arXiv subjects

Debmalya Sain

Publications and source records attributed to Debmalya Sain.

At least 73 records · Page 4Linked to original sources

arXiv2018

A complete characterization of Radon planes whose unit spheres are regular polygons

We study the structure of the unit sphere of polygonal Radon planes from a geometric point of view. In particular, we prove that a $ 2 $-dimensional real polygonal Banach space $ \mathbb{X} $ cannot be a Radon plane if the number of vertices of its unit sphere is $ 4n, $ for some $ n \in \mathbb{N}. $ We next obtain a complete characterization of polygonal Radon planes in terms of a tractable geometric concept introduced in this article. It follows from our characterization that every regular polygon with $ 4n+2 $ vertices, where $ n \in \mathbb{N}, $ is the unit sphere of a Radon plane. We further give example of a family of Radon planes for which the unit spheres are hexagons, but not regular ones.

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arXiv2018

Some remarks on Birkhoff-James orthogonality of linear operators

We study Birkhoff-James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff-James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.

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arXiv2018

On some geometric properties of operator spaces

In this paper we study some geometric properties like parallelism, orthogonality and semi-rotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X} $ and $\mathbb{Y}$, assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X} $ and $\mathbb{Y}.$ We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff-James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semi-rotund points (semi-rotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semi-rotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semi-rotund space which is not strictly convex, if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.

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arXiv2018

A complete characterization of Birkhoff-James orthogonality in infinite dimensional normed space

In this paper, we study Birkhoff-James orthogonality of bounded linear operators and give a complete characterization of Birkhoff-James orthogonality of bounded linear operators on infinite dimensional real normed linear spaces. As an application of the results obtained, we prove a simple but useful characterization of Birkhoff-James orthogonality of bounded linear functionals defined on a real normed linear space, provided the dual space is strictly convex. We also provide separate necessary and sufficient conditions for smoothness of bounded linear operators on infinite dimensional normed linear spaces.

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arXiv2018

A complete characterization of smoothness in the space of bounded linear operators

We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that Gâteaux differentiability and Fréchet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fréchet differentiable normed linear space, endowed with the usual operator norm.

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arXiv2018

On two extremum problems related to the norm of a bounded linear operator

We explore the norm attainment set and the minimum norm attainment set of a bounded linear operator between Hilbert spaces and Banach spaces. Indeed, we obtain a complete characterization of both the sets, separately for operators between Hilbert spaces and Banach spaces. We also study the interconnection between these two sets and prove that for operators between Hilbert spaces, these two sets are either equal or mutually orthogonal, provided both of them are non-empty. We also obtain separate complete characterizations of reflexive Banach spaces and Euclidean spaces in terms of the norm (minimum norm) attainment set, in order to illustrate the importance of our study.

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arXiv2018

Smooth points in operator spaces and some Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces

We introduce the notion of approximate norm attainment set of a bounded linear operator between Banach spaces and use it to obtain a complete characterization of smooth points in the space of compact linear operators, provided the domain space is reflexive and Kadets-Klee. We also apply the concept to characterize strong BPB property (sBPBp) of a pair of Banach spaces. We further introduce uniform $ ε- $BPB approximation of a bounded linear operator and uniform strong BPB property (uniform sBPBp) with respect to a given family of norm one linear operators and explore some of the relevant properties to illustrate its connection with earlier studies on Bishop-Phelps-Bollob$\acute{a}$s type theorems in Banach spaces. It is evident that our study has deep connections with the study of smooth points in operator spaces. We obtain a complete characterization of uniform sBPBp for a pair of Banach spaces, with respect to a given family of norm one bounded linear operators between them. As the final result of this paper, we prove that if $ \mathbb{X} $ is a reflexive Kadets-Klee Banach space and $ \mathbb{Y} $ is any Banach space, then the pair $ (\mathbb{X},\mathbb{Y}) $ has sBPBp for compact operators. Our results extend, complement and improve some of the earlier results in this context.

