arXiv subjects

Prakash A. Dabhi

Publications and source records attributed to Prakash A. Dabhi.

9 recordsLinked to original sources

arXiv2026

The BSE Property for Weighted Banach Algebras of Dirichlet Series

Let $ω: \mathbb{N} \to [1, \infty)$ be a weight on the multiplicative semigroup of natural numbers. We investigate the Bochner-Schoenberg-Eberlein (BSE) property for the weighted Banach algebra of Dirichlet series $A_ω= \ell^1(\mathbb{N}, ω)$ under Dirichlet convolution. By employing Bohr's correspondence, we establish a weak-$\ast$ approximation framework for $ω$-bounded semicharacters using scaled truncations that lie within the predual space $c_0(\mathbb{N}, 1/ω)$. Utilizing the canonical $L$-projection decomposition of the bidual space $A_ω^{\ast\ast} = A_ω\oplus c_0(\mathbb{N}, 1/ω)^\perp$ alongside the weak-$\ast$ compactness of the unit ball via the Banach-Alaoglu theorem, we prove that $A_ω$ is a BSE-algebra for any weight. Furthermore, we show that the intrinsic BSE-norm on the Gel'fand transforms is isometric to the natural weighted algebra norm $\|\cdot\|_ω$.

math.FA↗

arXiv2026

On Multiplicity of Uniform Norms and Maximal Spectral Substructures in Commutative Banach Algebras

Let $\mathcal A$ be a semisimple commutative Banach algebra. It is shown that either $\mathcal A$ has exactly one uniform norm or it admits uncountably many uniform norms. Further, it is shown that there always exists a largest closed subalgebra of $\mathcal A$ which is weakly regular, and that there always exist largest closed ideals in $\mathcal A$ having unique uniform norm property (UUNP) and spectral extension property (SEP) respectively.

math.FA↗

arXiv2026

Inverse-closedness of weighted Schur and BGS type quasi-Banach algebras

We prove that the weighted quasi-Banach algebras of operator valued matrices satisfying Schur and Baskakov-Gohberg-Sjöstrand (BGS) conditions are inverse-closed in the Banach algebra $B(\ell^2(X,\mathcal{H}))$ whenever the weight is admissible, where $\mathcal{H}$ is a Hilbert space and $X$ is a relatively separated subset of $\mathbb{R}^d$. Furthermore, we identify the Gel'fand space of weighted infinite variable group algebra $\ell^p_ω(\mathbb{Z^N})$ for $0<p\leq1$, and establish inverse-closedness of infinite variable analogue of BGS-type algebra in $B(\ell^2(\mathbb{Z^N},\mathcal{H}))$.

math.FA↗

arXiv2026

Wiener type theorems for countable limits of quasi-Beurling algebras and maximizing results on weights

We establish the vector-valued Wiener type theorems for countable projective and inductive limits of quasi-Banach algebras in a weighted setting for both finite and infinite dimensional cases. As an application, we extend the notions of rapidly decreasing and exponentially decreasing sequence spaces using quasi-Beurling algebras and show that they are inverse-closed; and obtain a hierarchy of inverse-closed vector-valued algebras using weights. In addition, we derive maximizing results on weights for the nonadmissible weighted version of Wiener's theorem in both discrete and continuous cases.

math.FA↗

arXiv2026

On the BSE and BED properties of the Beurling algebra $L^1(G,ω)$

Let $G$ be a locally compact abelian group, and let $ω:G \to [1,\infty)$ be a weight, i.e., $ω$ is measurable, $ω$ is locally bounded and $ω(s+t)\leq ω(s)ω(t)$ for all $s, t \in G$. If $ω^{-1}$ is vanishing at infinity, then we show that the Beurling algebra $L^1(G,ω)$ is both BSE- algebra and BED- algebra.

math.FA↗

arXiv2025

Weighted inversion of vector valued Dirichlet series

Let $Λ\subset[0,\infty)$ be an additive semigroup with $0\inΛ$, $ω$ be an admissible weight on $Λ$, $\mathcal A$ be a unital Banach algebra, and let $f(s)=\sum_{λ\inΛ} f_λe^{-λs}$ for $s\in\mathcal{H}=\{j+it\in\mathbb{C}:j\geq0\}$ be a generalized Dirichlet series satisfying $\|f\|_ω=\sum_{λ\inΛ}\|f_λ\|ω(λ)<\infty,$ where $f_λ\in\mathcal{A}$ for all $λ\inΛ$. We take $\mathcal{A}$ to be a commutative complex Banach algebra (with $Λ=\log\mathbb{N}$) and $M_d(\mathcal{X})$ - the Banach algebra of $d \times d$ matrices having entries from $\mathcal{X}$, where $\mathcal{X}$ is either the complex plane or the real algebra of bicomplex numbers or quaternions, and show that $f$ is invertible if and only if the closure of the image of $f$ is contained in the set of all invertible elements of $\mathcal{A}$.

