a-Numerical range on $$C^*$$-algebras

Bourhim, Abdellatif; Mabrouk, Mohamed

doi:10.1007/s11117-021-00825-6

Cited by 8 publications

(16 citation statements)

References 15 publications

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“…The next lemma shows that T ∈ B A 1∕2 (H) , then V A (T) can be expressed as an intersection of the closed discs D( , ‖T − I‖ a ), ( ∈ ℂ) . This property has recently been proven in [8]. However, here we present a quick and different proof.…”

Section: Accordingly T(n(a)) ⊂ N(a) Also Following the Prooflines Of [13 Theorem 3] It Is Easy To See That Cl(w A (T)) = Cl(w( T))mentioning

confidence: 72%

“…for operators in B A 1∕2 (H) that we are going to use frequently throughout the rest of this paper. The proof can be found in [3,8].…”

Section: Preliminaries and Auxiliary Resultsmentioning

confidence: 99%

“…Observe also that V e A (T) is a nonempty, convex subset of ℂ . This is due to the fact that endowed with the ‖T‖ e is a unital C * -algebra; see [8]. Moreover, V e A (T) = {0} if and only if AT ∈ K(H).…”

Section: It Is Clear That When a = I ‖T‖ Ementioning

confidence: 99%

“…For more information about the concepts discussed previously, we refer the reader to [1,3,8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Essential numerical ranges of operators in semi-Hilbertian spaces

Baklouti

Mabrouk

2021

Ann. Funct. Anal.

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Let A be a positive bounded operator on an infinite dimensional complex and separable Hilbert space (H, ⟨, ⟩) and ∶=B(H)∕K(H) be the Calkin algebra and ′ its dual; here K(H) denotes the closed ideal of B(H) consisting of all compact operators on H . For an operator T on H , let ‖T‖ A and V A (T) denote the A-operator semi-norm and the A-numerical range induced by A. The A-essential numerical range and the A-essential norm of T are defined by V e A (T) = V Â( T) andwhereIn this paper, we establish some properties of the A-essential numerical range.In particular we show that if T has an A-adjoint then

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Section: Accordingly T(n(a)) ⊂ N(a) Also Following the Prooflines Of [13 Theorem 3] It Is Easy To See That Cl(w A (T)) = Cl(w( T))mentioning

confidence: 72%

“…for operators in B A 1∕2 (H) that we are going to use frequently throughout the rest of this paper. The proof can be found in [3,8].…”

Section: Preliminaries and Auxiliary Resultsmentioning

confidence: 99%

Section: It Is Clear That When a = I ‖T‖ Ementioning

confidence: 99%

“…For more information about the concepts discussed previously, we refer the reader to [1,3,8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Essential numerical ranges of operators in semi-Hilbertian spaces

Baklouti

Mabrouk

2021

Ann. Funct. Anal.

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“…Recently, Bourhim and Mabrouk in [11] introduced and studied a-numerical range and a-numerical radius of elements in C * -algebras. Also, the authors in [4] continued the work on the a-numerical range and the a-numerical radius.…”

Section: Introductionmentioning

confidence: 99%

An extension of the $a$-numerical radius on $C^*$-algebras

Mabrouk¹,

Zamani²

2022

Preprint

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Let a be a positive element in a unital C * -algebra A. We define a semi-norm on A, which generalizes the a-operator semi-norm and the anumerical radius. We investigate basic properties of this semi-norm and prove inequalities involving it. Further, we derive new upper and lower bounds for the a-numerical radii of elements in A. Some other related results are also discussed.2010 Mathematics Subject Classification. 47A12; 47A30; 46L05. Key words and phrases. C * -algebra; numerical range; numerical radius; inequality.

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A-Spectral Permanence Property for $$C^*$$-Algebras

Mabrouk,

Zamani

2024

Mediterr. J. Math.

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a-Numerical range on $$C^*$$-algebras

Cited by 8 publications

References 15 publications

Essential numerical ranges of operators in semi-Hilbertian spaces

Essential numerical ranges of operators in semi-Hilbertian spaces

An extension of the $a$-numerical radius on $C^*$-algebras

A-Spectral Permanence Property for $$C^*$$-Algebras

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