arXiv subjects

Kallol Paul

Publications and source records attributed to Kallol Paul.

At least 37 records · Page 2Linked to original sources

arXiv2025

A unified approach to a family of optimization problems in Banach spaces

Our principal aim is to illustrate that the concept Birkhoff-James orthogonality can be applied effectively to obtain a unified approach to a large family of optimization problems in Banach spaces. We study such optimization problems from the perspective of Birkhoff-James orthogonality in certain suitable Banach spaces. In particular, we demonstrate the duality between the Fermat-Torricelli problem and the Chebyshev center problem which are important particular cases of the least square problem. We revisit the Fermat-Torricelli problem for three and four points and solve it using the same technique. We also investigate the behavior of the Fermat-Torricelli points under the addition or replacement of a new point, and present several new results involving the locations of the Fermat-Torricelli point and the Chebyshev center.

math.FA↗

arXiv2025

On approximate preservation of orthogonality and its application to isometries

Motivated by the famous Blanco-Koldobsky-Turnšek characterization of isometries, we study the \textit{approximate preservation of Birkhoff-James orthogonality by a linear operator between Banach spaces}. In particular, we investigate various geometric and analytic properties related to such preservation on finite-dimensional polyhedral Banach spaces. As an application of the results obtained here, we present refinements of the Blanco-Koldobsky-Turnšek characterization of isometries on certain Banach spaces.

math.FA↗

arXiv2025

On directional preservation of orthogonality and its application to isometries

We study the local preservation of Birkhoff-James orthogonality by linear operators between normed linear spaces, at a point and in a particular direction. We obtain a complete characterization of the same, which allows us to present refinements of the local preservation of orthogonality explored earlier. We also study the directional preservation of orthogonality with respect to certain special subspaces of the domain space, and apply the results towards identifying the isometries on a polyhedral normed linear space. In particular, we obtain refinements of the Blanco-Koldobsky-Turnšek Theorem for polyhedral normed linear spaces, including $ \ell_{\infty}^{n}, \ell_{1}^{n}. $

math.FA↗

arXiv2024

Orthogonality induced by norm derivatives : A new geometric constant and symmetry

In this article we study the difference between orthogonality induced by the norm derivatives (known as $ρ$-orthogonality) and Birkhoff-James orthogonality in a normed linear space $ \mathbb X$ by introducing a new geometric constant, denoted by $Γ(\mathbb{X}).$ We explore the relation between various geometric properties of the space and the constant $Γ(\mathbb{X}).$ We also investigate the left symmetric and right symmetric elements of a normed linear space with respect to $ρ$-orthogonality and obtain a characterization of the same. We characterize inner product spaces among normed linear spaces using the symmetricity of $ρ$-orthogonality. Finally, we provide a complete description of both left symmetric and right symmetric elements with respect to $ρ$-orthogonality for some particular Banach spaces.

math.FA↗

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} $.

math.FA↗

arXiv2024

$A$-Davis-Wielandt Radius Bounds of Semi-Hilbertian Space Operators

Consider $\mathcal{H}$ is a complex Hilbert space and $A$ is a positive operator on $\mathcal{H}.$ The mapping $\langle\cdot,\cdot\rangle_A: \mathcal{H}\times \mathcal{H} \to \mathbb {C}$, defined as $\left\langle y,z\right\rangle_{A}=\left\langle Ay,z\right\rangle $ for all $y,z$ $\in $ ${\mathcal{H}}$, induces a seminorm $ \left\Vert \cdot\right\Vert_{A}$. The $A$-Davis-Wielandt radius of an operator $S$ on $\mathcal{H}$ is defined as $dω_{A}\left( S\right) =\sup \left\{ \sqrt{\left\vert \left\langle Sz,z\right\rangle_{A}\right\vert ^{2}+\left\Vert Sz\right\Vert_{A}^{4}} :\left\Vert z\right\Vert_{A}=1\right\} \text{.} $ We investigate some new bounds for $dω_{A}\left( S\right)$ which refine the existing bounds. We also give some bounds for the $2\times 2$ off-diagonal block matrices.

