Formula omitted]-isometries and their harmonious applications to Hilbert-Schmidt operators

Abstract

Numerous works have been dedicated to the topic of m-isometries, including [2–6,14,18–20,27,47,48]. In this article, we introduce the concept of (m,NA)-isometry, where A is a non-zero operator and m is a positive integer, as an extension of the m-isometry class created by J. Alger and M. Stankus in the 1980s. We present some algebraic and spectral characteristics of (m,NA)-isometries. Additionally, we investigate the product of an (m,NA)-isometry by an (n,NB)-isometry, which enhances and broadens the previous work of Gu et al. on m-isometries [40]. Finally, we apply our main findings to elementary operators defined on the Hilbert-Schmidt class, which can be identified with a tensor product. This provides a new, less complicated, and non-combinatorial proof of Theorem 2.10 of [30].