## Mathematics Ph.D. Dissertations

#### Title

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, PhD

#### Second Advisor

Kyoo Kim, PhD (Committee Member)

#### Third Advisor

Kit C. Chan, PhD (Committee Member)

#### Fourth Advisor

J. Gordon Wade, PhD (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 {

*n*} lying on finitely many rays for suitably chosen

^{a}e^{2π ij/b}: 0≤j < b*a*∈ (0,1) and

*b*∈ ℕ, and more generally whose eigenvalues are the integer lattice points ℤ ×

*i*ℤ. 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.

#### Recommended Citation

Overmoyer, Kate, "Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators" (2011). *Mathematics Ph.D. Dissertations*. 6.

https://scholarworks.bgsu.edu/math_diss/6