arXiv subjects

Debmalya Sain

Publications and source records attributed to Debmalya Sain.

At least 37 records · Page 2Linked to original sources

arXiv2024

On strong orthogonality and strictly convex normed linear spaces

We introduce the notion of strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element $ x $ of the unit sphere $ S_{X}$ to be an exposed point of the unit ball $ B_X .$ We then prove that a normed linear space is strictly convex iff for each element x of the unit sphere there exists a bounded linear operator A on X which attains its norm only at the points of the form $ λx $ with $ λ\in S_{K} $.

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arXiv2024

On norm derivatives and the ball-covering property of Banach spaces

We study a local version of the ball-covering problem in Banach spaces, and obtain a complete solution to it in terms of the norm derivatives. We illustrate the advantage of the local approach by obtaining substantial refinements of several previously known results on this topic.

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arXiv2024

Orthogonality and smoothness induced by the norm derivatives

We study the concepts of orthogonality and smoothness in normed linear spaces, induced by the derivatives of the norm function. We obtain analytic characterizations of the said orthogonality relations in terms of support functionals in the dual space. We also characterize the related notions of local smoothness and establish its connection with the corresponding orthogonality set, which is analogous to the well-known relation between the Birkhoff-James orthogonality and the classical notion of smoothness. The similarities and the differences between the various notions of smoothness are illustrated by considering some particular examples, including $ \mathbb{K}(\mathbb{H}), $ the Banach space of all compact operators on a Hilbert space $ \mathbb{H}. $

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arXiv2024

On best coapproximations in subspaces of diagonal matrices

We characterize the best coapproximation(s) to a given matrix $ T $ out of a given subspace $ \mathbb{Y} $ of the space of diagonal matrices $ \mathcal{D}_n $, by using Birkhoff-James orthogonality techniques and with the help of a newly introduced property, christened the $ * $-Property. We also characterize the coproximinal subspaces and the co-Chebyshev subspaces of $ \mathcal{D}_n $ in terms of the $ * $-Property. We observe that a complete characterization of the best coapproximation problem in $ \ell_{\infty}^n $ follows directly as a particular case of our approach.

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arXiv2024

On the best coapproximation problem in $\ell_1^n$

We study the best coapproximation problem in the Banach space $ \ell_1^n, $ by using Birkhoff-James orthogonality techniques. Given a subspace $\mathbb{Y}$ of $\ell_1^n$, we completely identify the elements $x$ in $\ell_1^n,$ for which best coapproximations to $x$ out of $\mathbb{Y}$ exist. The methods developed in this article are computationally effective and it allows us to present an algorithmic approach to the concerned problem. We also identify the coproximinal subspaces and co-Chebyshev subspaces of $\ell_1^n$.

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arXiv2024

On k-smoothness of operators between Banach spaces

We explore the $k$-smoothness of bounded linear operators between Banach spaces, using the newly introduced notion of index of smoothness. The characterization of the $k$-smoothness of operators between Hilbert spaces follows as a direct consequence of our study. We also investigate the $k$-smoothness of operators between polyhedral Banach spaces. In particular, we show that the $k$-smoothness of rank $1$ operators between polyhedral spaces depends heavily on the dimension of the corresponding spaces rather than the geometry of the spaces. The results obtained in this article generalize and improve upon the existing results in the $k$-smoothness of operators between Banach spaces.

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arXiv2024

On isosceles orthogonality and some geometric constants in a normed space

We study the James constant $J(\mathbb{X})$, an important geometric quantity associated with a normed space $ \mathbb{X} $, and explore its connection with isosceles orthogonality $ \perp_I. $ The James constant is defined as $J(\mathbb{X}) := \sup\{\min \{\|x+y\|, \|x-y\|\}: x, y \in \mathbb{X},~ \|x\|=\|y\|=1 \}.$ We prove that if $J(\mathbb{X})$ is attained for unit vectors $x, y \in \mathbb{X},$ then $x\perp_I y.$ We also show that if $\mathbb{X}$ is a two-dimensional polyhedral Banach space then $J(\mathbb{X})$ is always attained at an extreme point $z$ of the unit ball of $\mathbb{X},$ so that $J(\mathbb{X}) = \|z+y\| = \|z-y\|,$ where $ \| y \| = 1 $ and $z\perp_I y.$ This helps us to explicitly compute the James constant of a two-dimensional polyhedral Banach space in an efficient way. We further study some related problems with reference to several other geometric constants in a normed space.

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arXiv2024

On some special subspaces of a Banach space, from the perspective of best coapproximation

We study the best coapproximation problem in Banach spaces, by using Birkhoff-James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets-Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.

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arXiv2024

Orthogonality of bilinear forms and application to matrices

We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to present an elementary proof of the well-known Bhatia-Šemrl Theorem in the real case.

