A Note on the Algebra Generated by a Subnormal Operator

Abstract

Let H be a separable Hilbert space, and let A ⊂ L(H) be a weak*-closed algebra. We say that A has property (/A₁) if every weak*-continuous functional f on A can be represented as

$$f(T) = (Tx,y),\,T\, \in \,A,$$

((1))

where x and y are vectors in H and (·,·) denotes the scalar product in H. We say that A has property (/A₁(r)) if, in addition, given s>r and f, we can choose x and y satisfying (1) and

$$\left\| x \right\|\left\| y \right\| \leqslant \,s\left\| f \right\|$$

((2))

.

The work in this paper was partially supported by grants from the National Science Foundation.