On operators with closed numerical ranges

Ji, Youqing; Liang, Bin

doi:10.1215/20088752-2017-0051

Cited by 7 publications

(4 citation statements)

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“…Our plan in this section is to study the strongly A-numerically closed class of operators. Following the work done in [24,34], we can easily show that A-normal and A-hyponormal operators are all strongly A-numericallly closed class of operators. Motivated by the notion of complex symmetric operators studied in [26] we here introduce the following definition.…”

Section: Anderson's Theorem On Semi-hilbertian Operatorsmentioning

confidence: 78%

“…An operator class C ⊆ B(H) is said to be strongly numerically closed, if for any T ∈ C and any ǫ > 0, there exists a compact operator K such that K < ǫ, T + K ∈ C and W (T +K) is closed. The study of strongly numerically closed operator class was motivated by Bourin [12] and then followed in [24,34]. Analogously, we define an operator class C ⊆ B A 1/2 (H) to be strongly A-numerically closed if for any T ∈ C and any ǫ > 0 there exists K ∈ K A 1/2 (H) with K A < ǫ such that T + K ∈ C and W A (T + K) = W A (T + K).…”

Section: Anderson's Theorem On Semi-hilbertian Operatorsmentioning

confidence: 99%

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Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Sen¹,

Goswami²

2023

Redefining Virtual Teaching Learning Pedagogy

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In this paper we explore the relation between the A-numerical range and the A-spectrum of A-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of A-normal operator and prove that closure of the A-numerical range of an A-normal operator is the convex hull of the A-spectrum. We further prove Anderson's theorem for the sum of A-normal and A-compact operators which improves and generalizes the existing result on Anderson's theorem for A-compact operators. Finally we introduce strongly A-numerically closed class of operators and along with other results prove that the class of A-normal operators is strongly A-numerically closed.

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Section: Anderson's Theorem On Semi-hilbertian Operatorsmentioning

confidence: 78%

Section: Anderson's Theorem On Semi-hilbertian Operatorsmentioning

confidence: 99%

Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Sen¹,

Goswami²

2023

Redefining Virtual Teaching Learning Pedagogy

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“…In our previous work [13], we proved that the class of triangular operators, the class of hyponormal operators and the class of weighted shift operators, etc., are all approximately strongly numerically closed. As we can see, all these results obtained are about operator classes which maybe not operator algebras.…”

Section: Introductionmentioning

confidence: 97%

“…Inspired by Bourin's theorem, in [13], we introduced the concept of being strongly numerically closed. But to be more precise, we realize that such an operator class should be said to be approximately strongly numerically closed.…”

Section: Introductionmentioning

confidence: 99%

Operators with closed numerical ranges in nest algebras

Liang

2018

Proc. Amer. Math. Soc.

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In the present paper, we continue our research on numerical ranges of operators. With newly developed techniques, we show thatLet N be a nest on a Hilbert space H and T ∈ T (N ), where T (N ) denotes the nest algebra associated with N . Then for given ε > 0, there exists a compact operator K with K < ε such that T + K ∈ T (N ) and the numerical range of T + K is closed.As applications, we show that the statement of the above type holds for the class of Cowen-Douglas operators, the class of nilpotent operators and the class of quasinilpotent operators.

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Characterization of numerical radius parallelism in $$C^*$$C∗-algebras

Zamani

2018

Positivity

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Let v(x) be the numerical radius of an element x in a C * -algebra A. First, we prove several numerical radius inequalities in A. Particularly, we present a refinement of the triangle inequality for the numerical radius in C *algebras. In addition, we show that if x ∈ A, then v(x) = 1 2 x if and only if x = Re(e iθ x) + Im(e iθ x) for all θ ∈ R. Among other things, we introduce a new type of parallelism in C * -algebras based on numerical radius. More precisely, we consider elements x and y of A which satisfy v(x+λx) = v(x)+v(y) for some complex unit λ. We show that this relation can be characterized in terms of pure states acting on A.

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On operators with closed numerical ranges

Cited by 7 publications

References 0 publications

Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Assessment of Modern Methods for Remote Teaching in Some Selected Educational Institutions in Kolkata City of West Bengal, India

Operators with closed numerical ranges in nest algebras

Characterization of numerical radius parallelism in $$C^*$$C∗-algebras

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