• Kyungpook Mathematical Journal
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• 제56권3호
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• pp.899-910
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• 2016

It is known that a semi-cubically hyponormal weighted shift need not satisfy the flatness property, in which equality of two weights forces all or almost all weights to be equal. So it is a natural question to describe all semi-cubically hyponormal weighted shifts $W_{\alpha}$ with first two weights equal. Let ${\alpha}$ : 1, 1, ${\sqrt{x}}$(${\sqrt{u}}$, ${\sqrt{v}}$, ${\sqrt{w}}$)^ be a backward 3-step extension of a recursively generated weight sequence with 1 < x < u < v < w and let $W_{\alpha}$ be the associated weighted shift. In this paper we characterize completely the semi-cubical hyponormal $W_{\alpha}$ satisfying the additional assumption of the positive determinant coefficient property, which result is parallel to results for quadratic hyponormality.

• 대한수학회보
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• 제44권3호
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• pp.517-522
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• 2007

In this paper we consider the hyponormality of Toeplitz operators $T_\varphi$ on the Bergman space $L_\alpha^2(\mathbb{D})$ with symbol in the case of function $f+\bar{g}$ with polynomials f and g. We present some necessary conditions for the hyponormality of $T_\varphi$ under certain assumptions about the coefficients of $\varphi$.

• 대한수학회보
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• 제39권4호
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• pp.527-534
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• 2002

The weighted shifts have been made a central role in the study of κ-hyponormality for hyponormal and subnormal operators in discrete notion. In this note we discuss the κ-hyponormality of weighted translation semigroups with a symbol function in continuous notion.

• 호남수학학술지
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• 제30권1호
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• pp.127-135
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• 2008

In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L^2_a(\mathbb{D})$ with symbol in the case of function f + $\overline{g}$ with polynomials f and g. We present some necessary conditions for the hyponormality of $T_{\varphi}$, under certain assumptions about the coefficients of ${\varphi}$.

• 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.

• Kyungpook Mathematical Journal
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• 제59권2호
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• pp.315-324
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• 2019

In this note we introduce a local-cubically hyponormal weighted shift of order ${\theta}$ with $0{\leq}{\theta}{\leq}{\frac{\pi}{2}}$, which is a new notion between cubic hyponormality and quadratic hyponormality of operators. We discuss the property of flatness for local-cubically hyponormal weighted shifts.

• 대한수학회논문집
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• 제34권2호
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• pp.477-486
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• 2019

Let ${\alpha}:1,(1,{\sqrt{x}},{\sqrt{y}})^{\wedge}$ be a weight sequence with Stampfli's subnormal completion and let $W_{\alpha}$ be its associated weighted shift. In this paper we discuss some properties of the region ${\mathcal{U}}:=\{(x,y):W_{\alpha}$ is semi-cubically hyponormal} and describe the shape of the boundary of ${\mathcal{U}}$. In particular, we improve the results of [19, Theorem 4.2].

• East Asian mathematical journal
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• 제7권1호
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• pp.61-70
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• 1991

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