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The Backward Shift On Dirichlet-Type Spaces, Stephan Ramon Garcia
The Backward Shift On Dirichlet-Type Spaces, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We study the backward shift operator on Hilbert spaces H_α (for α ≥ 0) which are norm equivalent to the Dirichlet-type spaces D_α. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on H_α in terms of a weighted superposition of backward shifts.
A *-Closed Subalgebra Of The Smirnov Class, Stephan Ramon Garcia
A *-Closed Subalgebra Of The Smirnov Class, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We study real Smirnov functions and investigate a certain *-closed subalgebra of the Smirnov class N^+ containing them. Motivated by a result of Aleksandrov, we provide an explicit representation for the space H^p ∩ H^p [overscore over the second H^p]. This leads to a natural analog of the Riesz projection on a certain quotient space of L^p for p ϵ (0, 1). We also study a Herglotz-like integral transform for singular measures on the unit circle ∂D.
Inner Matrices And Darlington Synthesis, Stephan Ramon Garcia
Inner Matrices And Darlington Synthesis, Stephan Ramon Garcia
Pomona Faculty Publications and Research
We describe and parameterize the solutions of the scalar valued Darlington synthesis problem. In the case of rational data we derive a simple procedure for producing all possible solutions.
Prolongations And Cyclic Vectors, William T. Ross, Harold S. Shapiro
Prolongations And Cyclic Vectors, William T. Ross, Harold S. Shapiro
Department of Math & Statistics Faculty Publications
For functions belonging to invariant subspaces of the backward shift operator Bf = (f − f(0))/z on spaces of analytic functions on the unit disk D, we explore, in a systematic way, the continuation properties of these functions.