math.FA↗

arXiv2024

On inversion of absolutely convergent weighted Dirichlet series in two variables

Let $0<p\leq 1$, and let $ω:\mathbb N^2 \to [1,\infty)$ be an almost monotone weight. Let $\mathbb H$ be the closed right half plane in the complex plane. Let $\widetilde a$ be a complex valued function on $\mathbb H^2$ such that $\widetilde a(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ with $\sum_{(m,n)\in \mathbb N^2} |a(m,n)|^pω(m,n)<\infty$. If $\widetilde a$ is bounded away from zero on $\mathbb H^2$, then there is an almost monotone weight $ν$ on $\mathbb N^2$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $ν$ is admissible if and only if $ω$ is admissible, the reciprocal $\frac{1}{\widetilde a}$ has the Dirichlet representation $\frac{1}{\widetilde a}(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\in \mathbb H^2$ and $\sum_{(m,n)\in \mathbb N^2}|b(m,n)|^pν(m,n)<\infty$. If $φ$ is holomorphic on a neighbourhood of the closure of range of $\widetilde a$, then there is an almost monotone weight $ξ$ on $\mathbb N^2$ such that $1\leq ξ\leq ω$, $ξ$ is constant if and only if $ω$ is constant, $ξ$ is admissible if and only if $ω$ is admissible, $φ\circ \widetilde a$ has the Dirichlet series representation $(φ\circ \widetilde a)(s_1,s_2)=\sum_{(m,n)\in \mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\;((s_1,s_2)\in \mathbb H^2)$ and $\sum_{(m,n)\in \mathbb N^2}|c(m,n)|^pξ(m,n)<\infty$. Let $ω$ be an admissible weight on $\mathbb N^2$, and let $\widetilde a$ have $p$-th power $ω$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\widetilde a$ has $p$-th power $ω$- absolutely convergent Dirichlet series if and only if $\widetilde a$ is bounded away from zero.

math.FA↗

arXiv2022

Vector valued Beurling algebra analogues of Wiener's Theorem

Let $0<p\leq 1$, $ω$ be a weight on $\mathbb Z$, and let $\mathcal A$ be a unital Banach algebra. If $f$ is a continuous function from the unit circle $\mathbb T$ to $\mathcal A$ such that $\sum_{n\in \mathbb Z} \|\widehat f(n)\|^p ω(n)^p<\infty$ and $f(z)$ is left invertible for all $z \in \mathbb T$, then there is a weight $ν$ on $\mathbb Z$ and a continuous function $g:\mathbb T \to \mathcal A$ such that $1\leq ν\leq ω$, $ν$ is constant if and only if $ω$ is constant, $g$ is a left inverse of $f$ and $\sum_{n\in \mathbb Z}\|\widehat g(n)\|^pν(n)^p<\infty$. We shall obtain a similar result when $ω$ is an almost monotone algebra weight and $1<p<\infty$. We shall obtain an analogue of this result on the real line. We shall apply these results to obtain $p-$power weighted analogues of the results of off diagonal decay of infinite matrices of operators.

math.FA↗

arXiv2022

Symmetry and inverse-closedness of some $p$-Beurling algebras

Let $(G,d)$ be a metric space with the counting measure $μ$ satisfying some growth conditions. Let $ω(x,y)=(1+d(x,y))^δ$ for some $0<δ\leq1$. Let $0<p\leq1$. Let $\mathcal A_{pω}$ be the collection of kernels $K$ on $G\times G$ satisfying $\max\{\sup_x\sum_y |K(x,y)|^pω(x,y)^p, \sup_y\sum_x |K(x,y)|^pω(x,y)^p\}<\infty$. Each $K \in \mathcal A_{pω}$ defines a bounded linear operator on $\ell^2(G)$. If in addition, $ω$ satisfies the weak growth condition, then we show that $\mathcal A_{pω}$ is inverse closed in $B(\ell^2(G))$. We shall also discuss inverse-closedness of $p$-Banach algebra of infinite matrices over $\mathbb Z^d$ and the $p$-Banach algebra of weighted $p$-summable sequences over $\mathbb Z^{2d}$ with the twisted convolution. In order to show these results, we prove Hulanicki's lemma and Barnes' lemma for $p$-Banach algebras.

math.FA↗