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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