arXiv subjects

Chaman Kumar Sahu

Publications and source records attributed to Chaman Kumar Sahu.

9 recordsLinked to original sources

arXiv2026

Multiplicative linear functionals on reproducing kernel Hilbert spaces

The classical Gleason--Kahane--Zelazko theorem characterizes multiplicative linear functionals on a unital Banach algebra through the scalar identity $\Lambda(x^{2})=\Lambda(x)^{2}$. We develop analogues of this theorem for bounded linear functionals on reproducing kernel Hilbert spaces of holomorphic functions on domains in $\mathbb{C}^{d}$, replacing conditions on the whole space by tractable conditions involving only kernel functions. Our first main result shows that if $k$ is a diagonal holomorphic kernel on a domain $\Omega\subseteq\mathbb{C}^{d}$ containing the origin, and if $k_{w}^{2}\in\mathcal H(k)$ for every $w\in\Omega$, then a bounded linear functional $\Lambda$ on $\mathcal H(k)$ satisfying $\Lambda(1)=1$ is multiplicative if and only if $\Lambda(k_{w}^{2})=\Lambda(k_{w})^{2}$ for all $w\in\Omega$. Kernels satisfying $2$-point Pick property and their powers furnish a natural class of examples. When $k$ arises as a Schur product or a tensor product of complete diagonal Pick kernels, we obtain a further, more algebraic characterization of multiplicativity, expressed in terms of the values of $\Lambda$ on kernel functions and their reciprocals. This framework subsumes the weighted Bergman and Dirichlet-type spaces, as well as the Hardy space on the polydisc. We extend the analysis to Hilbert spaces associated with diagonal Dirichlet series kernels on half-planes, encompassing in particular the Hardy space of Dirichlet series and its Riemann zeta reproducing kernel. Explicit examples demonstrate that the boundedness hypothesis on $\Lambda$ cannot be omitted. Finally, our characterization of multiplicative linear functionals leads to characterizations of weighted composition operators on a reproducing kernel Hilbert space associated with a diagonal holomorphic kernel.

math.FA↗

arXiv2026

Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces

Motivated by the recent developments of de Branges-Rovnyak spaces, we investigate the function theoretic aspects of finite rank de Branges-Rovnyak spaces $H(B)$ generated by row-valued Schur functions $B$. We provide a generalization of Sarason's fundamental work by characterizing finite rank $H(B)$-spaces as the domain of the adjoint of the Toeplitz operators $T_φ^*$ with symbol $φ= BA^{-1}$, where $A$ is an matrix-valued outer function satisfying $A^*A+B^*B = I$ a.e. on the unit circle. We derive a norm formula for functions in $H(B)$-space and provide a concrete realization of this norm in terms of the Taylor coefficients of the function and the symbol $φ$. As an application, we characterize all symbols $B$ for which $H^\infty \subseteq H(B)$ in terms of the boundary behavior of $I-BB^*$, thereby extending Sarason's criterion for the classical de Branges-Rovnyak spaces.

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arXiv2026

Multiplier varieties and multiplier algebras of CNP Dirichlet series kernels

We investigate isometric and algebraic isomorphism problems for multiplier algebras associated with Dirichlet series kernels that possess the complete Nevanlinna-Pick (CNP) property. A central aspect of our work is the explicit determination of the multiplier variety associated with each CNP Dirichlet series kernel, via polynomial equations derived from the arithmetic structure of the associated weight and frequency data. This description of multiplier varieties enables us to classify when the multiplier algebras of a signifincant class of CNP Dirichlet series kernels are isomorphic, or isometrically isomorphic. In this setting, a striking rigidity phenomenon emerges whereby the multiplier algebra determines the kernel up to natural equivalence. The results established for CNP Dirichlet series kernels also extend to classical CNP kernels, yielding new results for the associated multiplier algebras even in the classical setting. As an application, we resolve an open problem posed by McCarthy and Shalit ([19]).

math.FA↗

arXiv2026

Helson matrices induced by measures

We discuss the boundedness, Schatten-class properties and scattering theory of Helson matrices. We also discuss a class of Helson matrices induced by positive and signed measures. All the results of this paper are illustrated with several examples not considered earlier.

