Orthogonality of bounded linear operators

Ghosh, Puja; Sain, Debmalya; Paul, Kallol

doi:10.1016/j.laa.2016.03.009

Cited by 21 publications

(27 citation statements)

References 8 publications

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“…Recently in [5], Sain and Paul have characterized finite dimensional real Hilbert spaces among finite dimensional real Banach spaces in terms of operator norm attainment, using the notion of Birkhoff-James orthogonality. More recently, symmetry of Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Hilbert space H has been explored by Ghosh et al in [3]. However, it was remarked in [3] that analogous results corresponding to the far more general setting of Banach spaces remain unknown.…”

Section: Introductionmentioning

confidence: 99%

“…More recently, symmetry of Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Hilbert space H has been explored by Ghosh et al in [3]. However, it was remarked in [3] that analogous results corresponding to the far more general setting of Banach spaces remain unknown. The aim of the present paper is twofold: we characterize Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Banach space X and we also explore the symmetry of Birkhoff-James orthogonality of linear operators defined on X.…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, let us say that x is right symmetric (with respect to Birkhoff-James orthogonality) if y ⊥ B x implies x ⊥ B y for any y ∈ X. It was proved in [3] that if H is a finite dimensional real Hilbert space then T ∈ L(H) is a left symmetric point if and only if T is the zero operator and T ∈ L(H)…”

Section: Introductionmentioning

confidence: 99%

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Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces

Sain

2017

Journal of Mathematical Analysis and Applications

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In this paper we characterize Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Banach space X. We also explore the symmetry of Birkhoff-James orthogonality of linear operators defined on X. Using some of the related results proved in this paper, we finally prove that T ∈ L(l 2 p )(p ≥ 2, p = ∞) is left symmetric with respect to Birkhoff-James orthogonality if and only if T is the zero operator. We conjecture that the result holds for any finite dimensional strictly convex and smooth real Banach space X, in particular for the Banach spaces l n p (p > 1, p = ∞).2010 Mathematics Subject Classification. Primary 46C15, Secondary 47A30.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces

Sain

2017

Journal of Mathematical Analysis and Applications

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“…In [3], it was proved that if T is a compact linear operator on a real Hilbert space H then T is left symmetric if and only if T is the zero operator. Sain proved in [8] that if T ∈ ℓ 2 p (R), 1 < p < ∞, then T is left symmetric if and only if T is the zero operator.…”

Section: A Contradiction This Proves Our Claimmentioning

confidence: 99%

Orthogonality of bounded linear operators on complex Banach spaces

Paul¹,

Sain²,

Mal³

et al. 2018

Adv. Oper. Theory

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We study Birkhoff-James orthogonality of bounded linear operators on complex Banach spaces and obtain a complete characterization of the same. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of bounded (compact) linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case. We also study the left symmetric linear operators on complex two-dimensional lp spaces. We prove that T is a left symmetric linear operator on ℓ 2 p (C) if and only if T is the zero operator.2010 Mathematics Subject Classification. Primary 46B20, Secondary 47L05. 1 2 KALLOL PAUL, DEBMALYA SAIN, ARPITA MAL AND KALIDAS MANDALLet X, Y be complex Banach spaces. Let B X = {x ∈ X : x ≤ 1} and S X = {x ∈ X : x = 1} be the unit ball and the unit sphere of X, respectively. Let L(X, Y)(K(X, Y)) denote the Banach space of all bounded (compact) linear operators from X to Y, endowed with the usual operator norm. We write L(X, Y) = L(X) and K(X,For any two elements x, y ∈ X, x is said to be B-J orthogonal to y, written asSimilarly, for any two elements T, A ∈ L(X), T is said to be B-J orthogonal to A, written asFor a linear operator T defined on a Banach space X, let M T denote the collection of all unit vectors in X at which T attains norm, i.e,

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“…In [2] we proved that if H is a real finite-dimensional Hilbert space, T ∈ B(H) is right symmetric if and only if M T = S H and T ∈ B(H) is left symmetric if and only if T is the zero operator. It should be noted that if H is a complex Hilbert space then Theorem 2.5 of [11] gives a complete characterization of right symmetric bounded linear operators in B(H), in terms of isometry and coisometry.…”

Section: Introductionmentioning

confidence: 99%

On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces

Sain¹,

Ghosh²,

Paul³

2017

Operators and Matrices

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Abstract. We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space X, which answers a question raised recently by one of the authors in [7] [D. Sain, Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, J. Math. Anal. Appl. 447 (2017) [860][861][862][863][864][865][866]. We prove that T ∈ B(X) is left symmetric if and only if T is the zero operator. If X is two-dimensional then the same characterization can be obtained without the smoothness assumption. We also explore the properties of right symmetric linear operators defined on a finite dimensional real Banach space. In particular, we prove that smooth linear operators on a finite-dimensional strictly convex and smooth real Banach space can not be right symmetric.Mathematics subject classification (2010): Primary 47L05, secondary 46B20.

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Orthogonality of bounded linear operators

Cited by 21 publications

References 8 publications

Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces

Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces

Orthogonality of bounded linear operators on complex Banach spaces

On symmetry of Birkhoff-James orthogonality of linear operators on finite-dimensional real Banach spaces

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