ACTA issues

Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes

László Kérchy

Acta Sci. Math. (Szeged) 61:1-4(1995), 443-476

Abstract. Let $T\in{\cal L}({\cal H})$ be an absolutely continuous contraction such that the spectral multiplicity function of its unitary asymptote $T^a$ is at least $n (1\le n\le\aleph_0)$ almost everywhere on a subset $\gamma$ (with positive measure) of the unit circle $\bf T$. Let ${\cal G}_n$ be an $n$-dimensional Hilbert space and let $J_{n,\gamma}$ denote the canonical embedding of the Hilbert space $H^2({\cal G}_n)$ into $\chi_{\gamma}L^2({\cal G}_n)$. It is shown that, given any positive $\varepsilon$, there exists a factorization $J_{n,\gamma}=ZY$ such that the mappings $Y\in{\cal L}(H^2({\cal G}_n),{\cal H})$ and $Z\in{\cal L}({\cal H},\chi_{\gamma}L^2({\cal G}_n))$ intertwine $T$ with the operators of multiplication by the identical function on the corresponding spaces, and the product: $\|Y\| \|Z\|\le\sqrt2+\varepsilon$. As a consequence, we obtain that the unilateral shift $S_n$ of multiplicity $n$ can be completely injected into $T$ and that $T$ has the property $({\bf A}_{n,\aleph_0}(\gamma))$ introduced in the theory of dual algebras. It follows furthermore that in the special case $\gamma={\bf T}$ the contraction $T$ has an invariant subspace ${\cal H}'$ such that the restriction $T|{\cal H}'$ is similar to $S_n$.

AMS Subject Classification (1991): 47A45, 47A20, 47A15

Received April 27, 1995. (Registered under 5698/2009.)