• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.383-389
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• 2007

In this paper we examine the hyperinvariant subspace problem for quasi-n-hyponormal operators. The main result on this problem is as follows. If T = N + K is such that N is a quasi-n-hyponormal operator whose spectrum contains an exposed arc and K belongs to the Schatten p-ideal then T has a non-trivial hyperinvariant subspace.

• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.225-231
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• 2019

In a previous paper, the authors of this paper studied $2{\times}2$ matrices in upper triangular form, whose entries are operators on Hilbert spaces, and in which the the (1, 1) entry has a nontrivial hyperinvariant subspace. We were able to show, in certain cases, that the $2{\times}2$ matrix itself has a nontrivial hyperinvariant subspace. This generalized two earlier nice theorems of H. J. Kim from 2011 and 2012, and made some progress toward a solution of a problem that has been open for 45 years. In this paper we continue our investigation of such $2{\times}2$ operator matrices, and we improve our earlier results, perhaps bringing us closer to the resolution of the long-standing open problem, as mentioned above.

• Kyungpook Mathematical Journal
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• v.47 no.2
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• pp.221-226
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• 2007

In this note we set forth three possible definitions of the property of "almost commuting with a compact operator" and discuss an old result of W. Arveson that says that every operator on Hilbert space has the weakest of the three properties. Finally, we discuss some recent progress on the hyperinvariant subspace problem (see the bibliography), and relate it to the concept of almost commuting with a compact operator.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.489-494
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• 2018

The first purpose of this note is to generalize two nice theorems of H. J. Kim concerning hyperinvariant subspaces for certain classes of operators on Hilbert space, proved by him by using the technique of "extremal vectors". Our generalization (Theorem 1.2) is obtained as a consequence of a new theorem of the present authors, and doesn't utilize the technique of extremal vectors. The second purpose is to use this theorem to obtain the existence of hyperinvariant subspaces for a class of $2{\times}2$ operator matrices (Theorem 3.2).

• Kyungpook Mathematical Journal
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• v.53 no.2
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• pp.185-189
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• 2013

In this note we study a class of invertible weighted bilateral shifts on Hilbert space introduced by Haskell Rosenthal recently. We show that every Rosenthal shift is unitarily equivalent to its inverse, not quasisimilar to its adjoint, and has a nontrivial hyperinvariant subspace.

• Journal of the Korean Mathematical Society
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• v.51 no.5
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• pp.1089-1104
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• 2014

An operator $T{\in}L(H)$, is said to belong to k-quasi class $A_n^*$ operator if $$T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O$$ for some positive integer n and some positive integer k. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically k-quasi class $A_n^*$. Second, we consider the tensor product for k-quasi class $A_n^*$, giving a necessary and sufficient condition for $T{\otimes}S$ to be a k-quasi class $A_n^*$, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class $A_n^*$ operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and $(B^*)^{-1}$ are k-quasi class $A_n^*$ operators such that AX = XB, then $A^*X=XB^*$. Finally, we will prove the spectrum continuity of this class of operators.

• 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.