Absolutely norm attaining paranormal operators

Ramesh, Geetha

doi:10.1016/j.jmaa.2018.05.024

Cited by 20 publications

(29 citation statements)

References 11 publications

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“…The same is true for AN -operators. A sufficient condition to hold this result for AN -operators is given in [23], which depends on the essential spectrum. A similar condition works for AM-operators too.…”

Section: Am-operatorsmentioning

confidence: 98%

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Spectral properties of absolutely minimum attaining operators

Bala

Ramesh

2020

Banach J. Math. Anal.

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Let T : H 1 Ñ H 2 be a bounded linear operator defined between complex Hilbert spaces H 1 and H 2 . We say T to be minimum attaining if there exists a unit vector x P H 1 such that }T x} " mpT q, where mpT q :" inf t}T x} : x P H 1 , }x} " 1u is the minimum modulus of T . We say T to be absolutely minimum attaining (AM-operators in short), if for any closed subspace M of H 1 the restriction operator T | M : M Ñ H 2 is minimum attaining. In this paper, we give a new characterization of positive absolutely minimum attaining operators (AM-operators, in short), in terms of its essential spectrum. Using this we obtain a sufficient condition under which the adjoint of an AM-operator is AM. We show that a paranormal absolutely minimum attaining operator is hyponormal. Finally, we establish a spectral decomposition of normal absolutely minimum attaining operators. In proving all these results we prove several spectral results for paranormal operators. We illustrate our main result with an example.

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Section: Am-operatorsmentioning

confidence: 98%

“…A partial characterization of positive, absolutely norm attaining operators was given in [6], which was further improved in [20,22]. Another characterization of positive, absolutely norm attaining operators is discussed in [23].…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of absolutely minimum attaining operators

Bala

Ramesh

2020

Banach J. Math. Anal.

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“…Then by [4] and [5] the remaining part of the proof is analogous and this completes the proof.…”

Section: Resultsmentioning

confidence: 70%

“…The study of norm attaining operators has been interesting to many mathematicians and researchers over decades( see [1], [2] and [5]). The class of absolutely norm attaining operators between complex Hilbert spaces was introduced by [1] and they discussed several important examples and properties of these operators.…”

Section: Introductionmentioning

confidence: 99%

Characterization of Absolutely Norm Attaining Compact Hyponormal Operators

Okelo

2020

Proceedings of International Mathematical Sciences

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General characterization of Hilbert space operators has been a subject of interest to many mathematicians for decades. In this paper, we characterize absolute norm-attainability for compact hyponormal operators. We give necessary and sufficient conditions for a bounded linear compact hyponormal operator on an infinite dimensional complex Hilbert space to be absolutely norm attaining. Moreover, we discuss the structure of compact hyponormal operators when they are self-adjoint and normal. Lastly, we discuss in general, other properties of compact hyponormal operators when they are absolutely norm attaining and their commutators.

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“…We denote the set of all absolutely norm attaining operators by AN (H 1 , H 2 ). For more details of this class operators we refer to [24].…”

Section: Resultsmentioning

confidence: 99%

Spectral representation of absolutely minimum attaining unbounded normal operators

Kulkarni¹,

Ramesh²

2022

Preprint

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Let T : D(T ) → H 2 be a densely defined closed operator with domain D(T ) ⊂ H 1 . We say T to be absolutely minimum attaining if for every closed subspace M of H 1 , the restriction operatorIn this article show that T is absolutely minimum attaining and unbounded if and only if its null space is finite dimensional and its Moore-Penrose inverse is compact. We also prove a Spectral Theorem for self -adjoint unbounded operators of this class. We show that every operator with a compact generalized inverse has a nontrivial hyperinvariant subspace.

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Absolutely norm attaining paranormal operators

Cited by 20 publications

References 11 publications

Spectral properties of absolutely minimum attaining operators

Spectral properties of absolutely minimum attaining operators

Characterization of Absolutely Norm Attaining Compact Hyponormal Operators

Spectral representation of absolutely minimum attaining unbounded normal operators

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