On the structure of absolutely minimum attaining operators

Ganesh, Jadav; Ramesh, Geetha; Sukumar, D

doi:10.1016/j.jmaa.2015.03.016

Cited by 11 publications

(9 citation statements)

References 7 publications

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“…Again by applying Theorem (3.9), we get T˚T P AMpHq. (2) This follows by Case (1) and by[9, Corollary 4.9], that T P AMpHq if and only if T˚T P AMpHq.…”

mentioning

confidence: 84%

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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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“…Again by applying Theorem (3.9), we get T˚T P AMpHq. (2) This follows by Case (1) and by[9, Corollary 4.9], that T P AMpHq if and only if T˚T P AMpHq.…”

mentioning

confidence: 84%

“…. In general, if T P AMpHq, then T˚need not be an AM-operator (see [7,9] for more details). The same is true for AN -operators.…”

Section: Am-operatorsmentioning

confidence: 99%

Spectral properties of absolutely minimum attaining operators

Bala

Ramesh

2020

Banach J. Math. Anal.

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“…This class was defined in [7] and characterization of this class is discussed in [11]. For more details about AM-operators, we refer to [7,10,11].…”

Section: Am-operatorsmentioning

confidence: 99%

“…These two classes differ in spectra and many other properties. We refer to [6,10,11] for more details of the characterization and spectral properties of this class.…”

Section: Introductionmentioning

confidence: 99%

Hyperinvariant subspace for absolutely norm attaining and absolutely minimum attaining operators

Bala¹,

Ramesh²

2020

Preprint

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A bounded linear operator T : H1 → H2, where H1, H2 are Hilbert spaces, is called norm attaining if there exist x ∈ H1 with unit norm such that T x = T . If for every closed subspace M ⊆ H1, the operator T |M : M → H2 is norm attaining, then T is called absolutely norm attaining. If in the above definitions T is replaced by the minimum modulus, m(T ) := inf{ T x : x ∈ H1, x = 1}, then T is called minimum attaining and absolutely minimum attaining, respectively.In this article, we show the existence of a non-trivial hyperinvariant subspace for absolutely norm attaining operators as well as absolutely minimum attaining operators on a Hilbert space.

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“…This is exactly, the structure of absolutely minimum attaining operators (shortly AMoperators) in case when T is positive and one-to-one. We refer [11] for more details of the structure of these operators. We recall that A ∈ B(H 1 , H 2 ) is said to be minimum attaining if there exists x 0 ∈ S H 1 such that Ax 0 = m(A) and absolutely minimum attaining if A| M is minimum attaining for each non zero closed subspace M of H 1 .…”

Section: Positive An -Operatorsmentioning

confidence: 99%