Abstract
Inspired by recent works on (m, C)-isometric and [m, C]-isometric operators on Hilbert spaces studied respectively in Cho et al. (Complex Anal. Oper. Theory 10:1679-1694, 2016; Filomat 31:7, 2017), in this paper we introduce the class of (m, C)-isometries for tuple of commuting operators. This is a generalization of the class of (m, C)-isometric operators. A commuting tuples of operators T = (T-1, . . . ,T-d) is an element of B-(d)(H) is said to be (m, C)-isometric tuple if
Q(m)(T) : = Sigma(0 <= k <= m)(-1)(m-k)((m)(k)) (Sigma(vertical bar beta vertical bar=k) k!/beta!(TCTC)-C-*beta-C-beta) = 0
for some positive integer m and some conjugation C. We consider a multi-variable generalization of these single variable (m, C)-isometric operators and explore some of their basic properties.