• Kyungpook Mathematical Journal
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• 제48권4호
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• pp.721-727
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• 2008

Let $W_{\alpha}$ be a weighted shift with positive weight sequence ${\alpha}=\{\alpha_i\}_{i=0}^{\infty}$. The semi-cubical hyponormality of $W_{\alpha}$ is introduced and some flatness properties of $W_{\alpha}$ are discussed in this note. In particular, it is proved that if ${\alpha}_n={\alpha}_{n+1}$ for some $n{\geq}1$, ${{\alpha}_{n+k}}={\alpha}_n$ for all $k{\geq}1$.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제12권1호
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• pp.1-6
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• 2008

Duggal-Jeon-Kubrusly([2]) introduced Hilbert space operator T satisfying property ${\mid}T{\mid}^2{\leq}{\mid}T^2{\mid}$, where ${\mid}T{\mid}=(T^*T)^{1/2}$. In this paper we extend this property to general version, namely property B(n). In addition, we construct examples which distinguish the classes of operators with property B(n) for each $n{\in}\mathbb{N}$.

• 대한수학회보
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• 제37권1호
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• pp.53-62
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• 2000

In this paper we explore relations between joint Weyl and Browder spectra. Also, we give a spectral characterization of the Taylor-Browder spectrum for special classes of doubly commuting n-tuples of operators and then give a partial answer to Duggal's question.

• 대한수학회보
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• 제48권3호
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• pp.617-625
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• 2011

In [6], it was shown that hyponormality for Toeplitz operators with polynomial symbols can be reduced to classical Schur's algorithm in function theory. In [6], Zhu has also given the explicit values of the Schur's functions ${\Phi}_0$, ${\Phi}_1$ and ${\Phi}_2$. Here we explicitly evaluate the Schur's function ${\Phi}_3$. Using this value we find necessary and sufficient conditions under which the Toeplitz operator $T_{\varphi}$ is hyponormal, where ${\varphi}$ is a trigonometric polynomial given by ${\varphi}(z)$ = ${\sum}^N_{n=-N}a_nz_n(N{\geq}4)$ and satisfies the condition $\bar{a}_N\(\array{a_{-1}\\a_{-2}\\a_{-4}\\{\vdots}\\a_{-N}}\)=a_{-N}\;\(\array{\bar{a}_1\\\bar{a}_2\\\bar{a}_4\\{\vdots}\\\bar{a}_N}\)$. Finally we illustrate the easy applicability of the derived results with a few examples.

• 대한수학회보
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• 제57권1호
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• pp.81-93
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• 2020

Let m ∈ ℕ₀, p > 1 and $${\alpha}^{[m,p]}(x)\;:\;{\sqrt{x}},\;\{{\sqrt{\frac{(m+n-1)p-(m+n-2)}{(m+n)p-(m+n-1)}}}\}^{\infty}_{n=1}$$. In this paper, we consider the backward extensions of Bergman-type weighted shift W_α^[m,p](x). We consider its subnormality, k-hyponormality and positive quadratic hyponormality. Our results include all the results on Bergman weighted shift W_α(x) with m ∈ ℕ and $${\alpha}(x)\;:\;{\sqrt{x}},\;{\sqrt{\frac{m}{m+1}},\;{\sqrt{\frac{m}{m+2}},\;{\sqrt{\frac{m+2}{m+3}},{\cdots}$$.

• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.443-454
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• 2021

In this paper, we discuss the backward extensions of Bergman shifts W_α(m), where $${\alpha}(m)\;:\;\sqrt{\frac{m}{m+1}},\;{\sqrt{\frac{m+1}{m+2}}},\;{\cdots},\;(m{\in}\mathbb{N})$$. We obtained a complete description of the n-hynonormality for backward one, two and three step extensions.

• Journal of applied mathematics & informatics
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• 제40권1_2호
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• pp.15-24
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• 2022

Let p > 1 and α^[p](x) : $\sqrt{x}$, $\sqrt{\frac{p}{^2p-1}}$, $\sqrt{\frac{2p-1}{3p-2}}$, … , with 0 < x ≤ $\frac{p}{2p-1}$. In [10], the authors considered the subnormality, n-hyponormality and positive quadratic hyponormality of W_α^[p](x). By continuing to study, in this paper, we give a sufficient condition of quadratic hyponormality of W_α^[p](x). Finally, we give an example to characterize the gaps of W_α^[p](x) distinctively.

• 대한수학회논문집
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• 제11권3호
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• pp.657-663
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• 1996

In this paper we give some conditions under which the Weyl spectrum of an operator satisfies the spectral mapping theorem for analytic functions. Also we show that Weyl's theorem holds for p(T) where T is an operator of M-power class (N) and p is a polynomial on a neighborhood of $\sigam(T)$. Finally we answer an old question of Oberai.

• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.573-578
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• 2005

For a positive integer k and a positive number 0 < p$\le$1, an operator T is said to be (p, k)-quasiposinormal if $T^{{\ast}k}(c^2(T^{\ast}T)P - (TT^{\ast})^P)T^k {\ge} 0$ for some c > o. In this paper we consider a structure for (p, k)-quasiposinormal.

• 대한수학회보
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• 제51권1호
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• pp.237-252
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• 2014

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators $T_{\Phi}$, where ${\Phi}$ = $F+G^*$, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space $L^2_a(\mathbb{D},\mathbb{C}^n)$. We also show some necessary conditions for the hyponormality of $T_{F+G^*}$ with $F+G^*{\in}h^{\infty}{\otimes}M_{n{\times}n}$ on $L^2_a(\mathbb{D},\mathbb{C}^n)$.