Operator valued weighted sequence spaces
by
Novak Ivanovski
Department of Mathematics, University of Skopje
Coauthors: Marija Orovcanec (University of Skopje)

In this note we will give a generalization of weighted sequence spaces H²(\beta).

Definition. Let (B_n)^\infty_n=0 be a sequence of positive invertible self-adjoint operators on a Hilbert space H with the property 0 < m < B_n < M,   for alln in N. We consider the space of vectors H²(B)={f=(f₀, f₁, ...): f_i in H; \sum_i=0^\infty||B_if_i||² < \infty} with the inner product (f, g)_B=\sum_n=0^\infty (B_nf_n, B_ng_n).

The main results of this paper are as follows:

1. The space H²(B) is a Hilbert space.

2. Let (B_n)^\infty_n=0 and (C_n)^\infty_n=0 be sequences as in the the definition, then H²(B)=H²(C) and the norms are equivalent.

3. The unilateral shift U₊ on H²(B) is unitarily equivalent to an invertible operator valued weighted shift on the space l²(H).

4. The operators U₊ on H²(B) and H²(C) are similar.

5. We give a necessary and sufficient condition for the operators U₊ on H²(B) and H²(C) to be unitarily equivalent.

1991 Mathematics Subject Classification 47B37.

(T)

Date received: April 13, 2000