Some $A$-spectral radius inequalities for $A$-bounded Hilbert space operators

TL;DR

This paper investigates inequalities related to the $A$-spectral radius of operators on Hilbert spaces, providing bounds for products, sums, and series of $A$-bounded operators under certain conditions.

Contribution

It establishes new inequalities for the $A$-spectral radius of sums, products, and series of $A$-bounded operators, extending existing spectral radius bounds.

Findings

Derived bounds for the $A$-spectral radius of operator products and sums.

Established an inequality for the $A$-spectral radius of operator series.

Provided conditions under which the inequalities hold.

Abstract

Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is bounded with respect to the seminorm induced by a positive operator $A$ on a complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some $A$-spectral radius inequalities for products, sums and commutators of $A$-bounded operators. Moreover, under suitable conditions on $T$ and $A$ we show that \begin{equation*} r_A\left( \sum_{k=0}^{+\infty}c_{k}T^{k}\right) \leq \sum_{k=0}^{+\infty}|c_{k}|\left[r_A(T)\right]^{k}, \end{equation*} where $c_k$ are complex numbers for all $k\in \mathbb{N}$.