Applied Mathematics and Plasma Physics

Rick Chartrand, *Hilbert spaces of holomorphic functions: zero sets,
invariant subspaces, and Toeplitz operators.*, PhD thesis, University of California, Berkeley, 1999

Several problems are considered in the setting of Hilbert spaces of holomorphic functions on the unit disc. In Chapter 1, the main result is a characterization of the zero sets of a large class of such spaces. The characterization is in terms of Gram matrices of reproducing kernels associated with the points of a sequence in the disc. The construction involved in the proof is then applied to a smaller class of spaces to characterize the elements of a space having a fixed inner function as a factor. Specializing to the case of the Dirichlet space, the construction gives a characterization of the boundary-zero sets.

This construction gives rise to
a wandering vector of the shift operator of multiplication by
*z*. In the case of the Dirichlet space and certain
generalizations, the subspaces invariant under the shift
operator are generated by wandering vectors. Furthermore, the
wandering subspace of an invariant subspace is one-dimensional.
Therefore, the problem of describing the invariant subspaces of
Dirichlet-type spaces reduces to the problem of describing the
wandering vectors. For invariant subspaces of the Dirichlet
space determined by zero sets, inner divisors, and boundary-zero
sets, the aforementioned construction produces a wandering
vector and generator. It is not known whether other invariant
subspaces of the Dirichlet space exist.

In Chapter
2, variants of the classical Toeplitz operators on
*H ^{2}* are studied. A
characterization is obtained for the bounded, harmonic symbols
giving rise to a bounded Toeplitz operator on a Dirichlet-type
space. The relationship between the characterizing condition
and multipliers of the holomorphic and harmonic Dirichlet spaces
is examined.

@phdthesis{chartrand-1999-hilbert,

author = {Rick Chartrand},

title = {Hilbert spaces of holomorphic functions: zero sets,
invariant subspaces, and {T}oeplitz operators.},

year = {1999},

urlpdf = {http://math.lanl.gov/Research/Publications/Docs/chartrand-1999-hilbert.pdf},

school = {University of California, Berkeley}

}