• Honam Mathematical Journal
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• v.23 no.1
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• pp.29-39
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• 2001

Let $\mathcal{B}$ be a Banach space of analytic functions defined on the open unit disk. Assume S is a bounded operator defined on $\mathcal{B}$ such that S is in the commutant of $M_zn$ or $SM_zn=-M_znS$ for some positive integer n. We give necessary and sufficient condition between compactness of $SM_z+cM_zS$ where c = 1, -1, i, -i, and the structure of S. Also we characterize the commutant of $M_zn$ for some positive integer n.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.683-689
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• 2007

Let $\mathcal{B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain G in the complex plane. Let ${\varphi}$ be an analytic polynomial or a rational function and let $M_{\varphi}$ denote the operator of multiplication by ${\varphi}$. Under certain condition on ${\varphi}$ and G, we characterize the commutant of $M_{\varphi}$ that is the set of all bounded operators T such that $TM_{\varphi}=M_{\varphi}T$. We show that $T=M_{\Psi}$, for some function ${\Psi}$ in $\mathcal{B}$.

• Honam Mathematical Journal
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• v.26 no.3
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• pp.299-308
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• 2004

Let $\{{\beta}(n)\}^{\infty}_{n=0}$ be a sequence of positive numbers such that ${\beta}(0)=1$. We consider the space $H^2({\beta})$ of all power series $f(z)=^{Po}_{n=0}{\hat{f}}(n)z^n$ such that $^{Po}_{n=0}{\mid}{\hat{f}}(n){\mid}^2{\beta}(n)^2<{\infty}$. We link the ideas of subspaces of $H^2({\beta})$ and zero sets. We give some sufficient conditions for a vector in $H^2({\beta})$ to be cyclic for the multiplication operator $M_z$. Also we characterize the commutant of some multiplication operators acting on $H^2({\beta})$.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.523-533
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• 2007

Let $\cal{B}$ be a reflexive Banach space of functions analytic on the open unit disc and M be an invariant subspace of the multiplication operator by the independent variable, $M_z$. Suppose that $\varphi\;\in\;\cal{H}^{\infty}$ and $M_{\varphi}$ : M ${\rightarrow}$ M, defined by $M_{\varphi}f={\varphi}f$, is the operator of multiplication by ${\varphi}$. We would like to investigate the spectrum and the essential spectrum of $M_{\varphi}$ and we are looking for the necessary and sufficient conditions for $M_{\varphi}$ to be a Fredholm operator. Also we give a sufficient condition for a sequence $\{w_n\}$ to be an interpolating sequence for $\cal{B}$. At last the commutant of $M_{\varphi}$ under certain conditions on M and ${\varphi}$ is determined.