Preface

This monograph is a sequel to my earlier book on functional analysis and

two-point differential operators [24]. In the previous work we developed the basic

structure of an nth order differential operator L in the Hilbert space L2[a,b] that

is determined by an nth order formal differential operator £ and by independent

boundary values Bi,... ,Bn defined on the Sobolev space Hn[a,b]. As such L

has the structure of a Fredholm operator, as does the adjoint L*, which is the

differential operator determined by the formal adjoint £* and by adjoint boundary

values B{,... ,£*. The Green's function and generalized Green's function are

characterized in the third and fourth chapters of [24].

The current work is divided into two parts, with Chapters 1 and 2 comprising

the first part where the foundations of the spectral theory are laid in a general

Hilbert space setting. Chapter 1 introduces the closed linear operators, analytic

vector-valued and operator-valued functions, Cauchy's Theorem, and Taylor series

and Laurent series expansions. For the special case of a bounded linear operator, the

operational calculus is developed; it is one of major tools used to study the spectral

theory. Turning to the spectral theory, we introduce the resolvent set, spectrum,

and resolvent of a closed linear operator, illustrating these ideas with two-point

differential operators. Since our emphasis is on the non-self-adjoint operators, we

introduce the ascent and descent of an operator, and then the generalized eigenspace

and algebraic multiplicity corresponding to an eigenvalue. Of special importance is

the section on poles of the resolvent.

Chapter 2 introduces the Fredholm operators, with differential operators again

serving as models. Upon defining the nullity, defect, and index of a Fredholm

operator and forming the generalized inverse, the basic theorems for products and

perturbations are discussed, and the spectral theory is then studied in detail. The

local and global behavior of the algebraic multiplicity and ascent are determined,

and the spectrum is characterized. Special emphasis is placed on the spectral

theory for index zero, where again poles of the resolvent play a major role. In this

chapter the expansion problem for a vector in terms of the generalized eigenvectors

is discussed for the first time. After reviewing the Hilbert-Schmidt operators, a

very powerful completeness theorem is presented for the Hilbert-Schmidt discrete

operators; this theorem is a key component of the second part of the monograph.

Since most of the mathematics in the first part is well-known, we omit most

proofs and simply give references to the literature. The sole exception is the material

dealing directly with the spectral theory, where the results are presented in detail.

We hope that this approach allows the reader to get more quickly to the main topic

of this book: the spectral theory of non-self-adjoint two-point differential operators.

The second part consists of Chapters 3 through 6, where the spectral theory of

two-point differential operators is developed. In Chapter 3 the two-point differential

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