Abstract. In the first part of the paper we study the decompositions of a (bounded linear) operator similar to a normal operator in Hilbert space into the orthogonal sum of a normal (self-adjoint, unitary) part and of a part free of the given property, respectively. In the second part we investigate in a finite dimensional Hilbert space the minimal unitary power dilations (till the exponent $k$) of a contraction. We determine the general form of such dilations, examine their spectra, and the question of their isomorphy. The first step of the study here is also the decomposition of the contraction into unitary and completely non-unitary parts.

AMS Subject Classification (1991): 47A10, 47A20, 47A30

Keyword(s): operator similar to a normal, spectral operator of scalar type, unitary and completely non-unitary parts, bounded Boolean algebras of idempotents in Hilbert space, equivalent scalar product, minimal unitary power dilations (till the exponent $k$) of a contraction, characteristic function, matrix polynomial, isomorphy of dilations

Received November 16, 2012. (Registered under 91/2012.)