• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1027-1041
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• 2008

In this paper we consider the hyponormality of Toeplitz operators $T_{\varphi}$ on the Bergman space $L_a^2{(\mathbb{D})$ in the cases, where ${\varphi}\;:=f+\bar{g}$ (f and g are polynomials). We present some necessary or sufficient conditions for the hyponormality of $T_{\varphi}$ under certain assumptions about the coefficients of ${\varphi}$.

• Korean Journal of Mathematics
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• v.15 no.2
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• pp.185-193
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• 2007

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 ${\varphi}$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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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 $L^2_a(\mathbb{D},\mathbb{C}^n)$.

• Kyungpook Mathematical Journal
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• v.59 no.3
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• pp.415-431
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• 2019

In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator $T{\in}B(H)$ is said to be k-quasi-class A(s, t) if $$T^{*k}(({\mid}T^*{\mid}^t{\mid}T{\mid}^{2s}{\mid}T^*{\mid}^t)^{\frac{1}{t+s}}-{\mid}T^*{\mid}^{2t})T^k{\geq}0$$, where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop's property ${\beta}$ and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if $T^*$ is algebraically k-quasi-class A(s, t), then the generalized a-Weyl's theorem holds for T. Using these results we show that $T^*$ satisfies generalized the Weyl's theorem if and only if T satisfies the generalized Weyl's theorem if and only if T satisfies Weyl's theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.63-71
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• 2015

Let H be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on H into itself. Let $A,B{\in}\mathcal{L}(H)$ we define the generalized derivation ${\delta}_{A,B}:\mathcal{L}(H){\mapsto}\mathcal{L}(H)$ by ${\delta}_{A,B}(X)=AX-XB$, we note ${\delta}_{A,A}={\delta}_A$. If the inequality ${\parallel}T-(AX-XA){\parallel}{\geq}{\parallel}T{\parallel}$ holds for all $X{\in}\mathcal{L}(H)$ and for all $T{\in}ker{\delta}_A$, then we say that the range of ${\delta}_A$ is orthogonal to the kernel of ${\delta}_A$ in the sense of Birkhoff. The operator $A{\in}\mathcal{L}(H)$ is said to be finite [22] if ${\parallel}I-(AX-XA){\parallel}{\geq}1(*)$ for all $X{\in}\mathcal{L}(H)$, where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.