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arXiv2025

Nevanlinna-Pick interpolation in the right half-plane

The Szegö-Dirichlet kernel of the right half-plane $\mathbb H_{1/2}$ is given by ${\varkappa}(s, u) = ζ(s+\overline{u}),$ $s, u \in \mathbb H_{1/2},$ where $ζ$ denotes the Riemann zeta function. We show that none of the positive integer powers of $\varkappa$ has $2$-point scalar Pick property. Nevertheless, a network realization formula for the right half-plane $\mathbb H_0$ is obtained.

math.FA↗

arXiv2025

On the abscissas of a Dirichlet series and its subseries supported on prime factorization

For a sequence $\{a_n\}_{n \geq 1} \subseteq (0, \infty)$ and a Dirichlet series $f(s) = \sum_{n=1}^\infty a_n n^{-s},$ let $\sigma_a(f)$ denote the abscissa of absolute convergence of $f$ and let \begin{equation} \delta_a(f): = \inf\Bigg\{\Re(s) : \sum\limits_{\substack{j= 1 \\ \tiny{\mbox{gpf}}(j) \leq p_n }}^\infty a_j j^{-\Re(s)} < \infty ~\text{for all}~ n \geq 1\Bigg\}, \end{equation} where $\{p_j\}_{j \geq 1}$ is an increasing enumeration of prime numbers and $\text{\bf gpf}(n)$ denotes the greatest prime factor of an integer $n \geq 2.$ One significant aspect of these abscissas is their crucial role in analyzing the multiplier algebra of Hilbert spaces associated with diagonal Dirichlet series kernels. The main result of this paper establishes that $\sigma_a(f)- \delta_a(f)$ can be made arbitrarily large, meaning that it can be equal to any non-negative real number. As an application, we determine the multiplier algebra in some cases and, in others, gain insights into the structure of the multiplier algebra of certain Hilbert spaces of Dirichlet series.

math.NT↗

arXiv2023

Multipliers of the Hilbert spaces of Dirichlet series

For a sequence $\mathbf w = \{w_j\}_{j = 2}^\infty$ of positive real numbers, consider the positive semi-definite kernel $κ_{\mathbf w}(s, u) = \sum_{j = 2}^\infty w_j j^{-s - \overline{u}}$ defined on some right-half plane $\mathbb H_ρ$ for a real number $ρ.$ Let $\mathscr H_{\mathbf w}$ denote the reproducing kernel Hilbert space associated with $κ_\mathbf w.$ Let \begin{equation*} δ_{\mathbf w} = \inf\Bigg\{\Re(s) : \sum\limits_{\substack{j \geqslant 2 \\ \tiny{\textbf{gpf}}(j) \leqslant p_n }} w_j j^{- s} < \infty ~\text{for all}~ n \in \mathbb Z_+\Bigg\}, \end{equation*} where $\{p_j\}_{j \geqslant 1}$ is an increasing enumeration of prime numbers and $\textbf{gpf}(n)$ denotes the greatest prime factor of an integer $n \geqslant 2.$ If $\mathbf w$ satisfies \begin{equation*} \sum_{\substack{j \geqslant 2\\ j | n}} j^{-δ_\mathbf w} w_j μ\Big(\frac{n}{j}\Big) \geqslant 0,\quad n \geqslant 2, \end{equation*} where $μ$ is the M$\ddot{\mbox{o}}$bius function, then the multiplier algebra $\mathcal M(\mathscr H_\mathbf w)$ of $\mathscr H_\mathbf w$ is isometrically isomorphic to the space of all bounded and holomorphic functions on $\mathbb H_\frac{δ_{\mathbf w}}{2}$ that are representable by a convergent Dirichlet series in some right half plane. As a consequence, we describe the multiplier algebra $\mathcal M(\mathscr H_\mathbf w)$ when $\mathbf w$ is an additive function satisfying $δ_{\mathbf w} \leqslant 0$ and \begin{align*} \frac{w_{p^{j-1}}}{w_{p^j}} \leqslant p^{-δ_{\mathbf w}}~\text{for all integers} ~~ j \geqslant 2~\mbox{and all prime numbers}~p. \end{align*} Moreover, we recover a result of Stetler that classifies the multipliers of $\mathscr H_\mathbf w$ when $\mathbf w$ is multiplicative. The proof of the main result is a refinement of the techniques of Stetler.

