| Absztrakt: | Let T be a contraction acting on the Hilbert space H. Let us
assume that T is of class $C\sb{10}$ that is $\lim\sb{n\to \infty}\Vert
T\sp nh\Vert \ne 0=\lim\sb{n\to \infty}\Vert T\sp{*n}h\Vert$,
for every nonzero vector h. Then a new scalar product can be
introduced in H by $<h,k>\sb{\sim}=\lim\sb{n\to \infty}<T\sp
nh,T\sp nk>$. T acts as an isometry on the inner product space
$(H,<\cdot,\cdot >\sb{\sim})$. Let $\tilde T$ denote the minimal
unitary extension of this isometry acting on the Hilbert space
$\tilde H.$ The main result of the paper is a complete characterization
of the possible spectra of a $C\sb{10}$- contraction T and its
minimal unitary extension $\tilde T.$ The theorem is analogous
to the one obtained for $C\sb{11}$-contractions in the work [Proc.
Amer. Math. Soc. 95, 412-418 (1985)], jointly with {\it H. Bercovici};
however the proof is more difficult. |