• Korean Journal of Mathematics
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• v.31 no.1
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• pp.109-120
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• 2023

The Berezin symbol à of an operator A on the reproducing kernel Hilbert space 𝓗 (Ω) over some set Ω with the reproducing kernel k_λ is defined by $${\tilde{A}}(\lambda)=\,\;{\lambda}{\in}{\Omega}$$. The Berezin number of an operator A is defined by $$ber(A):=\sup_{{\lambda}{\in}{\Omega}}{\mid}{\tilde{A}}({\lambda}){\mid}$$. We study some problems of operator theory by using this bounded function Ã, including estimates for Berezin numbers of some operators, including truncated Toeplitz operators. We also prove an operator analog of some Young inequality and use it in proving of some inequalities for Berezin number of operators including the inequality ber (AB) ≤ ber (A) ber (B), for some operators A and B on 𝓗 (Ω). Moreover, we give in terms of the Berezin number a necessary condition for hyponormality of some operators.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1421-1441
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• 2017

We compute explicitly the matrix represented by the Toeplitz operator on the Hardy space over a smoothly finitely connected bounded domain in the plane with respect to special orthonormal bases consisting of the classical kernel functions for the space of square integrable functions and for the Hardy space. The Fourier coefficients of the symbol of the Toeplitz operator are obtained from zeroth row vectors and zeroth column vectors of the matrix. And we also find some condition for the product of two Toeplitz operators to be a Toeplitz operator in terms of matrices.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.149-163
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• 2018

In recent years, the study of slice Dirac operators has attracted more and more attention in the literature. In this paper, Almansitype decompositions for null solutions to the iterated slice Dirac operator and the generalized slice Dirac operator are obtained without a star-like domain centered at the origin. As applications, we investigate Riquier type problems and Dirichlet type problems in the theory of slice monogenic functions.

• East Asian mathematical journal
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• v.16 no.2
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• pp.169-174
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• 2000

We give an easy proof that ${\phi}(z)=\frac{z^2_1}{1-z^2_2}$ induces the bounded composition operator $C_{\phi}:\mathcal{B}{\rightarrow}{\bigcap}H^p(B_2)$ defined by $C_{\phi}f=f\;o\;{\phi}$.

• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.105-115
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• 2021

Generalized pseudo-differential operators (PDO) involving fractional Fourier transform associate with the symbol a(x, y) whose derivatives satisfy certain growth condition is defined. The product of two generalized pseudo-differential operators is shown to be a generalized pseudo-differential operator.

• Journal of the Korean Mathematical Society
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• v.33 no.2
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• pp.309-318
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• 1996

The Gross Laplacian $\Delta_G$ was introduced by Groww for a function defined on an abstract Wiener space (H,B) [1,7]. Suppose $\varphi$ is a twice H-differentiable function defined on B such that $\varphi"(x)$ is a trace class operator of H for every x \in B.in B.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.853-869
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• 2018

We give a new characterization of Browder's theorem through equality between the pseudo B-Weyl spectrum and the generalized Drazin spectrum. Also, we will give conditions under which pseudo B-Fredholm and pseudo B-Weyl spectrum introduced in [9] and [25] become stable under commuting Riesz perturbations.

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1219-1232
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• 2009

Let H(B) denote the space of all holomorphic functions on the unit ball B of $\mathbb{C}^n$. Let $\varphi$ = (${\varphi}_1,{\ldots}{\varphi}_n$) be a holomorphic self-map of B and $g{\in}2$(B) with g(0) = 0. In this paper we study the boundedness and compactness of the generalized composition operator $C_{\varphi}^gf(z)=\int_{0}^{1}{\mathfrak{R}}f(\varphi(tz))g(tz){\frac{dt}{t}}$ from generalized weighted Bergman spaces into Bloch type spaces.

• The Pure and Applied Mathematics
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• v.5 no.1
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• pp.17-21
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• 1998

In [5], Zhu introduces a bounded operator T from $L^{\infty}$(D) into Bloch space B. In this paper, we will consider the generalized Bloch spaces $B_{q}$ and find bounded operator from $L^{\infty}$(D) into $B_{q}$.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1391-1408
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• 2023

Let M and M^# be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators [M, b] and [M^#, b] in a general context of Banach function spaces when b belongs to BMO(?ⁿ) spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak-Orlicz spaces are also given.