Joint numerical ranges of operators in semi-Hilbertian spaces

Baklouti, Hamadi; Feki, Kais; Mahmoud, Sid Ahmed Ould Ahmed

doi:10.1016/j.laa.2018.06.021

Cited by 90 publications

(78 citation statements)

References 17 publications

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“…Remark 2.3. Note that the fact that W A (T ) = C in the case T (N (A)) ⊂ N (A) has recently been proved by H. Baklouti et al in [5]. However, our approach here is different from theirs.…”

Section: Resultsmentioning

confidence: 84%

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Spectral radius of semi-Hilbertian space operators and its applications

Feki

2020

Ann. Funct. Anal.

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In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space H, which are bounded with respect to the seminorm induced by a positive operator A on H. Mainly, we show that r A (T ) ≤ ω A (T ) for every A-bounded operator T , where r A (T ) and ω A (T ) denote respectively the A-spectral radius and the A-numerical radius of T . This allows to establish that r A (T ) = ω A (T ) = T A for every A-normaloid operator T , where T A is denoted to be the A-operator seminorm of T . Moreover, some characterizations of A-normaloid and A-spectraloid operators are given.

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“…Remark 2.3. Note that the fact that W A (T ) = C in the case T (N (A)) ⊂ N (A) has recently been proved by H. Baklouti et al in [5]. However, our approach here is different from theirs.…”

Section: Resultsmentioning

confidence: 84%

“…Recently, there are many papers that study operators defined on semi-Hilbertian spaces. One may see [5,6,21,18] and their references.…”

Section: Introductionmentioning

confidence: 99%

Spectral radius of semi-Hilbertian space operators and its applications

Feki

2020

Ann. Funct. Anal.

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“…This new concept is intensively studied (see [5,22]). Note that if T ∈ B(H) and satisfies T (N (A)) N (A), then ω A (T ) = +∞ (see [17,Theorem 2.2.]).…”

Section: Introductionmentioning

confidence: 99%

Davis–Wielandt shells of semi-Hilbertian space operators and its applications

Feki

Mahmoud

2020

Banach J. Math. Anal.

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In this paper we generalize the concept of Davis-Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator A is considered. Moreover, we investigate the parallelism of A-bounded operators with respect to the seminorm and the numerical radius induced by A. Mainly, we characterize A-normaloid operators in terms of their A-Davis-Wielandt radii. In addition, a connection between A-seminorm-parallelism to the identity operator and an equality condition for the A-Davis-Wielandt radius is proved. This generalizes the well-known results in [23,10]. Some other related results are also discussed.

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“…The A-numerical range of T ∈ B(H) is a subset of the set of complex numbers C and it is defined by W A (T ) = T x, x A : x ∈ H, x A = 1 . It is known as well that W A (T ) is a nonempty convex subset of C (not necessarily closed), and its supremum modulus, denoted by w A (T ) = sup |ξ| : ξ ∈ W A (T ) , is called the A-numerical radius of T (see [2]). It is a generalization of the concept of numerical radius of an operator: recall that the numerical radius of T ∈ B(H) is defined as w(T ) = sup | T x, x | : x ∈ H, x = 1 .…”

Section: Introductionmentioning

confidence: 99%

“…. For proofs and more facts about Anumerical radius of operators, we refer the reader to [2,9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A-Numerical Radius and Product of Semi-Hilbertian Operators

Zamani

2020

Bull. Iran. Math. Soc.

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Let A be a positive bounded operator on a Hilbert space H, ·, · . The semi-inner product x, y A := Ax, y , x, y ∈ H, induces a seminorm · A on H. Let w A (T ) denote the A-numerical radius of an operator T in the semi-Hilbertian space H, · A . In this paper, for any semi-Hilbertian operators T and S, we show that w A (T R) = w A (SR) for all (A-rank one) semi-Hilbertian operator R if and only if A 1/2 T = λA 1/2 S for some complex unit λ. From this result we derive a number of consequences.2010 Mathematics Subject Classification. Primary 47A05; Secondary 46C05, 47B65, 47A12.

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Joint numerical ranges of operators in semi-Hilbertian spaces

Cited by 90 publications

References 17 publications

Spectral radius of semi-Hilbertian space operators and its applications

Spectral radius of semi-Hilbertian space operators and its applications

Davis–Wielandt shells of semi-Hilbertian space operators and its applications

A-Numerical Radius and Product of Semi-Hilbertian Operators

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