arXiv subjects

Rounak Biswas

Publications and source records attributed to Rounak Biswas.

5 recordsLinked to original sources

arXiv2026

Finite-rank commutators of projections onto shift-invariant subspaces

In this article, using Halmos' two projections theorem, we completely characterize finite-rank commutators of orthogonal projections onto shift-invariant subspaces $ϕ_1 H^2(\mathbb{D})$ and $ϕ_2 H^2(\mathbb{D})$ of the Hardy space $H^2(\mathbb{D})$ corresponding to inner functions $ϕ_1, ϕ_2$ on the unit disc $\mathbb{D}$. Using our methods, we connect the finite-rank commutator with the Fredholmness of the projections $(P_{ϕ_1}, P_{ϕ_2})$ as introduced by Avron, Seiler and Simon. We conclude with several characterizations on the polydisc.

math.FA↗

arXiv2025

The generalized and pseudo $n$-strong Drazin inverse of the sum of elements in Banach algebras

In this paper, we begin by introducing some necessary and sufficient conditions for generalized $n$-strong Drazin invertibility (g$n$s-invertibility) and pseudo $n$-strong Drazin invertibility (p$n$s-invertibility) of an element in a Banach algebra for $n\in\mathbb{N}$. Subsequently, these results are utilized to prove some additive properties of g$n$s (p$n$s)-Drazin inverse in a Banach algebra. This process produces a generalization of some recent results of H Chen, M Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for g$n$s and p$n$s-Drazin inverse. Furthermore, we define and characterize weighted g$n$s and weighted p$n$s-Drazin inverse in a Banach algebra.

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arXiv2024

Drazin and group invertibility in algebras spanned by two idempotents

For two given idempotents $p\text{ and }q$ from an associative algebra $\mathcal{A},$ in this paper, we offer a comprehensive classification of algebras spanned by the idempotents $p\text{ and }q$. This classification is based on the condition that $p\text{ and }q$ are not tightly coupled and satisfies $(pq)^{m-1}=(pq)^{m}$ but $(pq)^{m-2}p\neq (pq)^{m-1}p$ for some $m(\geq2)\in\mathbb{N}.$ Subsequently, we categorized all the group invertible elements and established an upper bound for Drazin index of any elements in these algebras spanned by $p,q$. Moreover, we formulate a new representation for the Drazin inverse of $(αp+q)$ under two different assumptions, $(pq)^{m-1}=(pq)^m$ and $λ(pq)^{m-1}=(pq)^m,$ here $α$ is a non-zero and $λ$ is a non-unit real or complex number.

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arXiv2024

Drazin and g-Drazin invertibility of combinations of three Banach algebra elements

Consider a complex unital Banach algebra $\mathcal{A}.$ For $x_1,x_2,x_3\in\mathcal{A},$ in this paper, we establish that under certain assumptions on $x_1,x_2,x_3$, Drazin (resp. g-Drazin) invertibility of any three elements among $x_1,x_2,x_3$ and $x_1+x_2+x_3\text{ }(\text{or }x_1x_2+x_1x_3+x_2x_3)$ ensure the Drazin (resp. g-Drazin) invertibility of the remaining one. As a consequence for two idempotents $p,q\in\mathcal{A},$ this result indicates the equivalence between Drazin (resp. g-Drazin) invertibility of $$λ_1p+γ_1q-λ_1pq+λ_2\left(pqp-(pq)^2\right)+\cdots+λ_m\left((pq)^{m-1}p-(pq)^m\right)$$ and $$λ_1-λ_1pq+λ_2\left(pqp-(pq)^2\right)+\cdots+λ_m\left((pq)^{m-1}p-(pq)^m\right),$$ where $γ_1,λ_i\in\mathbb{C}$ for $i=1,2,\cdots,m,$ with $λ_1γ_1\neq0.$ Furthermore, for $x_1,x_2$, we establish that the Drazin (resp. g-Drazin) invertibility of any two elements among $x_1,x_2$ and $x_1+x_2$ indicates the Drazin (resp. g-Drazin) invertibility of the remaining one, provided that $x_1x_2=α(x_1+x_2)$ for some $α\in\mathbb{C}$. Additionally, if it exists, we furnish a new formula to represent the Drazin (resp. g-Drazin) inverse of any element among $x_1,x_2$ and $x_1+x_2$, by using the other two elements and their Drazin (resp. g-Drazin) inverse.

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arXiv2023

Additive and multiplicative properties of Drazin inverse under new weakly commutativity condition

Given a complex Banach space $X$, let $\mathcal{B}(X)$ be the collection of all bounded linear operators on $X.$ For $A,B\in\mathcal{B}(X)$ we define $A,B$ are $A$-weakly commutative if there exists $C\in\mathcal{B}(X)$ satisfying $$AB=CA\text{ and }BA=AC.$$ The objective of this paper is to study the Drazin invertibility of $A+B$ and $AB$, when $A, B\in\mathcal{B}(X)^D$ are $A\text{ and }B$-weakly commutative. Consequently, this extends some of the additive and multiplicative results established by Huanyin and Marjan Sheibani (Linear and Multilinear Algebra \textbf{70.1} (2022): 53-65) for a distinct family of elements. Moreover, we also establish these additive and multiplicative properties for g-Drazin inverses in a complex Banach algebra.

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