• East Asian mathematical journal
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• 제23권2호
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• pp.135-149
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• 2007

In 1975, Wenger [4] determined the fine spectra of $Ces{\grave{a}}ro$ operator $C_1$ on c, the space of convergent sequences. In [7], the spectrum of the Rhaly operators on $c_0$ and c, under the assumption that ${lim}\limits_{n{\rightarrow}{\infty}}(n+1)a_n\;=\;L\;{\neq}\;0$, has been determined. In this paper the author determine the fine spectra of the Rhaly matrix $R_a$ as an operator on the space c, with the same assumption.

• 호남수학학술지
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• 제29권4호
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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.

• 대한수학회논문집
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• 제28권3호
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• pp.487-500
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• 2013

In this paper we provide a brief introduction to the continuity of approximate point spectrum on the algebra B(X), using basic properties of Fredholm operators and the SVEP condition. Also, we give an example showing that in general it not holds that if the spectrum is continuous an operator T, then for each ${\lambda}{\in}{\sigma}_{s-F}(T){\setminus}\overline{{\rho}^{\pm}_{s-F}(T)}$ and ${\in}$ > 0, the ball $B({\lambda},{\in})$ contains a component of ${\sigma}_{s-F}(T)$, contrary to what has been announced in [J. B. Conway and B. B. Morrel, Operators that are points of spectral continuity II, Integral Equations Operator Theory 4 (1981), 459-503] page 462.

• Kyungpook Mathematical Journal
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• 제46권4호
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• pp.553-563
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• 2006

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set ${\sigma}_{B{\omega}}(A)$ of all ${\lambda}{\in}\mathbb{C}$ such that $A-{\lambda}I$ is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem ${\sigma}_{B{\omega}}(A)={\sigma}(A)$\E(A), and the B-Weyl spectrum ${\sigma}_{B{\omega}}(A)$ of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalizedWeyl's theorem holds for the case where A is an algebraically ($p,\;k$)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.

• Korean Journal of Mathematics
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• 제10권1호
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• pp.11-17
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• 2002

In this paper we give several necessary and sufficient conditions for an operator on the Hilbert space to obey Browder's theorem. And it is shown that if S has totally finite ascent and $T{\prec}S$ then $f(T)$ obeys Browder's theorem for every $f{\in}H({\sigma}(T))$, where $H({\sigma}(T))$ denotes the set of all analytic functions on an open neighborhood of ${\sigma}(T)$.

• 대한수학회논문집
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• 제12권3호
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• pp.597-602
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• 1997

Let H be a separable complex Hilbert space and L(H) be the *-algebra of all bounded linear operators on H. For $T \in L(H)$, we construct a pair of semi-positive definite operators $$ $\mid$T$\mid$_r = (T^*T)^{\frac{1}{2}} and $\mid$T$\mid$_l = (TT^*)^{\frac{1}{2}}. $$ An operator T is called a semi-hyponormal operator if $$ Q_T = $\mid$T$\mid$_r - $\mid$T$\mid$_l \geq 0. $$ In this paper, by using a technique introduced by Berberian [1], we show that the approximate point spectrum $\sigma_{ap}(T)$ of a semi-hyponomal operator T is empty.

• Kyungpook Mathematical Journal
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• 제58권4호
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• pp.637-650
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• 2018

In this paper first we show properties of isosymmetric operators given by M. Stankus [13]. Next we introduce an [m, C]-symmetric operator T on a complex Hilbert space H. We investigate properties of the spectrum of an [m, C]-symmetric operator and prove that if T is an [m, C]-symmetric operator and Q is an n-nilpotent operator, respectively, then T + Q is an [m + 2n - 2, C]-symmetric operator. Finally, we show that if T is [m, C]-symmetric and S is [n, D]-symmetric, then $T{\otimes}S$ is [m + n - 1, $C{\otimes}D$]-symmetric.

• 대한수학회논문집
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• 제13권3호
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• pp.481-487
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• 1998

Let (equation omitted) denote the closure of the set (equation omitted) of dominant operators in the norm topology. We show that the Weyl spectrum of an operator T $\in$ (equation omitted) satisfies the spectral mapping theorem for analytic functions, which is an extension of [5, Theorem 1]. Also we show that an operator approximately equivalent to an operator of class (equation omitted) is of class (equation omitted).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제5권2호
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• pp.109-114
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• 1998

Let H be a separable complex H be a space and let (equation omitted)(H) be the *-algebra of all bounded linear operators on H. An operator T in (equation omitted)(H) is said to be p-hyponormal if ($T^{\ast}T)^p - (TT^{\ast})^{p}\geq$ 0 for 0 ＜ p ＜ 1. If p = 1, T is hyponormal and if p = $\frac{1}{2}$, T is semi-hyponormal. In this paper, by using a technique introduced by S. K. Berberian, we show that the approximate point spectrum $\sigma_{\alpha p}(T) of a pure p-hyponormal operator T is empty, and obtains the compact perturbation of T.

• Korean Journal of Mathematics
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• 제23권2호
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• pp.249-257
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• 2015

For a positive integer k, an operator T is said to be k-quasi-*-paranormal if ${\parallel}T^{k+2}x{\parallel}{\parallel}T^kx{\parallel}{\geq}{\parallel}T^*T^kx{\parallel}^2$ for all x $\in$ H, which is a generalization of *-paranormal operator. In this paper, we give a necessary and sufficient condition for T to be a k-quasi-*-paranormal operator. We also prove that the spectrum is continuous on the class of all k-quasi-*-paranormal operators.