Subnormal operators, cyclic vectors and reductivity

Béla Nagy¹

¹ Department of Analysis Institute of Mathematics Budapest University of Technology and Economics H-1516 Budapest, Hungary

Studia Mathematica, Tome 216 (2013) no. 2, pp. 97-109

Résumé

Two characterizations of the reductivity of a cyclic normal operator in Hilbert space are proved: the equality of the sets of cyclic and $^*$-cyclic vectors, and the equality $L^2(\mu )={\bf P}^2(\mu )$ for every measure $\mu $ equivalent to the scalar-valued spectral measure of the operator. A cyclic subnormal operator is reductive if and only if the first condition is satisfied. Several consequences are also presented.

Détail

DOI : 10.4064/sm216-2-1

Keywords: characterizations reductivity cyclic normal operator hilbert space proved equality sets cyclic * cyclic vectors equality every measure equivalent scalar valued spectral measure operator cyclic subnormal operator reductive only first condition satisfied several consequences presented