Orthogonality of bilinear forms and application to matrices

Roy, Saikat; Senapati, Tanusri; Sain, Debmalya

doi:10.1016/j.laa.2020.12.032

Cited by 12 publications

(3 citation statements)

References 9 publications

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“…Our aim in this section is to obtain a characterization of Birkhoff-James orthogonality of weak * continuous functionals, which is of paramount importance in the context of our current work on best approximations. We would like to mention that the said characterization can also be obtained by modifying Theorem 2.1 of [13]. However, we present a complete proof of the same, for the convenience of the readers.…”

Section: Orthogonality Of Functionalsmentioning

confidence: 94%

On best approximations in Banach spaces from the perspective of orthogonality

Sain¹,

Roy²

2021

Preprint

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We study best approximations in Banach spaces via Birkhoff-James orthogonality of functionals. To exhibit the usefulness of Birkhoff-James orthogonality techniques in the study of best approximation problems, some algorithms and distance formulae are presented. As an application of our study, we obtain some crucial inequalities, which also strengthen the classical Hölder's inequality. The relevance of the algorithms and the inequalities are discussed through concrete examples.2010 Mathematics Subject Classification. Primary 46B28, Secondary 46B20. Key words and phrases. Birkhoff-James orthogonality; weak * continuous functionals; best approximations; inequalities.The research of Dr. Debmalya Sain is sponsored by DST-SERB N-PDF Fellowship under the mentorship of Professor Apoorva Khare. Dr. Sain feels grateful to have the opportunity to acknowledge the colossal contribution of Mr. Swapan Bandyopahyay, an extraordinarily devoted teacher, towards rightly shaping the philosophies of so many young learners like himself. The research of Mr. Saikat Roy is supported by CSIR MHRD in form of Senior Research Fellowship under the supervision of Professor Satya Bagchi.

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Section: Orthogonality Of Functionalsmentioning

confidence: 94%

On best approximations in Banach spaces from the perspective of orthogonality

Sain¹,

Roy²

2021

Preprint

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“…In particular, the space M n (C) d equipped with the joint operator norm is isometrically isomorphic to a closed subspace of C(S C n , (C n ) d ). In [14], this identification was used to give an alternate proof of orthogonality to one dimensional subspaces in M n (C) given in [2, Theorem 1]. We use this identification to prove the following result for the joint operator norm, analogous to Corollary 1.1.…”

Section: Proofsmentioning

confidence: 99%

Subdifferential of the joint numerical radius

Grover¹,

Singla²

2022

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“…In the year 1999, Bhatia-Šemrl [4], and Paul [16] independently characterized the Birkhoff-James orthogonality of bounded linear operators on a Hilbert space. It was subsequently revealed that the said characterization no longer remains intact when the underlying space is not a Hilbert space [3,13,15,20,21,25,29]. Since then, Birkhoff-James orthogonality techniques have been substantially used in determining various geometric aspects of operator spaces.…”

Section: Introductionmentioning

confidence: 99%

On the orthogonality of multilinear maps and application to smoothness

Roy¹

2022

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We characterize Birkhoff-James orthogonality of bounded multilinear maps in its full generality, probably for the first time. In the special case, the said result reduces to the Birkhoff-James orthogonality of bounded linear operators on an infinite-dimensional complex normed linear space, a new addition to the existing literature. The said characterization, whenever restricted to the setting of Hilbert space, leads to a multilinear generalization of the well-known Bhatia-Šemrl Theorem in both finite and infinite-dimensional cases. As a natural outcome of our investigations, we obtain necessary and sufficient conditions of smoothness for bounded multilinear maps. Contrary to most of the previous results that address the smoothness of linear operators in the context of real normed linear spaces, we study the smoothness of multilinear maps in the realm of complex normed linear spaces. Our results extend and improve some recent results related to the smoothness of linear operators.

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Orthogonality of bilinear forms and application to matrices

Cited by 12 publications

References 9 publications

On best approximations in Banach spaces from the perspective of orthogonality

On best approximations in Banach spaces from the perspective of orthogonality

Subdifferential of the joint numerical radius

On the orthogonality of multilinear maps and application to smoothness

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