ON THE JOINT (m, q)-PARTIAL ISOMETRIES AND THE JOINT m- INVERTIBLE TUPLES OF COMMUTING OPERATORS ON A HILBERT SPACE

Abstract

The study of tuples of commuting operators was the subject of intensive study by many authors.  Our aim in this work is to consider a generalization of the notions of m-partial isometries and (m, q)-partial isometries (resp. m-left inverse and m-right inverse) of a single operator done in [?] and [?] (resp. in [?],[?], [?]) to the multivariable operators.  We study some of the basic properties of these tuples of commuting operators.  A commuting d-tuple of operators T = (T₁, . . .  ,T_d) acting on a Hilbert space \(\mathcal{H}\) is called a joint (m; (q₁, . . .  ,q_d))-partial isometry, if

   \(\textbf{T}^{q}\left(\displaystyle \sum_{0≤k≤m} (-1)^{k} \binom m k \displaystyle \sum_{|a|=k} \frac{k!}{α!} \textbf{T}^{∗α} \textbf{T}^{α} \right) = 0. \)