## Subnormal operators, cyclic vectors and reductivity

### Volume 216 / 2013

Studia Mathematica 216 (2013), 97-109
MSC: Primary 47B20; Secondary 47B15.
DOI: 10.4064/sm216-2-1

#### Abstract

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.