Operators intertwining with isometries and Brownian parts of 2-isometries

For two operators A and T (A≥0) on a Hilbert space H satisfying T⁎AT=A and the A-regularity condition AT=A1/2TA1/2 we study the subspace N(A−A2) in connection with N(AT−TA), for T belonging to different classes. Our results generalize those due to C. Kubrusly concerning the case when T is a contraction and A=ST is the asymptotic limit of T. Also, the particular case of a 2-isometry in the sense of S. Richter as well as J. Agler and M. Stankus is considered. For such operators, under the same regularity condition we completely describe the reducing Brownian unitary and isometric parts, as well as the invariant Brownian isometric part. Some examples are provided in order to illustrate the limits of the theoretical results.