Quasinilpotent Part of class A or (p, k)-quasihyponormal Operators

Tanahashi, Kôtarô; Kim, In Hyoun; Uchiyama, Atsushi

doi:10.1007/978-3-7643-8893-5_13

Cited by 12 publications

(12 citation statements)

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“…For interesting properties of k-quasiclass A operators, see [20,30]. In [30], this class of operators is called quasi-class (A, k).…”

Section: An Operator T ∈ B(h) Is Called Weyl If It Is Fredholm Of Indmentioning

confidence: 99%

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Weyl's theorem for algebraically k-quasiclass A operators

Gao¹,

Fang²

2012

OpMath

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Abstract. If T or T * is an algebraically k-quasiclass A operator acting on an infinite dimensional separable Hilbert space and F is an operator commuting with T , and there exists a positive integer n such that F n has a finite rank, then we prove that Weyl's theorem holds for f (T ) + F for every f ∈ H(σ(T )), where H(σ(T )) denotes the set of all analytic functions in a neighborhood of σ(T ). Moreover, if T * is an algebraically k-quasiclass A operator, then α-Weyl's theorem holds for f (T ). Also, we prove that if T or T * is an algebraically k-quasiclass A operator then both the Weyl spectrum and the approximate point spectrum of T obey the spectral mapping theorem for every f ∈ H(σ(T )).

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“…For interesting properties of k-quasiclass A operators, see [20,30]. In [30], this class of operators is called quasi-class (A, k).…”

Section: An Operator T ∈ B(h) Is Called Weyl If It Is Fredholm Of Indmentioning

confidence: 99%

“…In [30], this class of operators is called quasi-class (A, k). We say that T is algebraically k-quasiclass A if there exists a nonconstant complex polynomial h such that h(T ) is k-quasiclass A.…”

Section: An Operator T ∈ B(h) Is Called Weyl If It Is Fredholm Of Indmentioning

confidence: 99%

Weyl's theorem for algebraically k-quasiclass A operators

Gao¹,

Fang²

2012

OpMath

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“…Theorem 1.2 [12,15] Let M be an invariant subspace of T . If T ∈ k-Q A or T ∈ * -k-Q A, and T | M is normal and injective, then M reduces T .…”

Section: Introductionmentioning

confidence: 99%

“…The class k-Q A(1) is equal to k-Q A [6,11,15] and 0-quasiclass A(n) means class A(n) [17]. It is well-known that, for each n, class A(n) includes every p-hyponormal operators [4,5].…”

Section: Introductionmentioning

confidence: 99%

Reducibility of Invariant Subspaces of Operators Related to k-Quasiclass-A(n) Operators

Yuan

Wang

2015

Complex Anal. Oper. Theory

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It is well-known that hyponormal operators have many interesting properties, for example, if the restriction T | M of the hyponormal operator T on its nontrivial closed invariant subspace M is normal, then M reduces T . In order to discuss the reducibility of invariant subspaces of an operator, four properties of invariant subspaces (R i , i = 1, . . . , 4) are introduced. Among others, it is proved that, for a k-quasi-A(n) operator T , if the restriction T | M is normal and injective, then M reduces T , thus the function σ : T −→ σ (T ) is continuous on the class of k-quasiclass-A(n) operators. Some examples related to class A(n) and n-paranormal operator are given which imply that the inclusion relations are strict.

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“…Let B(H) denote the algebra of bounded linear operators on an infinite dimensional complex Hilbert space H. For any operator T in B(H) set, as usual, |T | = (T * T ) (the self-commutator of T ), and consider the following standard definitions: T is hyponormal if |T * | 2 ≥ |T | 2 (i.e., if self-commutator [T * , T ] is non-negative or, equivalently, if T * x ≤ T x for every x in H), p-hyponormal if (T * T ) p ≥ (T T * ) p for some p ∈ (0, 1], and T is called paranormal if ||T 2 x|| ≥ ||T x|| 2 for all unit vector x ∈ H. Following [13] and [4] we say that T ∈ B(H) belongs to class A if |T 2 | ≥ |T | 2 . We shall denote classes of hyponormal operators, p-hyponormal operators, paranormal operators, and class A operators by H, H(p), PN , and A,…”

Section: Introductionmentioning

confidence: 99%