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Susmita Seal

Publications and source records attributed to Susmita Seal.

14 recordsLinked to original sources
arXiv2026

Bijections on the set of extreme points in a compact convex set

In a recent work, Roelands and Tiersma proved that, for a compact convex set $K$, the space $A(K)$ of all real-valued continuous affine functions on $K$, is a JB-algebra if and only if there is a gauge-reversing bijection on $A_c(K)$, the set of positive real-valued continuous affine functions on $K$. In this paper, we show that every such gauge-reversing bijection on $A_c(K)$ is completely determined by the induced bijection on the set of extreme points of $K$.

math.FA↗
arXiv2026

Semi denting points and related notions in Banach spaces

In this work, we study semi denting points and related notions in Banach spaces. We observe that $X$ has the Radon-Nikod\'ym Property if and only if every closed bounded convex set has a semi denting point. We also study the stability properties of semi denting, semi PC, and semi SCS points, as well as their $w^*$-analogues in Banach spaces, with respect to $l_p$-sums ( $1\leq p \leq \infty$), ideals, and projective tensor products.

math.FA↗
arXiv2025

Subdifferential of the $\mathcal{B(H,K)}$ norm, and approximate orthogonality

We present an expression for the right hand derivative of the $\mathcal{B(H,K)}$ norm generalizing the result for $\mathcal{K}=\mathcal{H}$ in [D. J. Ke$\check{\mathrm{c}}$ki$\grave{\mathrm{c}}$, Gateaux derivative of $B(H)$ norm, Proc. Amer. Math. Soc. 133 (2005): 2061--2067]. Using this, we obtain the subdifferential of the $\mathcal{B(H, K)}$ norm. For tuples of operators $\mathbf{A},\mathbf{X}\in$ $\mathcal{B(H, H}^d)$, we give a characterization for $\boldsymbol 0$ to be a best approximation to the subspace $\mathbb C^d \mathbf{X}$, generalizing a similar result for $\mathbb C^d \mathbf{I}$ in [P. Grover, S. Singla, A distance formula for tuples of operators, Linear Algebra Appl. 650 (2022): 267--285]. We define the concept of $\epsilon$-Birkhoff orthogonality to a subspace in a general normed space and derive a characterization in terms of the subdifferential set. Using this, we deduce interesting results for $A\in \mathcal{B(H,K)}$ to be $\epsilon$-Birkhoff orthogonal to a subspace of $\mathcal{B(H,K)}$, when $A$ is compact.

math.FA↗
arXiv2025

Geometric characterization of Generalized Property (II)

Let $\mathcal{A}$ be a compatible collection of bounded subsets of a Banach space $X$. In this paper, we introduce the notion of $\mathcal{A}$-Property (II) and prove that $X$ has $\mathcal{A}$-Property (II) if and only if every $f$ in $S_{X^*}$ is $\mathcal{A}$-semi PC of $B_{X^*}$.

math.FA↗
arXiv2025

Geometry of the unit ball of ${\mathcal L}(X,Y^*)$

In this work we study the geometry of the unit ball of the space of operators ${\mathcal L}(X,Y^*)$, by considering the projective tensor product $X\hat{\otimes}_{\pi} Y$ as a predual. We prove that if an elementary tensor (rank one operator) of the form $x_0^*\otimes y_0^* $ in the unit sphere $ S_{{\mathcal L}(X,Y^*)}$ is a weak$^*$-strongly extreme point of the unit ball, then $x_0^*$ is weak$^*$-strongly extreme point of unit ball of $X^*$ and $y_0^*$ is weak$^*$-strongly extreme point of the unit ball of $Y^*$. We show that a similar conclusion holds if the rank one operator is a Namioka point (equivalently, point of weak$^*$-weak continuity for the identity mapping) on the unit sphere of ${\mathcal L}(X,Y^*)$. We also study extremal phenomenon in the unit ball of ${\mathcal L}(X,Y^*)^*$. We partly solve the open problem, when does an elementary tensor, whose components are Namioka points is again a Namioka point? We show that if a point $z\in S_{{\mathcal L}(X,Y^*)^*}$ is a weak$^*$-strongly extreme point of the unit ball, then $z=x\otimes y$ for some weak$^*$-strongly extreme points $x\in S_X$ and $y\in S_Y$, provided the space of compact operators, $\mathcal{K}(X,Y^*)$ is separating for $X\hat{\otimes}_{\pi} Y$.

math.FA↗
arXiv2023

Two aspects of small diameter properties

In this short note, we study two different geometrical aspects of Banach spaces with small diameter properties, namely the Ball Dentable Property (BDP), Ball Huskable Property (BHP) and Ball Small Combination of slice Property (BSCSP). We show that BDP, BHP and BSCSP are separably determined properties. We also explore the stability of these properties over Kothe Bochner spaces.

math.FA↗
arXiv2023

Ball separation characterization of ball dentability and related properties

In Euclidean spaces, every closed, bounded, convex set can be characterized by two equivalent notions of separation properties. This is not true in general for arbitrary Banach spaces. In this work, we present a ball separation characterization for spaces where the unit ball is dentable. We also explore related properties.