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arXiv2022

Quasi-Invariance of the Dirichlet series kernels, Analytic symbols and Homogeneous operators

For a scalar matrix $\mathbf a=(a_{m, n})_{m, n=1}^{\infty},$ the Dirichlet series kernel $κ_{\mathbf a}$ is the double Dirichlet series $κ_{\mathbf a}(s, u) = \sum_{m, n =1}^{\infty} a_{m, n}m^{-s} n^{-\bar{u}}$ in the variables $s$ and $\bar{u},$ which is regularly convergent on some right half-plane $\mathbb H_ρ.$ The analytic symbols $A_{n, \mathbf a} = \sum_{m=1}^{\infty}a_{m, n}m^{-s},$ $n \geq 1$ play a central role in the study of the reproducing kernel Hilbert space $\mathscr H_{\mathbf a}$ associated with the positive semi-definite kernel $κ_{\mathbf a}.$ In particular, they form a total subset of $\mathscr H_{\mathbf a}$ and provide the formula $\sum_{n=1}^{\infty}\langle f, A_{n, \mathbf a} \rangle n^{-s},$ $s \in \mathbb H_ρ,$ for $f \in \mathscr H_{\mathbf a}.$ We combine the basic theory of Dirichlet series kernels with the Gelfond-Schneider theorem (Hilbert's seventh problem) to show that any quasi-invariant Dirichlet series kernel $κ_{\mathbf a}(s, u)$ factors as $f(s)\bar{f(u)}$ for some Dirichlet series $f$ on $\mathbb H_ρ.$ In particular, there is no quasi-invariant Dirichlet series kernel $κ_{\mathbf a}$ if the dimension of $\mathscr H_{\mathbf a}$ is bigger than one. This is in strict contrast with the case of the unit disc, where non-factorable quasi-invariant kernels exist in abundance. We further discuss the Dirichlet series kernels $κ_{\mathbf a}$ invariant under the group $\mathscr T$ of translation automorphisms of $\mathbb H_ρ$ and construct a family of densely defined $\mathscr T$-homogeneous operators in $\mathscr H_{\mathbf a},$ whose adjoints are defined only at the zero vector.

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arXiv2021

Dirichlet polynomials and a moment problem

Consider a linear functional $L$ defined on the space $\mathcal D[s]$ of Dirichlet polynomials with real coefficients and the set $\mathcal D_+[s]$ of non-negative elements in $\mathcal D[s].$ An analogue of the Riesz-Haviland theorem in this context asks: What are all $\mathcal D_+[s]$-positive linear functionals $L,$ which are moment functionals? Since the space $\mathcal D[s],$ when considered as a subspace of $C([0, \infty), \mathbb R),$ fails to be an adapted space in the sense of Choquet, the general form of Riesz-Haviland theorem is not applicable in this situation. In an attempt to answer the forgoing question, we arrive at the notion of a moment sequence, which we call the Hausdorff log-moment sequence. Apart from an analogue of the Riesz-Haviland theorem, we show that any Hausdorff log-moment sequence is a linear combination of $\{1, 0, \ldots, \}$ and $\{f(\log(n)\}_{n \geqslant 1}$ for a completely monotone function $f : [0, \infty) \rightarrow [0, \infty).$ Moreover, such an $f$ is uniquely determined by the sequence in question.

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