Some families of subnormal operators with finite rank self-commutators

In recent years, D. Xia has published several papers outlining the analytic model of a subnormal operator. In this dissertation, we will use Xia's analytic model to determine necessary and sufficient conditions for characterizing some families of pure subnormal operators S with finite rank self-commutators, i.e. rank [S*, S] = rank (S*S - SS*) < infinity.;Let F denote the family of pure subnormal operators S acting on a Hilbert space H with finite rank self-commutators and minimal normal extensions N where sigma(S) = {z : | z| ≤ 1} and sigma(N) = {z : |z| = 1} ∪ (a1,..., am}, |ai| < 1 for all i. This paper outlines five necessary and sufficient conditions for characterizing S ∈ F . Using these conditions, we will show that for S ∈ F and sigma(N) = {z : | z| = 1}, if dim M = m, M = [S*, S] H , then S is unitarily equivalent to U + ⊕ U+ ⊕ &cdots; ⊕ U+ (a direct sum of m copies). Thus, S is a unilateral shift of multiplicity m. We will next show that if S ∈ F with a rank two self-commutator, then either sigma p(N) = empty or sigmap(N) will contain a single point. We will also conclude that if S ∈ F , then the cardinal number of sigmap( N) is less than or equal to (dim M) - 1 and the maximum cardinal number of sigmap( N) is (dim M) - 1.;Suppose S is a pure subnormal operator with finite rank self-commutator. The pair of operators {Λ, C} are both defined on M = [S*, S] H , a finite dimensional space. According to Xia's analytic model, {Λ, C} is a complete unitary invariance of S. Giving the characterization of a complete unitary invariance for a family of subnormal operators is one of the central problems in operator theory. In this paper, we will present the conditions for any pair of operators {Λ, C} to be a complete unitary invariance for some pure subnormal operator S ∈ F .;Finally, using a generalization of Xia's analytic model of a subnormal operator, we will give some necessary conditions for characterizing a pure subnormal operator T with minimal normal extension N where sigma(T) is the closure of a simply-connected quadrature domain D with boundary L and sigma( N) = L ∪ {b1,..., bm}, bi ∈ D for all i.