Journal of Operator Theory

Volume 32, Issue 1, Summer 1994 pp. 47-75.

Equivalence classes of subnormal operators

Authors: James Zhijian Qiu
Author institution: Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, U.S.A.

Summary: Let G be a bounded simply connected domain with harmonic measure

$\omega$ and let

$P^2(\omega)$ be the closure in

$L^2(\omega)$ of

$\mathcal P$ , the set of analytic polynomials. Let

$S_\omega$ be the operator defined by

$S_\omega f = z f$ for each

$f \in P^2 (\omega)$ . We characterize all subnormal operators similar or quasisimilar to

$S_\omega$ and we describe the unitary equivalence class of

$S_\omega$ . We make the assumption in this study that G is a normal domain (we say G is normal if

$\mathcal P$ is dense in the Hardy space H^1 (G)). Some examples are given to show that the normality of G is necessary. We also give some characterizations of a domain (i.e., a connected open subset in the plane) that is the image of a weak-star generator of

$H^\infty (\mathbb D)$ .

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