arXiv subjects

Sreejith Siju

Publications and source records attributed to Sreejith Siju.

4 recordsLinked to original sources

arXiv2026

Differentiability of the operator norm on $\ell_p$ spaces

In this paper, we present a characterization of strong subdifferentiability of the norm of bounded linear operators on $\ell_p$ spaces, $1\leq p<\infty$. Furthermore, we prove that the set of all bounded linear operators in ${B}(\ell_p, \ell_q)$ for which the norm of ${B}(\ell_p, \ell_q)$ is strongly subdifferentiable is dense in ${B}(\ell_p, \ell_q)$. Additionally, we present a characterization of Frechet differentiability of the norm of bounded linear operators from $\ell_p$ to $\ell_q$, where $1 < p, q < \infty$. Applying this result, we will show that the Frechet differentiability and the Gateaux differentiability of the norm of bounded linear operators on $\ell_p$ spaces coincide, extending a known theorem regarding the operator norm on Hilbert spaces.

math.FA↗

arXiv2026

Ball Covering Property on Operators and Calkin Algebra

A Banach space $X$ is said to have the ball covering property (BCP) if the unit sphere of $X$ can be covered by countably many open balls $B(x_i, r_i)$ with $r_i\leq \|x_i\|$ for each $i\in\mathbb{N}$. If there are $R, \delta>0$ so that $r_i\leq R$ and $\|x_i\|-r_i>\delta$ for all $i\in\mathbb{N}$, then we say that $X$ has the uniform ball covering property (UBCP). In this paper, we show that if $X$ has an $1$-unconditional basis or $X$ is an $1$-complemented subspace of a Banach space with a shrinking $1$-unconditional basis, then the Calkin algebra $\mathcal{B}(X)/\mathcal{K}(X)$ fails the BCP. It is also shown that if $X$ has a shrinking unconditional basis with unconditional constant less than 2, then $\mathcal{B}(X)$ has the UBCP.

math.FA↗

arXiv2024

Analytic primes, $M$-ideals, and $p$-sets in $H^\infty(\mathbb{D})$

We investigate the structure of $p$-sets, $M$-ideals, and a newly introduced notion of analytic primes in $H^\infty(\mathbb{D})$, where $H^\infty(\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. We prove that $M$-ideals in $H^\infty(\mathbb{D})$ are analytic primes and are dense in the Hardy space. Outer functions play a key role in representing closed principal ideals in $H^\infty(\mathbb{D})$ that are $M$-ideals. Some of our results apply to the polydisc. The results presented in this paper offer some new perspectives on $H^\infty(\mathbb{D})$.

math.FA↗

arXiv2020

Ball proximinality of $M$-ideals of compact operators

In this article, we prove the proximinality of closed unit ball of $M$-ideals of compact operators. We also prove the ball proximinality of $M$-embedded spaces in their biduals. Moreover, we show that $\mathcal{K}(\ell_1)$, the space of compact operators on $\ell_1$, is ball proximinal in $\mathcal{B}(\ell_1)$, the space of bounded operators on $\ell_1$, even though $\mathcal{K}(\ell_1)$ is not an $M$-ideal in $\mathcal{B}(\ell_1)$.

math.FA↗