m-Isometric Commuting Tuples of Operators on a Hilbert Space

Abstract

We consider a generalization of isometric Hilbert space operators to the multivariable setting. We study some of the basic properties of these tuples of commuting operators and we explore several examples. In particular, we show that the d-shift, which is important in the dilation theory of d-contractions (or row contractions), is a d-isometry. As an application of our techniques we prove a theorem about cyclic vectors in certain spaces of analytic functions that are properly contained in the Hardy space of the unit ball of C d .

The 3-Isometric Lifting Theorem

“…In [2] the connection between Jordan operators and 3-symmetric operators was established in general. A sample of the more recent articles related to 3-isometries includes [3][4][5][6][7][8][9][10][11]15,16,18,20].…”

Products of m-isometries

“…Recently, the research about the m-isometries has received a great impulse. So, for example, in [14,18,19,26] different results about m-isometries are given. In [6,8,17] are proved some dynamic properties of m-isometries.…”

(m, p)-isometric and (m, ∞)-isometric operator tuples on normed spaces

“…This definition coincides with the definition of misometric tuples by Gleason and Richter if X is a Hilbert space (and p = 2) and has, in that context as an equivalent description, essentially already been presented in [14,Lemma 2.1].…”

“…Consequently, as one would expect, the basic theory of (m, p)-isometric tuples can be evolved in a similar fashion as in [14]. However, we will use a different approach, based on an idea described in [16].…”