arXiv subjects

Jaikishan

Publications and source records attributed to Jaikishan.

4 recordsLinked to original sources

arXiv2026

A Gleason-Kahane-\.Zelazko Theorem for $H^p_{\alpha, \beta}$ spaces

We study two entities that have proved to be of interest and importance in their own right viz. the $H^p_{\alpha, \beta}$ spaces and the classical Gleason-Kahane-\.Zelazko (GKZ) theorem. We establish a GKZ-type theorem on the $H^p_{\alpha, \beta}$ spaces by identifying a natural class of functions serving as the counterpart of invertible elements, and proving that every continuous linear functional nonvanishing on this class is a point evaluation. As an application, we characterize weighted composition operators on these spaces. It must be noted that the $H^p_{\alpha, \beta}$ spaces do not possess the various structural advantages of the classical Hardy spaces where the GKZ theorem already exists and thus we have had to modify and establish methods that rely on suitable extensions and refinements of the known techniques. Additionally, we provide examples that illustrate the natural class of functions arising in our GKZ-type theorem and demonstrate the sharpness of our characterization of weighted composition operators under the assumption of surjectivity.

math.FA↗

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↗

arXiv2024

A several variables Kowalski-S\lodkowski theorem for topological spaces

In this paper, we provide a version of the classical result of Kowalski and S\l odkowski that generalizes the famous Gleason-Kahane-$\dot{\rm Z}$elazko (GKZ) theorem by characterizing multiplicative linear functionals amongst all complex-valued functions on a Banach algebra. We first characterize maps on $\mathcal{A}$-valued polynomials of several variables that satisfy some conditions, motivated by the result of Kowalski and S\l odkowski, as a composition of a multiplicative linear functional on $\mathcal{A}$ and a point evaluation on the polynomials, where $\mathcal{A}$ is a complex Banach algebra with identity. We then apply it to prove an analogue of Kowalski and S\l odkowski's result on topological spaces of vector-valued functions of several variables. These results extend our previous work from \cite{jaikishan2024multiplicativity}; however, the techniques used differ from those used in \cite{jaikishan2024multiplicativity}. Furthermore, we characterize weighted composition operators between Hardy spaces over the polydisc amongst the continuous functions between them. Additionally, we register a partial but noteworthy success toward a multiplicative GKZ theorem for Hardy spaces.

math.FA↗

arXiv2023

Multiplicativity of linear functionals on function spaces on an open unit disc

This paper presents a fairly general version of the well-known Gleason-Kahane-$\dot{\text{Z}}$elazko (GKZ) theorem in the spirit of a GKZ type theorem obtained recently by Mashreghi and Ransford for Hardy spaces. In effect, we characterize a class of linear functionals as point evaluations on the vector space of all complex polynomials $\cl P$. We do not make any topological assumptions on $\cl P$. We then apply this characterization to present a version of the GKZ theorem for a vast class of topological spaces of complex-valued functions including the Hardy, Bergman, Dirichlet, and many more well-known function spaces. We obtain this result under the assumption of continuity of the linear functional, which we show, with the help of an example, to be a necessary condition for the desired conclusion. Lastly, we use the GKZ theorem for polynomials to obtain a version of the GKZ theorem for strictly cyclic weighted Hardy spaces.

math.FA↗