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arXiv2018

On the norm attainment set of a bounded linear operator and semi-inner-products in normed spaces

We obtain a complete characterization of the norm attainment set of a bounded linear functional on a normed space, in terms of a semi-inner-product defined on the space. Motivated by this result, we further apply the concept of semi-inner-products to obtain a complete characterization of the norm attainment set of a bounded linear operator between any two real normed spaces. In particular, this answers an open question raised recently in [Sain, D., \textit{On the norm attainment set of a bounded linear operator}, J. Math. Anal. Appl., \textbf{457} (2018), 67-76.]. Our results illustrate the applicability of semi-inner-products towards a better understanding of the geometry of normed spaces.

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arXiv2018

On extreme contractions between real Banach spaces

We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and $ μ- $compatible point pair ($ μ- $CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in $ \mathbb{L}(\ell_4^2, \ell_4^2) $ and $ \mathbb{L}(\ell_4^2, \mathbb{H}) $, where $\mathbb{H}$ is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in $ \mathbb{L}(\mathbb{H}, \ell_{p}^2), $ whenever $ p $ is even and $\mathbb{H}$ is any Hilbert space. We further study extreme contractions between infinite-dimensional Banach spaces and obtain some analogous results. Finally, we characterize real Hilbert spaces among real Banach spaces in terms of CPP, that substantiates our motivation behind introducing these new geometric notions.

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arXiv2017

Orthogonality of bounded linear operators on complex Banach spaces

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 $l_p$ spaces. We prove that $ T $ is a left symmetric linear operator on $ \ell_p^2{(\mathbb{C})}$ if and only if $ T $ is the zero operator.

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arXiv2017

On extreme contractions and the norm attainment set of a bounded linear operator

In this paper we completely characterize the norm attainment set of a bounded linear operator on a Hilbert space. This partially answers a question raised recently in [\textit{D. Sain, On the norm attainment set of a bounded linear operator, accepted in Journal of Mathematical Analysis and Applications}]. We further study the extreme contractions on various types of finite-dimensional Banach spaces, namely Euclidean spaces, and strictly convex spaces. In particular, we give an elementary alternative proof of the well-known characterization of extreme contractions on a Hilbert space, that works equally well for both the real and the complex case. As an application of our exploration, we prove that it is possible to characterize real Hilbert spaces among real Banach spaces, in terms of extreme contractions on two-dimensional subspaces of it.

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arXiv2017

Approximate Birkhoff-James orthogonality in the space of bounded linear operators

There are two notions of approximate Birkhoff-James orthogonality in a normed space. We characterize both the notions of approximate Birkhoff-James orthogonality in the space of bounded linear operators defined on a normed space. A complete characterization of approximate Birkhoff-James orthogonality in the space of bounded linear operators defined on Hilbert space of any dimension is obtained which improves on the recent result by Chmieliński et al. [ J. Chmieliński, T. Stypula and P. Wójcik, \textit{Approximate orthogonality in normed spaces and its applications}, Linear Algebra and its Applications, \textbf{531} (2017), 305--317.], in which they characterized approximate Birkhoff-James orthogonality of linear operators on finite dimensional Hilbert space and also of compact operators on any Hilbert space.

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arXiv2017

On approximate Birkhoff-James orthogonality and normal cones in a normed space

We study two notions of approximate Birkhoff-James orthogonality in a normed space, from a geometric point of view, and characterize them in terms of normal cones. We further explore the interconnection between normal cones and approximate Birkhoff-James orthogonality to obtain a complete characterization of normal cones in a two-dimensional smooth Banach space. We also obtain a uniqueness theorem for approximate Birkhoff-James orthogonality set in a normed space.