math.FA↗
arXiv2023

Nonrough norms in spaces with small diameter

In this work, we study the non rough norms in Banach spaces with small diameter properties, namely the Ball Dentable Property (BDP), the Ball Huskable Property (BHP) and the Ball Small Combination of Slice Property (BSCSP). We introduce two more notions of non rough norms, namely the weakly average non rough norms and average non rough norms in Banach spaces. We prove the duality between these three versions of non rough norms and the small diameter properties in Banach spaces. We also prove that each of the three non rough norms is a three space property under certain assumptions.

math.FA↗
arXiv2022

Non-rough norms and dentability in spaces of operators

In this work, we study non-rough norms in L(X,Y), the space of bounded linear operators between Banach spaces X and Y. We prove that L(X,Y) has non-rough norm if and only if X* and Y have non-rough norm. We show that the injective tensor product of X and Y has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of L(X,Y)*. These results leads to a discussion on stability of the small diameter properties for projective and injective tensor product spaces.

math.FA↗
arXiv2021

Small Diameter Properties In Ideals of Banach Spaces

A Banach space has the ball huskable property ($BHP$) if the closed unit ball has weakly open sets of arbitrarily small diameter. We can analogously define $w^*$-$BHP$ in the dual space. In this short note, we study these properties in the context of ideals in Banach spaces. The notion of an ideal, was introduced by Godefroy, Kalton and Saphar. We show that if a Banach space $X$ has $BHP,$ then any $M$-ideal of $X$ also has $BHP.$ We further show that if $Y$ is an $M$-ideal of $X,$ then $Y^*$ has $w^*$-$BHP$ implies $X^*$ has $w^{*}$-$BHP.$ We use this result to prove that for a compact Hausdorff space $K$ which has an isolated point, $X$ has $BHP$ whenever $C(K,X)$ has $BHP$ and $X^*$ has $w^{*}$-$BHP$ implies $C(K,X)^*$ has $w^{*}$-$BHP.$ We also prove that $w^*$-$BHP$ can be lifted from $Y^*$ to $X^*$ provided $Y$ is a strict ideal of $X$. Lastly, we show that if $Y$ is an almost isometric ideal of $X,$ then $BHP$ can be lifted from $Y$ to $X.$ We obtain similar results for ball dentable property ($BDP$) and ball small combination of slices Property ($BSCSP$) as well.

math.FA↗
arXiv2021

Small Combination of Slices, Dentability and Stability Results Of Small Diameter Properties In Banach Spaces

In this work we study three different versions of small diameter properties of the unit ball in a Banach space and its dual. The related concepts for all closed bounded convex sets of a Banach space was initiated and developed in \cite{B3}, \cite{BR} ,\cite{EW}, \cite{GM} was extensively studied in the context of dentability, huskability, Radon Nikodym Property and Krein Milman Property in \cite{GGMS}. We introduce the the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related properties, $BSCSP$ namely, the unit ball has convex combination of slices of arbitrarily small diameter and $BDP$ namely, the closed unit ball has slices of arbitrarily small diameter. We show $BDP$ implies $BHP$ which in turn implies $BSCSP$ and none of the implications can be reversed. We prove similar results for the $w^*$-versions. We prove that all these properties are stable under $l_p$ sum for $1\leq p \leq \infty, c_0$ sum and Lebesgue Bochner spaces. Finally, we explore the stability of these with properties in the light of three space property. We show that $BHP$ is a three space property provided $X/Y$ is finite dimensional and same is true for $BSCSP$ when $X$ has $BSCSP$ and $X/Y$ is strongly regular (\cite{GGMS}).

math.FA↗
arXiv2021

Stability Results Of Small Diameter Properties In Banach Spaces

The geometric notion of huskability initiated and developed in [B3], [BR] ,[EW], [GM] was subsequently extensively studied in the context of dentability and Radon Nikodym Property in [GGMS]. In this work, we introduce a new geometric property of Banach space, the Ball Huskable Property ($BHP$), namely, the unit ball has relatively weakly open subsets of arbitrarily small diameter. We compare this property to two related geometric properties, $BSCSP$ namely, the unit ball has convex combination of slices of arbitrarily small diameter and $BDP$ namely, the closed unit ball has slices of arbitrarily small diameter. We show $BDP$ implies $BHP$ which in turn implies $BSCSP$ and none of the implications can be reversed. We prove similar results for the $w^*$-versions. We prove that all these properties are stable under $l_p$ sum for $1\leq p \leq \infty$. These stability results lead to a discussion in the context of ideals of Banach spaces. We prove that $BSCSP$ (respectively $BHP$, $BDP$) can be lifted from an M-Ideal to the whole space. We also show similar results for strict ideals. We note that the space $C(K,X)^*$ has $w^*$-$BSCSP$ (respectively $w^*$-$BHP$, $w^*$-$BDP$) when K is dispersed and $X^*$has the $w^*$-$BSCSP$ (respectivley $w^*$-$BHP$, $w^*$-$BDP$).

math.FA↗
arXiv2020

Characterization of $M$-compact sets via statistically convergent sequences

In this paper, we study stability of $M$-compactness for $l^p$ sum of Banach spaces for $1\leq p<\infty$. We also obtain a characterization of $M$-compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of $\mathcal{I}$-$M$-compactness of a bounded subset $M$ of a normed linear space $X$ with respect to an ideal $\mathcal{I}$ and show that it is equivalent to $M$-compactness for non-trivial admissible ideals.

math.FA↗