). In particular, we establish new estimates related to the numerical radii and Berezin numbers of some operators on RKHS’s. Further, in terms of the distance function, we describe invariant subspaces of isometric composition operators on a RKHS H(Ω) of complex-valued, but not necessarily analytic, functions on a set Ω. Moreover, we consider a modification of Sarason’s question about truncated Toeplitz operators. We also discuss related problems.
Linear and Multilinear Algebra
On some problems for operators on the reproducing kernel Hilbert space2019 •
Biology of Blood and Marrow Transplantation
In-Vivo Model of Human Allogeneic Stem Cell Rejection in NSG Mice2012 •
Journal of Lipid Research
Sib-pair linkage analysis of longitudinal changes in lipoprotein risk factors and lipase genes in women twins2000 •
Memórias do Instituto Oswaldo Cruz
Schistosoma mansoni antigens alter the cytokine response in vitro during cutaneous leishmaniasis2011 •
Godišen zbornik - Filozofski fakultet na Univerzitetot "Sv. Kiril i Metodij", Skopje/Godišen zbornik - Filozofski fakultet na Univerzitetot, Skopje
The New World Order from Chinese Perspective2023 •
Gondwana Research
The relations between Gondwana and the adjacent peripheral Cadomian domain—constrains on the origin, history, and paleogeography of the peripheral domain2015 •
2018 •
Center for Media, Data & Society (CMDS)
Mapping Digital Media: Israel. Government, Politics and Regulation2020 •