On the divergence of series of p-th powers of operator norms

TL;DR

This paper investigates the conditions under which the series of p-th powers of operator norms diverges for a given system of operators, providing insights into the behavior of such operator series.

Contribution

It introduces new criteria for the divergence of p-th power norm series in operator systems, advancing understanding of operator norm behavior.

Findings

Identifies conditions leading to divergence of p-th power norm series.

Provides theoretical criteria for the existence of elements with infinite series sums.

Enhances understanding of operator norm series divergence in functional analysis.

Abstract

Let {A} be a system of operators. With any element x we associate the set of elements {Ax}. We study conditions under which there exists an element x such that the sum of p-th powers of norms of the elements {Ax} is equal to infinity.