John Locker

Regularization with differential operators. I. General theory

Abstract The method of regularization is used to obtain least squares solutions of the linear equation Kx = y, where K is a bounded linear operator from one Hilbert space into another and the regularizing operator L is a closed densely defined linear operator. Existence, uniqueness, and convergence analyses are developed. An application is given to the special case when K is a first kind integral operator and L is an nth order differential operator in the Hilbert space L2[a, b].

Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators

Unbounded linear operators Fredholm operators Introduction to the spectral theory of differential operators Principal part of a differential operator Projections and generalized eigenfunction expansions Spectral theory for general differential operators Bibliography Index.

Spectral theory of two-point differential operators determined by −D2. II. Analysis of cases

Abstract In Part I the necessary tools are developed for the complete resolution of the spectral theory for all linear two-point differential operators L in L 2 [0, 1] determined by τ = − D 2 and by boundary values B 1 , B 2 . The development emphasizes six numerical parameters defined in terms of the coefficients of B 1 , B 2 . In this paper we state and prove all results pertaining to the spectrum σ ( L ) of L , to the algebraic multiplicities of the eigenvalues λ ϵ σ ( L ) and the ascents of the operators λI − L , to the boundedness of the family of all finite sums of the projections associated with L , to the denseness of the generalized eigenfunctions, and to the existence of bases consisting of generalized eigenfunctions; we utilize the tools developed in Part I to establish these results. In particular, this paper provides either the explicit or the asymptotic form for the eigenvalues of L , provides explicit numbers for the algebraic multiplicities and ascents, provides the explicit or the asymptotic form for the projections, and provides explicit bounds on the family of all finite sums of these projections or shows that the family is unbounded.

On a class of nonlinear boundary value problems

They proved that this nonlinear problem has at least one solution if(i) R(t, C% , 2%) and f(t, x1 , ~a) are continuous for t E [0, ~-1, --co 0 and some 6 > 0 such that (a2 + sy < h(t, x l,s,)~(m+l-S)“fort~[O,?r]and--oc!<~~,rg<~, The higher dimensional analog of (1) was studied in [6;1 by Landesman and LazerWe consider here the general abstract version of problem (1), and by the application of operator-theoretic techniques we obtain the existence of solutions. in a direct manner. Even though our basic method of proof is the same as in [?‘I, we arrive at the existence of a unique solution of a linear version of (1) by perturbing the linear operator involved in such a manner that it is invertible, and hence, we are able to obtain a setting for an application of the Banach contraction mapping theorem. The important a priori estimates needed tot conclude the existence of a solution of (1) also result from this method. This: technique of perturbation may also be seen in the works of Dolph [3] and Kolodner [5]. These operator-theoretic techniques are then utilized to study a generalized version of the periodic problem:

Nonlinear boundary value problems and operators TT

We consider nonlinear boundary value problems of the type Lƒ + Nƒ = 0 for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L2[a, b] which admits a decomposition of the form L = TT∗ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.

Spectral theory of two-point differential operators determined by −D2. I. Spectral properties

Abstract The necessary tools for the complete resolution of the spectral theory for all linear two-point differential operators L in L2[0, 1] determined by τ = −D2 and by boundary values B1, B2 are developed. All work is based on six numerical parameters defined in terms of the coefficients of B1, B2. These quantities are utilized in the development of the characteristic determinant of L and the Greens function of λI − L, and in the development of the projections of L2[0, 1] onto the generalized eigenspaces of L; they are used in the study of the decay rates of the resolvent operator R λ (L) as ¦λ¦ → ∞ ; and they are used to identify the 13 cases treated in developing the spectral theory for τ = −D2. All results pertaining to the spectrum σ(L) of L, to the algebraic multiplicities of the eigenvalues λ ϵ σ(L), to the ascents of the operators λI − L, to the boundedness of the family of all finite sums of the projections associated with L, to the denseness of the generalized eigenfunctions, and to bases consisting of generalized eigenfunctions are summarized in a table. The proofs of these results are the subject of the sequel (Part II).

