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Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators
Overmoyer, Kate

2011, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics.
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.
Steven M. Seubert, PhD (Advisor)
Kyoo Kim, PhD (Committee Member)
Kit C. Chan, PhD (Committee Member)
J. Gordon Wade, PhD (Committee Member)
97 p.

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Overmoyer, K. (2011). Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Overmoyer, Kate. "Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators." Electronic Thesis or Dissertation. Bowling Green State University, 2011. OhioLINK Electronic Theses and Dissertations Center. 26 Sep 2017.

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Overmoyer, Kate "Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators." Electronic Thesis or Dissertation. Bowling Green State University, 2011. https://etd.ohiolink.edu/

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