Bases from orthogonal subspaces obtained by evaluation of the reproducing kernel

Du\v san Georgijevi\'c

Abstract: 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 $r$, $0\leq r<1$, 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.

Classification (MSC2000): 46E40

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