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:10.1007/s00025-018-0810-0], we introduce the class of $(A,m)$-isometric tuples of commuting operators. A d-tuple $T=(T_1,\dots ,T_d)\in \mathcal{L}(\mathcal{H}{)}^d$ is said to be an $(A,m)$-isometric tuple of operators if $\sum _{k=0}^m(-1{)}^{m-k}\left(\genfrac{}{}{0ex}{}{m}{k}\right)\sum _{|\alpha |=k}\frac{k!}{\alpha !}{T}^{\ast \alpha }A{T}^{\alpha }=0$ 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].

在本文中，我们考虑一个泛化$(\mathrm{一个},米)$-多变量设置的等距希尔伯特空间算子。受作品启发 [Sid Ahmed OAM, Cho M, Lee JE. 在希尔伯特空间上的 (m,C)-等距交换元组上的算子。水库数学。2018 年；73:51。Doi:10.1007/s00025-018-0810-0]，我们介绍一下类$(\mathrm{一个},米)$- 通勤运算符的等距元组。一个d元组$吨=({吨}_1,\dots ,{吨}_d)\in \mathcal{大号}(\mathcal{H}{)}^d$据说是一个$(\mathrm{一个},米)$- 等距运算符元组 if$\sum _{ķ=0}^{米}(-1{)}^{米-ķ}\left(\genfrac{}{}{0ex}{}{米}{ķ}\right)\sum _{|\alpha |=ķ}\frac{ķ！}{\alpha ！}{吨}^{*\alpha }\mathrm{一个}{吨}^{\alpha }=0$对于一些正整数m和一些正运算符A。我们研究了这些交换算子元组的一些基本性质，这些性质概括了在顾[希尔伯特空间上算子的m-等距元组的例子]中建立的那些。J 韩国数学学会 2018;55(1):225–251]，Gleason 和 Richter [m-Isometric commuting tuples of operator on a Hilbert space。Int Equ 操作理论。2006;56(2):181–196]，以及 Sid Ahmed 和 Saddi [半希尔伯特空间中的 Am-Isometric 算子。线性代数应用程序。2012;436(10): 3930–3942]。