Powers of m-isometries

Abstract

A bounded linear operator T on a Banach space X is called an (m, p)isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X,We prove that any power of an (m, p)-isometry is also an (m, p)-isometry. In general the converse is not true. However, we prove that if T r and T r+1 are (m, p)-isometries for a positive integer r, then T is an (m, p)-isometry. More precisely, if T r is an (m, p)isometry and T s is an (l, p)-isometry, then T t is an (h, p)-isometry, where t = gcd(r, s) and h = min(m, l).2010 Mathematics Subject Classification: Primary 47B99.

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 …”

“…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].…”

Products of m-isometries

“…For example, if T = S is an m-isometry, then T 2 is an m-isometry [7]. So the following question is natural: Is Theorem 4.2 optimal?…”

“…Patel in [24, Theorem 2.1] proves that any power of a 2-isometry is again a 2-isometry. Recently, in [7] was proved that if T is an m-isometry then any power T r is also an m-isometry. In general the converse is not true; however, in [7] it is proved that if the power T r is an m-isometry and the power T s is an l-isometry, then T t is an h-isometry, where t is the greatest common divisor of r and s, and h is the minimum of m and l.…”

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.…”