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arXiv2024

On local preservation of orthogonality and its application to isometries

We investigate the local preservation of Birkhoff-James orthogonality at a point by a linear operator on a finite-dimensional Banach space and illustrate its importance in understanding the action of the operator in terms of the geometry of the concerned spaces. In particular, it is shown that such a study is related to the preservation of k-smoothness and the extremal properties of the unit ball of a Banach space. As an application of the results obtained in this direction, we obtain a refinement of the well-known Blanco-Koldobsky-Turnsek characterization of isometries on some polyhedral Banach spaces, including $ \ell_{\infty}^n, \ell_1^n. $

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arXiv2024

On uniform Bishop-Phelps-Bollobás type approximations of linear operators and preservation of geometric properties

We study uniform $ε-$BPB approximations of bounded linear operators between Banach spaces from a geometric perspective. We show that for sufficiently small positive values of $ε,$ many geometric properties like smoothness, norm attainment and extremality of operators are preserved under such approximations. We present examples of pairs of Banach spaces satisfying non-trivial norm preserving uniform $ε-$BPB approximation property in the global sense. We also study these concepts in case of bounded linear operators between Hilbert spaces. Our approach in the present article leads to the improvement and generalization of some earlier results in this context.

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arXiv2024

Extreme contractions on finite-dimensional Banach spaces

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein-Milman Theorem, we prove that a \emph{rank one} norm one linear operator between such spaces can be expressed as a convex combination of \emph{rank one} extreme contractions, whenever the domain is two-dimensional. We establish that the same result holds true in the space of all linear operators from $\ell_{\infty}^n(\mathbb{C}) $ to $ \ell_1^n (\mathbb{C}). $ Furthermore, we present a geometric characterization of extreme contractions between finite-dimensional polyhedral Banach spaces.

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arXiv2024

On T-orthogonality in Banach spaces

Let $\mathbb{X}$ be a Banach space and let $\mathbb{X}^*$ be the dual space of $\mathbb{X}.$ For $x,y \in \mathbb{X},$ $ x$ is said to be $T$-orthogonal to $y$ if $Tx(y) =0,$ where $T$ is a bounded linear operator from $\mathbb{X}$ to $\mathbb{X}^*.$ We study the notion of $T$-orthogonality in a Banach space and investigate its relation with the various geometric properties, like strict convexity, smoothness, reflexivity of the space. We explore the notions of left and right symmetric elements w.r.t. the notion of $T$-orthogonality. We characterize bounded linear operators on $\mathbb{X}$ preserving $T$-orthogonality. Finally we characterize Hilbert spaces among all Banach spaces using $T$-orthogonality. \end{abstract}

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arXiv2024

A numerical range approach to Birkhoff-James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff-James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia-Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of $Z$. As a consequence, if $X$ is a Banach space containing strongly exposed points and $Y$ is a smooth Banach space with dimension at least two, then there are no spear operators from $X$ to $Y$. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

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arXiv2022

Geometry of nowhere vanishing, point separating sub-algebras of $\mathcal{H}ol(Γ\cup\text{Int}(Γ))$ and zeros of Holomorphic functions

We study $ \mathcal{H}ol(Γ\cup\text{Int}(Γ)) $, the normed algebra of all holomorphic functions defined on some simply connected neighbourhood of a simple closed curve $Γ$ in $\mathbb{C} $, equipped with the supremum norm on $ Γ$. We explore the geometry of nowhere vanishing, point separating sub-algebras of $ \mathcal{H}ol(Γ\cup \text{Int}(Γ)) $. We characterize the extreme points and the exposed points of the unit balls of the said sub-algebras for $Γ$ analytic. We also characterize the smoothness of an element in these sub-algebras by using Birkhoff-James orthogonality techniques without any restriction on $Γ$. As a culmination of our study, we assimilate the geometry of the aforesaid sub-algebras with some classical concepts of complex analysis and establish a connection between Birkhoff-James orthogonality and zeros of holomorphic functions.

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arXiv2022

Approximate smoothness in normed linear spaces

We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.

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arXiv2022

On subspaces of $\ell_\infty$ and extreme contraction in $\mathbb{L}(\mathbb{X}, \ell_{\infty}^n)$

We investigate different possiblities of subspaces of the space $\ell_{\infty}$ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of $\ell_{\infty}$ which are of the form $\ell_\infty^n$ form some $ n \geq 2.$ As an application of the results we compute the number of extreme contractions for a class of the space of bounded linear operators. In particular we find the number of extreme contractions of $\mathbb{L}(\mathbb{X}, \ell_{\infty}^n),$ where $\mathbb{X}$ is a finite-dimensional polyhedral space.

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arXiv2022

Birkhoff-James Orthogonality and Its Local Symmetry in Some Sequence Spaces

We study Birkhoff-James orthogonality and its local symmetry in some sequence spaces namely $\ell_p,$ for $1\leq p\leq\infty$, $p\neq2$, $c$, $c_0$ and $c_{00}$. Using the characterization of the local symmetry of Birkhoff-James orthogonality, we characterize isometries of each of these spaces onto itself and obtain the Banach-Lamperti theorem for onto operators on the sequence spaces.

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