ACTA issues

## On the quasi-similarity of compressed shift operators

György Michaletzky

Acta Sci. Math. (Szeged) 69:1-2(2003), 223-239
2895/2009

 Abstract. In this paper we analyse the compressed shift operators belonging to various subspaces arising in the theory of stochastic realizations without reconstructing the whole picture used in the analysis of stationary processes. Let $H$ be a separable Hilbert space with a bilateral shift operator $U$ of finite multiplicity. Assume that $H=H_1\oplus H_2$, where $H_1$ and $H_2$ are double-invariant subspaces. Let $S\subset H$ be a subspace for which $U^{-1}S\subset S$. We show that under some weak regularity conditions the compressed shift operators defined on the subspaces $\Pr_{H_1}S\ominus\left (S\cap H_1\right )$ and $\Pr_{H_2}S\ominus\left (S\cap H_2\right )$ are quasi-similar. Using Hardy space approach and working directly with inner functions and spectral factors under some more restricted circumstances this result was proven by P. Fuhrmann and A. Gombani [2]. AMS Subject Classification (1991): 45C07, 47A15, 47A40, 60G10, 93E08 Keyword(s): compressed shift operators, quasi-similarity, stochastic realization theory Received June 8, 2001, and in final form June 5, 2002. (Registered under 2895/2009.)