# 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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