Mathematics Ph.D. Dissertations

Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators

Date of Award

2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Steven M. Seubert

Second Advisor

Kyoo Kim (Committee Member)

Third Advisor

Kit C. Chan (Committee Member)

Fourth Advisor

J. Gordon Wade (Committee Member)

Abstract

A diagonal operator acting on the space H(B(0,R)) of functions analytic on the disk B(0,R) where 0 < R ≤ ∞ is defined to be any continuous linear map on H(B(0,R)) having the monomials z n as eigenvectors. In this dissertation, examples of diagonal operators D acting on the spaces H(B(0,R)) where 0 <R< ∞, are constructed which fail to admit spectral synthesis; that is, which have invariant subspaces that are not spanned by collections of eigenvectors. Examples include diagonal operators whose eigenvalues are the points {nae2π ij/b: 0≤j < b} lying on finitely many rays for suitably chosen a ∈ (0,1) and b ∈ ℕ, and more generally whose eigenvalues are the integer lattice points ℤ × iℤ. Conditions for removing or perturbing countably many eigenvalues of a non-synthetic operator which yield another non-synthetic operator are also given. In addition, sufficient conditions are given for a diagonal operator on the space H(B(0,R)) of entire functions (for which R=∞) to admit spectral synthesis.

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