• 대한수학회지
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• 제44권4호
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• pp.855-861
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• 2007

Let A and B be commuting bounded linear operators on a Hilbert space. In this paper, we study spectral area estimates for norms of $A^*B-BA^*$ when A is subnormal or p-hyponormal.

• Journal of applied mathematics & informatics
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• 제41권2호
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• pp.273-286
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• 2023

In this paper we show that if T ∈ L(X) and S ∈ L(X) is a Riesz operator commuting with T and X_S(F) ∈ Lat(S), where F = {0} or F ⊆ ℂ ⧵ {0} is closed then T|X_S(F) and T|X_T(F) + S|X_S(F) share the local spectral properties such as SVEP, Dunford's property (C), Bishop's property (𝛽), decomopsition property (𝛿) and decomposability. As a corollary, if T ∈ L(X) and Q ∈ L(X) is a quasinilpotent operator commuting with T then T is Riesz if and only if T + Q is Riesz. We also study some spectral properties of Riesz operators acting on Banach spaces. We show that if T, S ∈ L(X) such that TS = ST, and Y ∈ Lat(S) is a hyperinvarinat subspace of X for which 𝜎(S|Y ) = {0} then 𝜎_*(T|Y + S|Y ) = 𝜎_*(T|Y ) for 𝜎_* ∈ {𝜎, 𝜎_loc, 𝜎_sur, 𝜎_ap}. Finally, we show that if T ∈ L(X) and S ∈ L(Y ) on the Banach spaces X and Y and T is similar to S then T is Riesz if and only if S is Riesz.

• 대한수학회보
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• 제52권5호
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• pp.1481-1487
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• 2015

Let H be a separable complex Hilbert space. A commuting tuple $T=(T_1,{\cdots},T_n)$ of bounded linear operators on H is called a spherical isometry if $\sum_{i=1}^{n}T^*_iT_i=I$. The tuple T is called a toral isometry if each $T_i$ is an isometry. In this paper, we show that for each $n{\geq}1$ there is a supercyclic n-tuple of spherical isometries on $\mathbb{C}^n$ and there is no spherical or toral isometric tuple of operators on an infinite-dimensional Hilbert space.

• 대한수학회보
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• 제54권2호
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• pp.633-646
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• 2017

Recently, Curto, Lee and Yoon considered the properties (such as, hyponormality, subnormality, and flatness, etc.) for 2-variable weighted shifts and constructed several families of commuting pairs of subnormal operators such that each family can be used to answer a conjecture of Curto, Muhly and Xia negatively. In this paper, we consider the weak and quadratic hyponormality of 2-variable weighted shifts ($W_1,W_2$). In addition, we detect the weak and quadratic hyponormality with some interesting 2-variable weighted shifts.

• Journal of applied mathematics & informatics
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• 제40권3_4호
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• pp.785-794
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• 2022

In this paper we show that if A ∈ L(X) and R ∈ L(X) is a quasinilpotent operator commuting with A then X_A(F) = X_A+R(F) for all subset F ⊆ ℂ and 𝜎_loc(A) = 𝜎_loc(A + R). Moreover, we show that A and A + R share many common local spectral properties such as SVEP, property (C), property (𝛿), property (𝛽) and decomposability. Finally, we show that quasisimility preserves local spectrum.

• 충청수학회지
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• 제29권4호
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• pp.543-552
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• 2016

In this paper we investigate the local spectral properties of quasidecomposable operators. We show that if $T{\in}L(X)$ is quasi-decomposable, then T has the weak-SDP and ${\sigma}_{loc}(T)={\sigma}(T)$. Also, we show that the quasi-decomposability is preserved under commuting quasi-nilpotent perturbations. Moreover, we show that if $f:U{\rightarrow}{\mathbb{C}}$ is an analytic and injective on an open neighborhood U of ${\sigma}(T)$, then $T{\in}L(X)$ is quasi-decomposable if and only if f(T) is quasi-decomposable. Finally, if $T{\in}L(X)$ and $S{\in}L(Y)$ are asymptotically similar, then T is quasi-decomposable if and only if S does.

• 대한수학회보
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• 제60권6호
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• pp.1555-1566
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• 2023

A Hilbert space operator A ∈ B(H) is a generalised n-projection, denoted A ∈ (G-n-P), if A^*n = A. (G-n-P)-operators A are normal operators with finitely countable spectra σ(A), subsets of the set $\{0\}\,{\cup}\,\{\sqrt[n+1]{1}\}.$ The Aluthge transform à of A ∈ B(H) may be (G - n - P) without A being (G - n - P). For doubly commuting operators A, B ∈ B(H) such that σ(AB) = σ(A)σ(B) and ${\parallel}A{\parallel}\,{\parallel}B{\parallel}\;{\leq}\;{\parallel}{\tilde{AB}}{\parallel},$ ${\tilde{AB}}\;{\in}\;(G\,-\,n\,-\,P)$ if and only if $A\;=\;{\parallel}{\tilde{A}}{\parallel}\,(A_{00}\,{\oplus}\,(A_0\,{\oplus}\,A_u))$ and $B\;=\;{\parallel}{\tilde{B}}{\parallel}\,(B_0\,{\oplus}\,B_u),$ where A₀₀ and B₀, and A₀ ⊕ A_u and B_u, doubly commute, A₀₀B₀ and A₀ are 2 nilpotent, A_u and B_u are unitaries, A^*n_u = A_u and B^*n_u = B_u. Furthermore, a necessary and sufficient condition for the operators αA, βB, αà and ${\beta}{\tilde{B}},\;{\alpha}\,=\,\frac{1}{{\parallel}{\tilde{A}}{\parallel}}$ and ${\beta}\,=\,\frac{1}{{\parallel}{\tilde{B}}{\parallel}},$ to be (G - n - P) is that A and B are spectrally normaloid at 0.

• 대한수학회논문집
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• 제32권3호
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• pp.653-659
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• 2017

Let $p{\geq}1$ be a real number. A tuple $T=(T_1,{\ldots},T_n)$ of commuting bounded linear operators on a Banach space X is called an ${\ell}^p$-spherical isometry if ${\sum_{i=1}^{n}}{\parallel}T_ix{\parallel}^p={\parallel}x{\parallel}^p$ for all $x{\in}X$. The tuple T is called a toral isometry if each Ti is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n{\geq}1$, there is a supercyclic ${\ell}^2$-spherical isometric n-tuple on ${\mathbb{C}}^n$ but there is no supercyclic ${\ell}^2$-spherical isometry on an infinite-dimensional Hilbert space. In this article, we investigate the supercyclicity of ${\ell}^p$-spherical isometries and toral isometries on Banach spaces. Also, we introduce the notion of semicommutative tuples and we show that the Banach spaces ${\ell}^p$ ($1{\leq}p$ < ${\infty}$) support supercyclic ${\ell}^p$-spherical isometric semi-commutative tuples. As a result, all separable infinite-dimensional complex Hilbert spaces support supercyclic spherical isometric semi-commutative tuples.