Bounded point evaluations for cyclic operators

A. Bourhim


Let T be a cyclic bounded linear operator on a Hilbert space H with cyclic vector x that is the finite linear combinations of the vectors x, Tx, T²x, . . . are dense. A complex number l Î C is said to be a bounded point evaluation of T if there is a constant M>0 such that

|p(l)| £ M||p(T)x||


The set of all bounded point evaluations of T will be denoted by B(T). Note that it follows from Riesz Representation Theorem that l Î B(T) if and only if there is a unique vector denoted k(l) ÎH such that p(l) = <p(T)x, k(l)> for every complex polynomial p. An open subset O of C is said to be an analytic set for T if it is contained in B(T) such that for every y Î H, the complex function ŷ defined on B(T) by ŷ(l) = <y, k(l)>, is analytic on O. The largest analytic set for T will denoted by B_a(T) and its points will be called analytic bounded point evaluations for T. In 1979, T. T. Trent proved that if T is subnormal operator then


s(T)\s_ap(T) = B_a(T) (*)


where s(T) and s_ap(T) denote respectively the spectrum and approximate point spectrum of T. In 1994, L. R. Williams proved that for every arbitrary cyclic operator T, s(T)\s_ap(T) Ì B_a(T) and asked if (*) remain valid for arbitrary cyclic operators Dynamic systems and Applications 3(1994) 103-112. In this talk, we first give necessary and sufficient conditions for weighted shifts to satisfy (*) and exhibit an operator which provides a negative answer to the question of L. R. Williams. Furthermore, using a different method of T. T. Trent, we prove that every arbitrary cyclic hyponormal operator satisfies (*).