arXiv subjects

Soumitra Daptari

Publications and source records attributed to Soumitra Daptari.

6 recordsLinked to original sources

arXiv2025

On relationship among three types of Birkhoff-James orthogonality

In this paper, we study three types of Birkhoff-James orthogonality in Hilbert $C^*$-modules, that is, the strong, quasi-strong, and original Birkhoff-James orthogonality. In general, the strong Birkhoff-James orthogonality is stronger than the quasi-strong Birkhoff-James orthogonality, and the quasi-strong Birkhoff-James orthogonality is stronger than the original Birkhoff-James orthogonality. Meanwhile, each reverse implication in this chain requires additional conditions. As the main results, we show that the strong and quasi-strong Birkhoff-James orthogonality are equivalent in a full Hilbert $C^*$-module if and only if the underlying $C^*$-algebra is commutative, and that the equivalence of the quasi-strong and original Birkhoff-James orthogonality in a full Hilbert $C^*$-module implies the primeness of the underlying $C^*$-algebra. Moreover, two examples, explaining the complexity of conditions for full Hilbert $C^*$-modules in which the quasi-strong and original Birkhoff-James orthogonality are equivalent, are given in the $C^*$-algebra settings.

math.FA↗

arXiv2025

A study on state spaces in classical Banach spaces

Let $X$ be a real or complex Banach space. Let $S(X)$ denote the unit sphere of $X$. For $x\in S(X)$, let $S_{x}=\{x^*\in S(X^*):x^*(x)=1\}$. A lot of Banach space geometry can be determined by the `quantum' of the state space $S_{x}$. In this paper, we mainly study the norm compactness and weak compactness of the state space in the space of Bochner integrable function and $c_{0}$-direct sums of Banach spaces. Suppose $X$ is such that $X^*$ is separable and let $\mu$ be the Lebesgue measure on $[0,1]$. For $f\in L^1(\mu,X)$, we demonstrate that if $S_{f}$ is norm compact, then $f$ is a smooth point. When $\mu$ is the discrete measure, we show that if $ (x_i) \in S(\ell^{1}(X))$ and $ \|x_{i}\|\neq 0$ for all $i\in{\mathbb{N}}$, then $ S_{(x_i)}$ is weakly compact in $ \ell^\infty(X^*) $ if and only if $ S_{\frac{x_i}{\|x_i\|}} $ is weakly compact in $X^*$ for each $i\in{\mathbb{N}}$ and $\text{diam}\left(S_{\frac{x_i}{\|x_i\|}}\right) \to 0 $. For discrete $c_{0}$-sums, we show that for $(x_{i})\in c_{0}(X)$, $S_{(x_{i})}$ is weakly compact if and only if for each $i_{0}\in \mathbb{N}$ such that $\|x_{i_{0}}\|=1$, the state space $S_{x_{i_{0}}}$ is weakly compact.

math.FA↗

arXiv2024

On Hahn-Banach smoothness and related properties in Banach spaces

In this paper, we study several variants of Hahn-Banach smoothness, viz., property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed that $L_1(μ)$ has property-$(wU)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ but it does not have property-$(U)$ in $L_1(μ,(\mathbb{R}^2,\|.\|_2))$ for a non-atomic measure $μ$. We derive a sufficient condition when property-$(wU)$ is equivalent to property-$(U)$ of a subspace. It is observed that these properties are separably determined. Finally, finite-dimensional and finite co-dimensional subspaces of $c_0$, $\ell_p$ ($1\leq p<\infty$) having these properties are characterized.

math.FA↗

arXiv2023

New $\mathbb{A}$-numerical radius equalities and inequalities for certain operator matrices and applications

The main goal of this article is to establish several new $\mathbb{A}$-numerical radius equalities and inequalities for $n\times n$ cross-diagonal, left circulant, skew left circulant operator matrices, where $\mathbb{A}$ is the $n\times n$ diagonal operator matrix whose diagonal entries are positive bounded operator $A$. Also, we introduce two new matrices called left imaginary circulant operator matrix and left imaginary skew circulant operator matrix and present their $\mathbb{A}$-numerical radii. Certain $\mathbb{A}$-numerical radii of general $n\times n$ operator matrices are obtained. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Applications of our results are established through some interesting examples. We also provide a concluding section by posing a problem for future research.

math.FA↗

arXiv2022

Uniqueness of Hahn--Banach extensions and some of its variants

In this study, we analyze the various strengthening and weakening of the uniqueness of the Hahn--Banach extension. In addition, we consider the case in which $Y$ is an ideal of $X$. In this context, we study the property-$(U)/ (SU)/ (HB)$ and property-$(k-U)$ for a subspace $Y$ of a Banach space $X$. We obtain various new characterizations of these properties. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is observed that a hyperplane in $c_0$ has property-$(HB)$ if and only if it is an $M$-summand. Considering $X, Z$ as Banach spaces and $Y$ as a subspace of $Z$, by identifying $(X\widehat{\otimes}_πY)^*\cong \mathcal{L}(X,Y^*)$, we observe that an isometry in $\mathcal{L}(X,Y^*)$ has a unique norm-preserving extension over $(X\widehat{\otimes}_πZ)$ if $Y$ has property-$(SU)$ in $Z$. It is observed that a finite dimensional subspace $Y$ of $c_0$ has property-$(k-U)$ in $c_0$, and if $Y$ is an ideal, then $Y^*$ is a $k$-strictly convex subspace of $\ell_1$ for some natural $k$.

math.FA↗

arXiv2020

Uniqueness of Hahn-Banach extension and related norm-$1$ projections in dual spaces

In this paper we study two properties viz. property-$U$ and property-$SU$ of a subspace $Y$ of a Banach space which correspond to the uniqueness of the Hahn-Banach extension of each linear functional in $Y^*$ and in addition to that this association forms a linear operator of norm-1 from $Y^*$ to $X^*$. It is proved that, under certain geometric assumptions on $X, Y, Z$ these properties are stable with respect to the injective tensor product; $Y$ has property-$U$ ($SU$) in $Z$ if and only if $X\otimes_\e^\vee Y$ has property-$U$ ($SU$) in $X\otimes_\e^\vee Z$. We prove that when $X^*$ has the Radon-Nikod$\acute{y}$m Property for $1 3$ is a Hilbert space if and only if for any two subspaces $Y, Z$ with property-$SU$ in $X$, $Y+Z$ has property-$SU$ in $X$ whenever $Y+Z$ is closed. We characterize all hyperplanes in $c_0$ which have property-$SU$.

math.FA↗