A CLASS OF OPERATOR-VALUED MEROMORPHIC FUNCTIONS ON THE UNIT DISK

Let L(H) denote the algebra of all bounded linear operators on a separable Hilbert space H. The first objective of this note is to study the class D0(L(H)) of all L(H)-valued functions F meromorphic in the open disc D and holomorphic at 0, admitting a representation of the form [GRAPHICS] where g(j) = g(-j)BAR is-an-element-of C, j = 0,...N, SIGMA is a positive operator measure and h(j) = -(h(-j))* is-an-element-of L(H), j = 0,..., N. Evidently, the class D0(L(H)) contains the holomorphic operator-valued functions in D with non-negative real part. It also contains the Krein-Langer classes C(kappa), kappa = 0, 1,..., of meromorphic operator functions [16]. If J is-an-element-of L(H) J = J-1 = J* , and U is a definitizable unitary operator in the Krein space (H, (J.,.)H), one can define the spectra of non-negative and non-positive type of U and construct the spectral function of U with the help of the function z bar arrow pointing right J(U + zI)(U - zI)-l , which belongs to D0(L(H)). For an arbitrary F is-an-element-of D0(L(H)) we define analogues of the spectra of non-negative and non-positive type of definitizable unitary operators in the same way. Multiplicity functions for these sets are defined with the help of an analogue of the spectral function. The second objective of this paper is to study the behaviour of the ''spectra of non-negative and non-positive type of F'' under perturbations of F. For F in a fixed class C(kappa) these sets depend continuously on F with respect to the uniform weak convergence.