Products of m-isometries

Abstract

An operator T on a Banach space X is called anIn this paper we prove that if T is an (n, p)-isometry, S is an (m, p)-isometry and they commute, then TS is an (m + n − 1, p)-isometry.This result applied to elementary operators of length 1 defined on the Hilbert-Schmidt class proves a conjecture in [11].

Tensor product of left n-invertible operators

“…Furthermore, if T 1 is strictly essentially left (resp., right) m-invertible, then T 1 ⊗ T 2 is: (i) essentially left (resp., right) (m + n − 1)-invertible if and only if T 2 is essentially left (resp., right) n-invertible; (ii) strictly essentially left (resp., right) (m + n − 1)-invertible if and only if T 2 is strictly essentially left (resp., right) n-invertible. This generalizes some results of Botelho et al [7,8], Bermúdez et al [6,5], and those of one of the authors on the tensor product of m-isometric operators [9,10,11]. We remark that these results have a natural interpretation for the left-right multiplication operator ST …”

“…A limited version of Theorem 2.15 has been considered by Sid Ahmed [17, Theorems 3.1 and 3.2], and versions of the theorem for m-isometric operators on the ideal C 2 (H) of Hilbert-Schmidt class operators have been considered in [6,7,8,9,10].…”

“…An interesting example of a left m-invertible Hilbert space operator is that of an m-isometric operator T for which m i=0 (−1) i ( m i )T * m−i T m−i = 0, where T * denotes the Hilbert space adjoint of T . A study of m-isometric operators has been carried out by Agler and Stankus in a series of papers [1,2,3]; more recently a generalization of these operators to Banach spaces has been obtained by Bayart [4], Bermúdez et al [6,5] and Hoffmann et al [13].…”

“…A proof of the theorem may be obtained using a combinatorial argument similar to that in the papers [9,10], or by using an argument similar to the one used to prove [11, Corollary 2.2] (see also [6]). However, we follow here an argument using double sequences satisfying certain properties.…”

Structures of left n-invertible operators and their applications

“…The concept of left n-invertible operators is motivated by the m-isometries studied earlier in [2]- [6], [24] on Hilbert spaces and more recently in [9], [11]- [13], [15], [26] on Hilbert spaces and [7], [8], [10], [16], [22] on Banach spaces. An operator T on a Hilbert space is an n-isometry if β n (T * , T ) = 0, that is, T * is a left n-inverse of T.…”