Linearly similar Szökefalvi-Nagy–Foias model in a domain

Abstract

x0. Introduction In this paper we construct a new linearly similar functional model for linear operators and study its elementary properties. This model generalizes the Sz.-Nagy{Foia s model for C0-contractions and also forC0-dissipative operators. We shall not restrict ourselves to the disk or the half-plane: the model will be constructed in a fairly arbitrary domain. The reduction of an operator to an \almost diagonal model form will be written directly via the resolvent of the operator. Attention will be focused on the case of C00-operators. The main results of this paper were announced in [76]. Let be a union of nitely many piecewise-smooth contours in the complex plane C. Suppose that the sets int ,ext are open and have empty intersection,C =int[ [ext, and = @int = @ext (here we omit a certain technical condition on ). The main objects of this paper are model spacesH() that will be associated with operator-valued bounded analytic functions dened on int and having some special properties. Consider the special case where 2 H 1 int;L(R;R) , 1 is a meromorphic function on int ,a ndk 1 () kC a.e. on . Here R and R are Hilbert spaces, and L(R;R) denotes the space of all bounded linear operators from R to R .W e always consider only separable Hilbert spaces. Let E 2 (ext;R) denote the Smirnov class of functions on ext whose values are vectors in the space R (seex2). Suppose that int and ext are unbounded. In this special case the model spaceH() consists of all functions f meromorphic in int, holomorphic in ext, and satisfying fjext2 E 2 (ext;R) ;fjint2 E 2 (int;R) ;f int = fext a.e. on : This is a Hilbert space. The general denition ofH() will be given inx2. Ap air (A;J) of linear operators (possibly, unbounded) will be called a 2-system if 1) A is a closed operator in a Hilbert space H with nonempty eld of regularity (A )= Cn(A) and with domainD(A )( here(A) is the spectrum of the operator A); 2) J : D(J)! R ,w hereD(J )= D(A) H and J is bounded in the graph norm kxkG def = kxk 2 +kAxk 2 1=2 inD(A). Here R is a Hilbert space.