Thesis (M. Phil.)--University of Hong Kong, 1994. / Includes bibliographical references (leaves 47-49).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1980. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaf 29).

Thesis (Ph. D.)--University of Wisconsin--Madison,1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 182-184).

Thesis--University of Florida. / Description based on print version record. Typescript. Vita. Includes bibliographical references (leaves 80-83).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1985. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1970. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2007. / Title from title screen (site viewed July 9, 2007). PDF text: 96 p. UMI publication number: AAT 3252832. Includes bibliographical references. Also available in microfilm and microfiche formats.

This thesis investigates the properties of the L₂-eigenvalues of singular, elliptic, second order operators, primarily the operator L defined by [formula omitted]. Here the "potential function", V(x), is such that [formula omitted] is a norm on [formula omitted] being the usual norm in the Sobolev space W¹̕²(G) and [formula omitted] is the completion of [formula omitted] in the metric from this norm, identified with a subset L₂(G) ; Δ is the Laplacian and G is an arbitrary open domain of E[superscript n] . Several sufficient conditions are given on V and on G in order that L have spectrum satisfying [formula omitted] , for some real number [formula omitted] denote the spectrum and point spectrum of L , respectively). The properties of these lower eigenvalues are investigated by examining the eigenvalues of a coercive bilinear form corresponding to the operator. Such a form B , having domain [symbol omitted] , say, is defined to have eigenvalueλє¢ with corresponding eigenfunction [symbols omitted] if B[u,f] = λ (u,f) for all f є [symbol omitted] . Variational properties are discussed in detail; In particular, a condition is given which ensures that the numbers sup inf B[u,u] (the sup and inf being over appropriate sets involving [symbol omitted] and n ) are eigenvalues of B . These properties are applied to L to generalize the well-known classical property (G bounded) of monotonic dependence of the eigenvalues on the underlying domain G : G [symbol omitted] G* implies [formula omitted] for corresponding eigenvalues, with strict inclusion implying strict inequality. A few miscellaneous properties of the eigenvalues and eigenfunctions then follow from this dependence. / Science, Faculty of / Mathematics, Department of / Graduate

We consider a formally self-adjoint elliptic differential operator in IRⁿ, denoted by τ. T₀ and T are operators given by τ with specific domains. We determine conditions under which T₀ is essentially self-adjoint, introducing the topic by means of a brief historical survey of some results in this field. In Part I, we consider an operator of order 4, and in Part II, we generalise the results obtained there to ones for an operator of order 2m. Thus, the two parts run parallel. In Chapter 1, we determine the domain of T₀*, denoted by D(T₀*), where T₀* denotes the adjoint of T₀, and introduce operators <u>T</u>₀ and <u>T</u> which are modifications of T₀ and T. In Chapter 2, we use a theorem of Schechter to give conditions under which <u>T</u>₀ is essentially self-adjoint. Working with the operator <u>T</u>, in Chapter 3 ve show that we can approximate functions u in D(T₀*) by a particular sequence of test-functions, which enables us to derive an identity involving u, Tu and the coefficient functions of the operator concerned. In Chapter 4, we determine an upper bound for the integral of a function involving a derivative of u in D(T₀*) whose order is half the order of the operator concerned, and we use the identity from the previous chapter to reformulate this upper bound. In Chapter 5, we give conditions which are sufficient for the essential self-adjointness of T₀. In the main theorem itself, the major step is the derivation of the integral of the function involving the particular derivative of u in D(T₀*) whose order is half the order of the operator concerned, referred to above, itself as a term of an upper bound of an integral we wish to estimate. Hence, we can employ the upper bound from Chapter 4. This "sandwiching" technique is basic to the approach we have adopted. We conclude with a brief discussion of the operators we considered, and restate the examples of operators which we showed to be essentially self-adjoint.

In this thesis, we study weighted composition operatorsT(xn)=(£fnX£m(n)) between sequence spaces(c0,c,l1,lp), and more precisely, the sufficient and necessary condition that they are compact. First,we obtain some results of weighted composition operators beingcompact, weakly compact and completely continuous on c0 spaces. Then, we extend then to c,l1,and lp(1<p<¡Û) spaces. Finally, we obtain the condition that an operator from c0, c or lp into c0, c, or lq is compact, weakly compact or completely continuous.