math.FA↗

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 $\ell_1^n$.

math.FA↗

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.

math.FA↗

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.

math.FA↗

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.

math.FA↗

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. $

math.FA↗

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.

math.FA↗

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.

math.FA↗

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 note on the $A$-numerical range of semi-Hilbertian operators

In this paper we explore the relation between the $A$-numerical range and the $A$-spectrum of $A$-bounded operators in the setting of semi-Hilbertian structure. We introduce a new definition of $A$-normal operator and prove that closure of the $A$-numerical range of an $A$-normal operator is the convex hull of the $A$-spectrum. We further prove Anderson's theorem for the sum of $A$-normal and $A$-compact operators which improves and generalizes the existing result on Anderson's theorem for $A$-compact operators. Finally we introduce strongly $A$-numerically closed class of operators and along with other results prove that the class of $A$-normal operators is strongly $A$-numerically closed.

math.FA↗

arXiv2023

Euclidean operator radius and numerical radius inequalities

Let $T$ be a bounded linear operator on a complex Hilbert space $\mathscr{H}.$ We obtain various lower and upper bounds for the numerical radius of $T$ by developing the Euclidean operator radius bounds of a pair of operators, which are stronger than the existing ones. In particular, we develop an inequality that improves on the inequality $$ w(T) \geq \frac12 {\|T\|}+\frac14 {\left|\|Re(T)\|-\frac12 \|T\| \right|} + \frac14 { \left| \|Im(T)\|-\frac12 \|T\| \right|}.$$ Various equality conditions of the existing numerical radius inequalities are also provided. Further, we study the numerical radius inequalities of $2\times 2$ off-diagonal operator matrices. Applying the numerical radius bounds of operator matrices, we develop the upper bounds of $w(T)$ by using $t$-Aluthge transform. In particular, we improve the well known inequality $$ w(T) \leq \frac12 {\|T\|}+ \frac12{ w(\widetilde{T})}, $$ where $\widetilde{T}=|T|^{1/2}U|T|^{1/2}$ is the Aluthge transform of $T$ and $T=U|T|$ is the polar decomposition of $T$.

math.FA↗

arXiv2023

Estimations of Euclidean operator radius

We develop several Euclidean operator radius bounds for the product of two $d$-tuple operators using positivity criteria of a $2\times 2$ block matrix whose entries are $d$-tuple operators. From these bounds, by using the polar decomposition of operators, we obtain Euclidean operator radius bounds for $d$-tuple operators. Among many other interesting bounds, it is shown that \begin{eqnarray*} w_e(\mathbf{A}) &\leq&\frac1{\sqrt2} \mathbf{A}\|^{1/2}\sqrt{\left\|\sum_{k=1}^{d} (|A_k|+|A_k^*|)\right\|}, \end{eqnarray*} where $w_e(\mathbf{A})$ and $\|\mathbf{A}\|$ are the Euclidean operator radius and the Euclidean operator norm, respectively, of a $d$-tuple operator $\mathbf{A}=(A_1,A_2, \ldots,A_d).$ Further, we develop an upper bound for the Euclidean operator radius of $n\times n$ operator matrix whose entries are $d$-tuple operators. In particular, it is proved that if $\begin{bmatrix} \mathbf{A_{ij}} \end{bmatrix}_{n\times n}$ is an $n\times n$ operator matrix then $$ w_e\left( \begin{bmatrix} \mathbf{A_{ij}} \end{bmatrix}_{n\times n}\right)\leq w \left(\begin{bmatrix} a_{ij} \end{bmatrix}_{n\times n}\right),$$ where each $\mathbf{A_{ij}}$ is a $d$-tuple operator, $1\leq i,j\leq n$, $a_{ij}=w_e(\mathbf{A_{ij}})\, \textit{ if i=j}$, $a_{ij}= \sqrt{w_e\left(|\mathbf{A_{ji}|}+|\mathbf{A_{ij}^*}|\right)w_e\left(|\mathbf{A_{ij}|}+|\mathbf{A_{ji}^*}|\right)}\,\textit{ if $i j$}.$ Other related applications are also discussed.

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