Bohr operator on opertor valued polyanalytic functions on simply connected domains

TL;DR

This paper investigates the Bohr operator for operator-valued polyanalytic functions on simply connected domains, establishing subordination results, a von Neumann-type inequality, and Bohr inequalities with specific radius estimates.

Contribution

It extends scalar-valued subordination and Bohr inequalities to operator-valued polyanalytic functions, providing new results and bounds in this operator-theoretic setting.

Findings

Established subordination results for operator-valued functions.

Derived a von Neumann-type inequality for self-analytic mappings.

Obtained Bohr radius estimates for operator-valued polyanalytic functions.

Abstract

In this article, we study the Bohr operator for the operator valued subordination class $S(f)$ consisting of holomorphic functions subordinate to $f$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}$, where $f:\mathbb{D} \rightarrow \mathcal{B}(\mathcal{H})$ is holomorphic and $\mathcal{B}(\mathcal{H})$ is the algebra of bounded linear operators on a complex Hilbert space $\mathcal{H}$. We establish several subordination results, which can be viewed as the analogues of a couple of interesting subordination results from scalar valued settings. We also obtain a von Neumann-type inequality for the class of self-analytic mappings of the unit disk $\mathbb{D}$ which fix the origin. Furthermore, we extensively study Bohr inequalities for operator valued polyanalytic functions in certain proper simply connected domains in $\mathbb{C}$. We obtain Bohr radius for the operator valued polyanalytic functions of the form $F(z)= \sum_{l=0}^{p-1} \overline{z}^l \, f_{l}(z) $, where $f_{0}$ is subordinate to an operator valued convex biholomorphic function, and operator valued starlike biholomorphic function in the unit disk $\mathbb{D}$.