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On reproducing kernels and invariant subspaces of the Bergman shift.

Thesis information

*Author:   Chailos, George.;
*Advisor(s):   Richter, Stefan
*Degree:   Ph.D.
*School:   The University of Tennessee.
*Year:   2002

Full Abstract


We denote by L2aD the classical Bergman space of all square integrable analytic functions with respect to the Lebesgue area measure on the unit disc. We set ζ( z) = z, z ∈ D and by Mζ we denote the operator of multiplication by ζ on L2aD . Additionally, we suppose that M is a multiplier invariant subspace of L2aD ; that is, Mζf ∈ M for all f ∈ M and moreover we assume that ind M ≡ dim (M ⊖ ζM) = 1.;If G is a unit vector in M ⊖ ζM, then there is a positive definite sesquianalytic kernel llz defined on D×D such that 1-lzll z1-l z2 is the reproducing kernel for MG (this space is the closure of the analytic polynomials in L2aG 2D ). As a consequence, one checks that llz defines the space M uniquely. Hence, it is natural to ask about the properties, the boundary behavior and the structure of llz . It is within this context that this study has been undertaken.;We define the rank of a positive definite sesquianalytic kernel and we study its properties for a larger class of Hilbert spaces which contains L2aD . In the case of L2aD , we set sM*z&vbm0; M⊥ to be the spectrum of M*z restricted to M⊥ and we consider a conjecture which is due to H. Hedenmalm and which states that rank ll equals the cardinality of sM*z&vbm0; M⊥ . We show that; cardinalitys M*z&vbm0;M⊥ ∩D ≤rankll≤cardinality sM*z&vbm0; M⊥ .;Additionally, we prove that the conjecture is true whenever M is a nontrivial, zero based invariant subspace of L2aD .;Furthermore, it is shown that if I = T&bsolm0;sM* z&vbm0;M⊥ ≠∅ , then ll for fixed l ∈ D , has a meromorphic continuation across I. We also provide examples when some extra hypotheses are imposed on ll and we obtain information for the invariant subspaces related to them.  
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