Let B(H) denote the C^*-algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B(H) and c = 〖(c1, . . . , cn)〗^t ∈ C^n with n being a positive integer such that n ≤ dim H, the c-numerical range and c-numerical radius of A are defined by W_e (A)= {∑_(i=1)^n▒c_i 〈〖Ax〗_i, x_i 〉 : {x_1, …, x_n } is an orthonormal set in H} and W_C (A)={|z| :z ∈W_(c ) (A)} respectively. When c = 〖(1, 0, . . . , 0)〗^t, Wc(A) reduces to the classical numerical range W(A). Preserver problems concern the characterization of maps between spaces of bounded linear operators that leave invariant certain functions, subsets, or relations etc. In this thesis, several preserver problems related to the numerical range or its generalizations were studied. For A ∈ B(H), the diameter of its numerical range is d_w(A) = sup{|a - b| : a, b ∈ W(A)}. The first result in this thesis was a characterization of linear surjections on B(H) preserving the diameter of the numerical range, i.e., linear surjections T : B(H) → B(H) satisfying d_w(T(A)) =d_w(A) for all A ∈ B(H) were characterized. Let Mn be the set of n × n complex matrices and Tn the set of upper triangular matrices in Mn. Suppose c = 〖(c1, . . . , cn)〗^t ∈ R^n. When wc(·) is a norm on Mn, mappings T on Mn (or Tn) satisfying wc(T(A) - T(B)) = wc(A - B) for all A,B were characterized. Let V be either B(H) or the set of all self-adjoint operators in B(H). Suppose V^n is the set of n-tuples of bounded operators  = (A1, . . . ,An), with each Ai ∈ V. The joint numerical radius of  is defined by w(Â) = sup{||(⟨A1x, x⟩, . . . , ⟨Anx, x⟩)∥ : x ∈ H, ∥x∥ = 1}, where ∥ · ∥ is the usual Euclidean norm on F^n with F = C or R. When H is infinite-dimensional, surjective linear maps T : V^n→V^n satisfying w(T(Â)) = w(Â) for all  ∈ V^n were characterized. Another generalization of the numerical range is the Davis-Wielandt shell. For A ∈ B(H), its Davis-Wielandt shell is DW(A) = {(⟨Ax, x⟩, ⟨Ax, Ax⟩): x ∈ H and∥x∥= 1}. Define the Davis-Wielandt radius of A by dw(A) = sup{(√(|⟨Ax, x⟩ |^2 + |⟨Ax, Ax⟩ |^2) : x ∈ H and ∥x∥= 1}. Its properties and relations with normaloid matrices were investigated. Surjective mappings T on B(H) satisfying dw(T(A) - T(B))= dw(A - B) for all A,B ∈ B(H) were also characterized. A characterization of real linear surjective isometries on B(H) by Dang was used to prove the preserver result about the Davis-Wielandt radius. The result of Dang is proved by advanced techniques and is applicable on a more general setting than B(H). In this thesis, the characterization of surjective real linear isometries on B(H) was re-proved using elementary operator theory techniques. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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Vibrational spectra techniques such as IR, Raman and SFG all carry molecular orientation information. Extracting the orientation information from the vibrational spectra often involves creating model spectra with known orientation details to match the experimental spectra. The running time for the exhaustive approach is O(n!). With the help of linear programming, the running time is pseudo O(n). The linear programming approach is with out a doubt far more superior than exhaustive approach in terms of running time. We verify the accuracy of the answer of the linear programming approach by creating mock experimental data with known molecular orientation distribution information of alanine, isoleucine, methionine, lysine, valine and threonine. Linear programming returns the correct orientation distribution information when the mock experimental spectrum consisted of different amino acids. As soon as the mock experimental spectrum consisted of same amino acids, different conformer with different orientation distribution, linear programming fails to give the correct answer albeit the species population is roughly correct. / Graduate

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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