arXiv subjects

Kallol Paul

Publications and source records attributed to Kallol Paul.

114 records · Page 7Linked to original sources

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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arXiv2015

Characterization of inner product spaces

We prove that the existence of best coapproximation to any element of the normed linear space out of any one dimensional subspace and its coincidence with the best approximation to that element out of that subspace characterizes a real inner product space of dimension $ > 2 $. We conjecture that a finite dimensional real smooth normed space of dimension $ >2 $ is an inner product space iff given any element on the unit sphere there exists a strongly orthonormal Hamel basis in the sense of Birkhoff-James containing that element. This is substantiated by our result on the spaces $(R^n,\|.\|_p).$

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arXiv2014

Strictly convex space : Strong orthogonality and Conjugate diameters

In a normed linear space X an element x is said to be orthogonal to another element y in the sense of Birkhoff-James, written as $ x \perp_{B}y, $ iff $ \| x \| \leq \| x + λy \| $ for all scalars $ λ.$ We prove that a normed linear space X is strictly convex iff for any two elements x, y of the unit sphere $ S_X$, $ x \perp_{B}y $ implies $ \| x + λy \| > 1~ \forall~ λ\neq 0. $ We apply this result to find a necessary and sufficient condition for a Hamel basis to be a strongly orthonormal Hamel basis in the sense of Birkhoff-James in a finite dimensional real strictly convex space X. Applying the result we give an estimation for lower bounds of $ \| tx+(1-t)y\|, t \in [0,1] $ and $ \| y + λx \|, ~\forall ~λ$ for all elements $ x,y \in S_X $ with $ x \perp_B y. $ We find a necessary and sufficient condition for the existence of conjugate diameters through the points $ e_1,e_2 \in ~S_X $ in a real strictly convex space of dimension 2. The concept of generalized conjuagte diameters is then developed for a real strictly convex smooth space of finite dimension.

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arXiv2014

On rectangular constant in normed linear spaces

We study the properties of rectangular constant $ μ(\mathbb{X}) $ in a normed linear space $\mathbb{X}$. We prove that $ μ(\mathbb{X}) = 3$ iff the unit sphere contains a straight line segment of length 2. In fact, we prove that the rectangular modulus attains its upper bound iff the unit sphere contains a straight line segment of length 2. We prove that if the dimension of the space $\mathbb{X}$ is finite then $μ(\mathbb{X})$ is attained. We also prove that a normed linear space is an inner product space iff we have sup$\{\frac{1+|t|}{\|y+tx\|}$: $x,y \in S_{\mathbb{X}}$ with $x\bot_By\} \leq \sqrt{2}$ $\forall t$ satisfying $|t|\in (3-2\sqrt{2},\sqrt{2}+1)$.

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arXiv2013

Computation of antieigenvalues of bounded linear operators via centre of mass

We introduce the concept of theta-antieigenvalue and theta-antieigenvector of a bounded linear operator on complex Hilbert space. We study the relation between theta-antieigenvalue and centre of mass of a bounded linear operator and compute antieigenvalue using the relation. This follows the notion of symmetric antieigenvalues introduced by Hossein et al. in \cite{19}. We show that the concept of real antieigenvalue, imaginary antieigenvalue and symmetric antieigenvalue follows as a special case of theta-antieigenvalue. We also show how the concept of total antieigenvalue is related to the $θ$-antieigenvalue. In fact, we show that all the concepts of antieigenvalues studied so far follows from the concept of theta-antieigenvalue. We illustrate with example how to calculate the $θ$-antieigenvalue for an operator acting on a finite dimensional Hilbert space.

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arXiv2010

Cosine of angle and center of mass of an operator

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.

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