On (A, m)-isometric commuting tuples of operators on a Hilbert space.

In this paper, we consider a generalization of (A , m) -isometric Hilbert space operators to the multivariable setting. Inspired by the work [Sid Ahmed OAM, Chō M, Lee JE. On (m,C)-isometric commuting tuples of operators on a Hilbert space. Res Math. 2018;73:51. Doi:], we introduce the class of (A , m) -isometric tuples of commuting operators. A d-tuple T = (T 1 , ... , T d) ∈ L (H) d is said to be an (A , m) -isometric tuple of operators if ∑ k = 0 m (− 1) m − k m k ∑ | α | = k k ! α ! T ∗ α A T α = 0 for some positive integer m and some positive operator A. We study some basic properties of these tuples of commuting operators which generalize those established in Gu [Exapmles of m-isometric tuples of operators on a Hilbert space. J Korean Math Soc. 2018;55(1):225–251], Gleason and Richter [m-Isometric commuting tuples of operators on a Hilbert space. Int Equ Oper Theory. 2006;56(2):181–196], and Sid Ahmed and Saddi [A-m-Isometric operators in semi-Hilbertian spaces. Linear Algebra Appl. 2012;436(10): 3930–3942]. [ABSTRACT FROM AUTHOR]