On expansive three-isometries

The sub-Brownian 3-isometries in Hilbert spaces are the natural counterparts of the 2-isometries, because all of them have Brownian-type extensions in the sense of J. Agler and M. Stankus. We show that all powers Tn for n ≥ 2 of every expansive 3-isometry T are sub-Brownian, even if T does not have such a property. This fact induces some useful relations between the corresponding covariance operators of T. We analyze two matrix representations of T in order to get some conditions under which T is sub-Brownian, or T admits the Wold-type decomposition in the sense of S. Shimorin. We show that the restriction of T to its range is sub-Brownian of McCullough’s type, and that under some conditions on N(T∗), T itself is sub-Brownian, and it admits the Wold-type decomposition.

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)