Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.

To solve the linear equation Ax = f, where f is an element of Hilbert space H and A is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate -1 the inverse operator A by an operator V which is a polynomial in A. Using the spectral theory of bounded normal operators the problem is reduced to that of approximating a function of the real variable by polynomials of best uniform approximation. We apply two different techniques of evaluating A-1 so that the operator V is chosen either as a polynomial P (A) when P (A) approximates the n n function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn (A) approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A) satisfy three point recurrence relations, thus the approximate solution vectors P (A)f n and Q (A)f can be evaluated iteratively. We compare the procedures involving n Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite. We also show that the technique can be applied to an operator which is not selfadjoint, but close, in the sense of operator norm, to a selfadjoint operator. The iterative techniques we develop are used to solve linear systems arising from the discretization of Freedholm integral equations of the second kind. Both smooth and weakly singular kernels are considered. We show that earlier work done on the approximation of linear functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse operator and are thus special cases of a general result involving an approximation of arbitrary degree to A -1 . / Thesis (Ph.D.)-University of Natal, 1994.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Computer Science." Includes bibliographical references (p. 86-91).

In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that the dimension of the eigenspace at ω is 1 for all ω Ω then the map ω ker(T - ω) admits a non-zero holomorphic section, say γ, and therefore defines a line bundle on Ω. As is well known, the curvature defined by the formula is a complete invariant for the line bundle . On the other hand, define and note that NT (ω)2 = 0. It follows that if T is unitarily equivalent to T˜, then the corresponding operators NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω. However, Cowen and Douglas prove the non-trivial converse, namely that if NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω then T and T˜ are unitarily equivalent. What does this have to do with the line bundles and .To answer this question, we must ask what is a complete invariant for the unitary equivalence class of the operator NT (ω). To find such a complete invariant we represent NT (ω) with respect to the orthonormal basis obtained from the two linearly independent vectors γ(ω),∂γ(ω) by Gram-Schmidt orthonormalization process. Then an easy computation shows that It then follows that is a complete invariant for NT (ω), ω Ω. This explains the relationship between the line bundle and the operator T in an explicit manner. Subsequently, in the paper ”Operators Possesing an Open Set of Eigenvalues”, Cowen and Douglas define a class of commuting operators possessing an open set of eigenvalues and attempt to provide similar computations as above. However, they give the details only for a pair of commuting operators. While the results of that paper remain true in the case of an arbitrary n tuple of commuting operators, it requires additional effort which we explain in this thesis.

A pair of commuting bounded operators (S; P ) acting on a Hilbert space, is called a -contraction, if it has the symmetrised bides = f(z1 + z2; z1z2) : jz1j 1; jz2j 1g C2 as a spectral set. For every -contraction (S; P ), the operator equation S S P = DP F DP has a unique solution F 2 B(DP ) with numerical radius, denoted by w(F ), no greater than one, where DP is the positive square root of (I P P ) and DP = RanDP . This unique operator is called the fundamental operator of (S; P ). This thesis constructs an explicit normal boundary dilation for -contractions. A triple of commuting bounded operators (A; B; P ) acting on a Hilbert space with the tetra block E = f(a11; a22; detA) : A = a11 a12 with kAk 1g C 3 a21 a22 as a spectral set, is called a tetra block contraction. Every tetra block contraction possesses two fundamental operators and these are the unique solutions of A B P = DP F1DP ; and B A P = DP F2DP : Moreover, w(F1) and w(F2) are no greater than one. This thesis also constructs an explicit normal boundary dilation for tetra block contractions. In these constructions, the fundamental operators play a pivotal role. Both the dilations in the symmetrised bidisc and in the tetra block are proved to be minimal. But unlike the one variable case, uniqueness of minimal dilations fails in general in several variables, e.g., Ando's dilation is not unique, see [44]. However, we show that the dilations are unique under a certain natural condition. In view of the abundance of operators and their complicated structure, a basic problem in operator theory is to find nice functional models and complete sets of unitary invariants. We develop a functional model theory for a special class of triples of commuting bounded operators associated with the tetra block. We also find a set of complete unitary invariants for this special class. Along the way, we find a Burling-Lax-Halmos type of result for a triple of multiplication operators acting on vector-valued Hardy spaces. In both the model theory and unitary invariance, fundamental operators play a fundamental role. This thesis answers the question when two operators F and G with w(F ) and w(G) no greater than one, are admissible as fundamental operators, in other words, when there exists a -contraction (S; P ) such that F is the fundamental operator of (S; P ) and G is the fundamental operator of (S ; P ). This thesis also answers a similar question in the tetra block setting.

No description available.

No description available.

Whether it is designing and controlling super-efficient high speed transport systems or understanding environmental fluid flows, a key question that arises is: what state does the fluid take and why? An answer to this question lies in understanding the hydrodynamic stability properties of the flow as a function of parameters. While much work has been done in this area in the past, there are many open questions that need to be addressed. Here we study the effect of spatial domain size, number of modes, non-hermitianness and non-normality on the finite time and infinite time stability properties of a standing, viscous shock flow problem. It has been shown that the above problems are not only non-normal but also non-hermitian, when the base flow has shear. The eigenvalue problems corresponding to infinite spatial domain, finite spatial domain, Forward and L2 adjoint problems are solved exactly by converting the linear partial differential equations into nonlinear Riccati equations. In the finite domain case, the full time dependent solutions are obtained analytically using bi-orthogonal basis functions. In the infinite domain case, the point spectrum of the forward operator is shown to be unbounded and that of the adjoint operator to be empty. In the unbounded case, the spectrum fills the entire area on one side of a parabola in the complex plane and is connected. As the fluid viscosity decreases the width of the parabola increases and in the limit of zero viscosity covers almost entire left half plane(LHP). On the other hand, as the fluid viscosity increases the width of parabola decreases and in the limit of infinite viscosity becomes negative real axis, which is the spectrum of heat equation. The spectrum of adjoint problem is empty for all values of the viscosity and prescribed velocity. In the finite spatial domain case, the point spectrum lies in the open left half plane for all Reynolds numbers and hence asymptotically stable. The results obtained showed that perturbations grow substantially large for finite time before they decay at large times. It is also found that retainig right number of modes is crucial for observing transient growth phenomena. Finally, the linear results are compared with the nonlinear finite amplitude simulation results. The relevance of current results to other fluid flows is presented.