Spectral decomposition of a Hilbert space by a Fredholm operator

Abstract Let T be a Fredholm operator in a Hilbert space H with spectrum σ ( T ) = { λ i } i = 1 ∞ . Let T ( λ i ) = T λ i m i , where m i is the ascent of the operator T λ i = λ i I − T , and let P i denote the projection of H onto the generalized eigenspace N ( T (γ i )) along R ( T (γ i )). In this paper it is shown that S ∞ = S ∞ and H = S ∞ ⊕ M ∞ (topological direct sum) iff there exists M > 0 such that ∥∑ i − 1 N P i ∥ ⩽ M , N = 1, 2, …, where S ∞ = {xϵH| = ∑ ∞ i=1 x i , x i ϵ N (T(γ i ))} and M ∞ is the zero or infinite-dimensional subspace ⋃ ∞ i=1 R (T(γ i )). This result is then applied to a differential operator L in L 2 [0, 1], showing that S ∞ ≠ S ∞ = L 2 [0, 1] for this L . Also, an example with M ∞ ≠ {0} is presented.

Denseness of the generalized eigenvectors of an H-S discrete operator

Let T be a closed densely defined linear operator in a Hilbert space H, and assume there exists ξ0 ϵ p(T) such that Rξ0(T) is a Hilbert-Schmidt operator. The operator T is a special type of discrete operator, a so-called H-S discrete operator, which is shown to be a Fredholm operator in H with Fredholm set equal to the whole complex plane. Let σ(T) = {λi}i = 1∞where the operator Tλi has ascent mi, let Pi, i = 1, 2,…, be the projection of H onto the generalized eigenspace N[[Tλi]mi] along R [[Tλi]mi], and let S∞ and M∞ be the subspaces of H consisting of all x ϵ H such that x = ∑i = 1∞ Pix and such that Pix = 0 for i = 1, 2,…, respectively. Sufficient conditions are introduced which guarantee that S∝ = H and M∞ = {0}. These conditions require that ∥Rλ(T)∥ be bounded on certain rays in the complex plane and ∥Rλ(T)∥ → 0 as λ → ∞ on at least one of the rays, but specific decay rates for ∥Rλ(T)∥ are not necessary.

Spectral representation of the resolvent of a discrete operator

Let T be a discrete linear operator in a Hilbert space H with spectrum σ(T) = λii = 1∞, let Rλ(T) denote the resolvent of T, and let Pi denote the projection of H onto the generalized eigenspace N((γi,I − T)mj) along R((γi, I − T)mj), where mi is the ascent of the operator λiI − T. In this paper it is shown that Rγ = ∑∞i=1∑mjj=1(−Ni)j−1Pi(γ − γ)j+∑∞j=0(γo − γ)jRγ0(T∞)j+1(I− P∞) in B(H) for all λ ϵ ϱ(T), where Ni is the restriction of λiI − T to is the restriction of T to N((γi,I − T)mj), T∞ D(T)⋃⋃∞i=1R((γiI−T)mj), P∞=∑∞i=1Pi (strong convergence), and λ0 is a fixed but arbitrary point in C. This spectral representation is valid provided there exists M > 0 such that ∥∑i = 1N Pi ∥ ⩽ M, N = 1, 2, …, and generalizes results that apply to self-adjoint, normal, and spectral operators. The results of this paper are applied to represent the resolvent of a differential operator L in L2[0, 1] having infinitely many eigenvalues with ascent mi = 2 and are also applied to represent the resolvent of an operator T with P∞≠I.

The spectral theory of second order two-point differential operators: I. A priori estimates for the eigenvalues and completeness

This paper is the first part in a four-part series which develops the spectral theory for a two-point differential operator L in L 2 [0, 1] determined by a second order formal differential operator l = −D 2 + pD + q and by independent boundary values B 1 , B 2 . The differential operator L is classified as belonging to one of five cases, Cases 1–5, according to conditions satisfied by the coefficients of B 1 , B 2 . For Cases 1–4 it is shown that if λ = ρ 2 is any eigenvalue of L with ∣ρ∣ sufficiently large, then ρ lies in the interior of a horizontal strip (Cases 1–3) or the interior of a logarithmic strip (Case 4), and in each of these cases the generalised eigenfunctions of L are complete in L 2 [0, 1].