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

Overmoyer, Kate
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1305826657

Abstract Details

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.
Mathematics
invariant subspaces; diagonal operators; spectral synthesis; cyclicity

Recommended Citations

Citations

  • Overmoyer, K. (2011). Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1305826657

    APA Style (7th edition)

  • Overmoyer, Kate. Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators. 2011. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1305826657.

    MLA Style (8th edition)

  • Overmoyer, Kate. "Applications of Entire Function Theory to the Spectral Synthesis of Diagonal Operators." Doctoral dissertation, Bowling Green State University, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1305826657

    Chicago Manual of Style (17th edition)

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© 2011, all rights reserved.
This open access ETD is published by Bowling Green State University and OhioLINK.