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arXiv2016

On Symmetry of Birkhoff-James Orthogonality of Linear Operators on Finite-dimensional Real Banach Spaces

We characterize left symmetric linear operators on a finite dimensional strictly convex and smooth real normed linear space $ \mathbb{X},$ which answers a question raised recently by one of the authors in \cite{S} [D. Sain, \textit{Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces, Journal of Mathematical Analysis and Applications, accepted, $ 2016 $}]. We prove that $ T\in B(\mathbb{X}) $ is left symmetric if and only if $ T $ is the zero operator. If $ \mathbb{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.

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arXiv2016

Chebyshev centers that are not farthest points

In this paper we address the question whether in a given Banach space, a Chebyshev center of a nonempty bounded subset can be a farthest point of the set. Our exploration reveals that the answer depends on the convexity properties of the Banach space. We obtain a characterization of two-dimensional real strictly convex spaces in terms of Chebyshev center not contributing to the set of farthest points. We explore the scenario in uniformly convex Banach spaces and further study the roles played by centerability and M-compactness in the scheme of things to obtain a step by step characterization of strictly convex Banach spaces. We also illustrate with examples the optimality of our results.

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arXiv2016

On the norm attainment set of a bounded linear operator

In this paper we explore the properties of a bounded linear operator defined on a Banach space, in light of operator norm attainment. Using Birkhoff-James orthogonality techniques, we give a necessary condition for a bounded linear operator attaining norm at a particular point of the unit sphere. We prove a number of corollaries to establish the importance of our study. As part of our exploration, we also obtain a characterization of smooth Banach spaces in terms of operator norm attainment and Birkhoff-James orthogonality. Restricting our attention to $ l_{p}^{2} (p \in \mathbb{N}\setminus \{ 1 \})$ spaces, we obtain an upper bound for the number of points at which any linear operator, which is not a scalar multiple of an isometry, may attain norm.

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arXiv2016

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

In this paper we characterize Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Banach space $ \mathbb{X}. $ We also explore the symmetry of Birkhoff-James orthogonality of linear operators defined on $ \mathbb{X}. $ Using some of the related results proved in this paper, we finally prove that $ T \in \mathbb{L}(l_{p}^2) (p \geq 2, p \neq \infty) $ 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 $ \mathbb{X}, $ in particular for the Banach spaces $ l_{p}^{n} (p > 1, p \neq \infty). $

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arXiv2016

Birkhoff-James orthogonality and smoothness of bounded linear operators

We present a sufficient condition for smoothness of bounded linear operators on Banach spaces for the first time. Let $T, A \in B(\mathbb{X}, \mathbb{Y}),$ where $\mathbb{X}$ is a real Banach space and $\mathbb{Y}$ is a real normed linear space. We find sufficient condition for $ T \bot_{B} A \Leftrightarrow Tx \bot_{B} Ax $ for some $ x \in S_{\mathbb{X}}$ with $ \|Tx\| = \|T\|, $ and use it to show that $T$ is a smooth point in $ B(\mathbb{X}, \mathbb{Y}) $ if $T$ attains its norm at unique (upto muliplication by scalar) vector $ x \in S_{\mathbb{X}},$ $Tx$ is a smooth point of $\mathbb{Y} $ and {\em sup}$_{y \in C} \|Ty\| < \|T\|$ for all closed subsets $C$ of $S_{\mathbb{X}}$ with $d(\pm x,C) > 0.$ For operators on a Hilbert space $ \mathbb{H}$ we show that $ T \bot_{B} A \Leftrightarrow Tx \bot_{B} Ax $ for some $ x \in S_{\mathbb{H}}$ with $ \|Tx\| = \|T\| $ if and only if the norm attaining set $M_T = \{ x \in S_{\mathbb{H}} : \|Tx\| = \|T\| \} = S_{H_0}$ for some finite dimensional subspace $H_0$ and $ \|T\|_{{H_o}^{\bot}} < \|T\|.$ We also characterize smoothness of compact operators on normed spaces and bounded linear operators on Hilbert spaces.

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