• 대한수학회보
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• 제55권2호
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• pp.673-674
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• 2018

Corrections are made for some examples following Theorem 2.2 in Posinormal terraced matrices, Bull. Korean Math. Soc. 46 (2009), no. 1, 117-123.

• 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 $${\alpha}(x)\;:\;{\sqrt{x}},\;{\sqrt{\frac{m}{m+1}},\;{\sqrt{\frac{m}{m+2}},\;{\sqrt{\frac{m+2}{m+3}},{\cdots}$$.

• 충청수학회지
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• 제23권3호
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• pp.425-434
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• 2010

Here we see that the $p-Ces{\grave{a}}ro$ operators, the generalized $Ces{\grave{a}}ro$ operators of order one, the discrete generalized $Ces{\grave{a}}ro$ operators, and their adjoints are all posinormal operators on ${\ell}^2$, but many of these operators are not dominant, not normaloid, and not spectraloid. The question of dominance for $C_k$, the generalized $Ces{\grave{a}}ro$ operators of order one, remains unsettled when ${\frac{1}{2}}{\leq}k<1$, and that points to some general questions regarding terraced matrices. Sufficient conditions are given for a terraced matrix to be normaloid. Necessary conditions are given for terraced matrices to be dominant, spectraloid, and normaloid. A very brief new proof is given of the well-known result that $C_k$ is hyponormal when $k{\geq}1$.

• 대한수학회보
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• 제46권1호
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• pp.117-123
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• 2009

This paper is a study of some properties of a collection of bounded linear operators resulting from terraced matrices M acting through multiplication on ${\ell}^2$; the term terraced matrix refers to a lower triangular infinite matrix with constant row segments. Sufficient conditions are found for M to be posinormal, meaning that $MM^*=M^*PM$ for some positive operator P on ${\ell}^2$; these conditions lead to new sufficient conditions for the hyponormality of M. Sufficient conditions are also found for the adjoint $M^*$ to be posinormal, and it is observed that, unless M is essentially trivial, $M^*$ cannot be hyponormal. A few examples are considered that exhibit special behavior.

• Kyungpook Mathematical Journal
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• 제50권1호
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• pp.89-100
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• 2010

Let g = (V, E, ${\mu}$) be a weighted directed tree, where V is a vertex set, E is an edge set, and ${\mu}$ is ${\sigma}$-finite measure on V. The tree g induces a composition operator C on the Hilbert space $l^2$(V). Hand-type directed trees are defined and characterized the weak hyponormalities of such C in this note. Also some additional related properties are discussed. In addition, some examples related to directed hand-type trees are provided to separate classes of weak-hyponormal operators.

• Kyungpook Mathematical Journal
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• 제49권2호
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• pp.203-209
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• 2009

In this paper we study some spectral properties of the class (y) operators and we will investigate on the relation between this class and other usual classes of operators.

• 대한수학회지
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• 제54권2호
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• pp.599-612
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• 2017

If ${\psi}$ is analytic on the open unit disk $\mathbb{D}$ and ${\varphi}$ is an analytic self-map of $\mathbb{D}$, the weighted composition operator $C_{{\psi},{\varphi}}$ is defined by $C_{{\psi},{\varphi}}f(z)={\psi}(z)f({\varphi}(z))$, when f is analytic on $\mathbb{D}$. In this paper, we study normal, cohyponormal, hyponormal and normaloid weighted composition operators on the Hardy and weighted Bergman spaces. First, for some weighted Hardy spaces $H^2({\beta})$, we prove that if $C_{{\psi},{\varphi}}$ is cohyponormal on $H^2({\beta})$, then ${\psi}$ never vanishes on $\mathbb{D}$ and ${\varphi}$ is univalent, when ${\psi}{\not\equiv}0$ and ${\varphi}$ is not a constant function. Moreover, for ${\psi}=K_a$, where |a| < 1, we investigate normal, cohyponormal and hyponormal weighted composition operators $C_{{\psi},{\varphi}}$. After that, for ${\varphi}$ which is a hyperbolic or parabolic automorphism, we characterize all normal weighted composition operators $C_{{\psi},{\varphi}}$, when ${\psi}{\not\equiv}0$ and ${\psi}$ is analytic on $\bar{\mathbb{D}}$. Finally, we find all normal weighted composition operators which are bounded below.

• Journal of applied mathematics & informatics
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• 제37권1_2호
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• pp.73-83
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• 2019

In this article, we introduce a new kind of subnormal weighted shifts, which is a generalized form of Bergman shift, and discuss the k-hyponormality of its backward extensions.

• 호남수학학술지
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• 제35권2호
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• pp.311-317
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• 2013

In this note we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Weighted Bergman space $A^2_{\alpha}(\mathbb{D})$ with symbol in the class of functions $f+\bar{g}$ with polynomials $f$ and $g$ of degree 2.