Bases From Orthogonal Subspaces Obtained by Evaluation of the Reproducing Kernel

Dušan Georgijević

Publications de l'Institut Mathématique, _N_S_37 (1985) no. 51, p. 93

Résumé

Every inner operator function $\theta$ with values in $B(E,E)$, $E$ -- a fixed (separable) Hilbert space, determines a co-invariant subspace $H(\theta)$ of the operator of multiplication by $z$ in the Hardy space $H^2_E$. ``Evaluating'' the reproducing kernel of $\,H(\theta)$ at ``U-points'' of the function $\theta$ ($U$ is unitary operator) we obtain operator functions $\gamma_t(2)$ and subspaces $\gamma_tE$. The main result of the paper is: Let the operator $I-\theta(z)U^*$ have a bounded inverse for every $z$ $|z|1$. If $(1-r)^{-1}\Re\varphi(rt)$ for definition of $\varphi$ see (1) is uniform bounded in $r$, $0łeq r1$, for all $t$, $|t|=1$, except for a countable set, then the familly of subspaces $\gamma_tE$ is orthogonal and complete in $H(\theta)$. This generalizes an analogous result of Clark [3] in the scalar case.

Détail

Classification : 46E40