Thesis (D.A.)--Illinois State University, 1991. / Title from title page screen, viewed December 16, 2005. Dissertation Committee: Albert D. Otto (chair), John Dossey, Lotus Hershberger, Michael Plantholt, Gary Ramseyer, Beverly Rich. Includes bibliographical references (leaves 131-135) and abstract. Also available in print.

Thesis--Copenhagen.

Thesis--Faculté des Sciences de Paris.

Thesis (Ph. D.)--Johns Hopkins University, 1894. / Biographical sketch.

Thesis (Ph. D.)--University of Chicago, 1920. / Vita.

Inaug.-dis.--Göttingen. / Lebenslauf.

Thèse--Faculté des sciences de Paris.

Thesis (PH.D)--Johns Hopkins University, 1894. / Biographical sketch.

Let X be a square matrix of order k over a field F. The permanent of X is given by [Formula omitted] where σ ranges over all the permutations of 1,2,...,k. The original object of this investigation was to characterize those linear maps which leave the permanent unaltered ; that is, per(X) = per(T(X)), all X. Let M[subscript m,n] denote the vector space of all matrices having m rows and n columns with entries taken from F. Fix an integer r, 2 ≤ r ≤ min(m,n). The r-th permanental compound of X ε M[subscript m,n] is defined in an analogous way to the r-th compound of X, and is denoted by P[subscript r](X) ε M[subscript (m over r) [comma] (n over r)]. Subject to mild restrictions on F, the following theorem can be proved. Let T be a linear map on M[subscript m,n] into itself, let S[subscript r] be a non-singular linear map on M[subscript (m over r) [comma] (n over r)] onto itself. Suppose that P[subscript r](T(X)) = S[subscript r](P[subscript r](X)), all X ε M[subscript m,n]. Then for max(m,n) > 2, we have T(X) = DPXQK when m ≠ n ; when m = n , we have either T(X) = DPXQK, allX, or T(X) = DPX'QK, all X. Here P,Q are permutation matrices and D,K are diagonal matrices, of appropriate orders. For the case r = m = n = 2 , there is a certain non-singular linear map B on M[subscript 2,2] onto itself such that BTB(X) = UXV, all X, or BTB(X) = UX'V, all X. Here U,V are non-singular. The original problem arises in the case r = m = n , with S[subscript r] =1, the unit of F. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis is concerned with the problem of unitary equivalence of certain kinds of non-normal operators. Suppose [m, K, G, g ↦ U [subscript g]] is an ergodic and free C-system, with G abelian. Let m = m [symbol omitted] 1, n = R(U[subscript g] [symbol omitted] V[subscript g] : g є G), and let a = R(m, n) = R[m, K, G, g ↦ U[subscript g]] be the von Neumann algebra constructed from [m, K, G, g ↦U[subscript g]] according to von Neumann. We compute: (1) the group A(α; m, n) of all automorphisms of α which keep m pointwise fixed and keep n invariant, and (2) the group A(m, α; n) (resp. G(m, α; n)) of all automorphisms of m which extend to automorphisms (resp. inner automorphisms) of α keeping n pointwise fixed. These calculations lead us to compute G' [symbol omitted] [G] and G' (where [symbol omitted] is the full group generated by G). We show that for an abelian and ergodic G on an abelian m G' [symbol omitted] [G] = G . In the course of this investigation we obtain several interesting results. For example we see that such [symbol omitted] G is automatically free on m. For a large class of tensor algebras (and in particular for a large class of multiplication algebras) we succeed in determining G'. (For the particular cases of multiplication algebras we only use measure-theoretical arguments.) These results are applied to solve the problem of unitary equivalence of certain kinds of non-normal operators. Finally for most of the interesting thick subalgebras E in the literature, we construct numerous unitarily non-equivalent operators A ,with R(Re A) = E. / Science, Faculty of / Mathematics, Department of / Graduate