Spectral theory of non-self-adjoint two-point differential operators / John Locker.

Format:

Book

Author/Creator:

Locker, John, author.

Series:

Mathematical surveys and monographs ; no. 73.

Mathematical surveys and monographs, 0076-5376 ; volume 73

Language:

English

Subjects (All):

Nonselfadjoint operators.

Spectral theory (Mathematics).

Physical Description:

1 online resource (266 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Language Note:

English

Summary:

This monograph develops the spectral theory of an $n$th order non-self-adjoint two-point differential operator $L$ in the Hilbert space $L2[0,1]$. The mathematical foundation is laid in the first part, where the spectral theory is developed for closed linear operators and Fredholm operators. An important completeness theorem is established for the Hilbert-Schmidt discrete operators. The operational calculus plays a major role in this general theory. In the second part, the spectral theory of the differential operator $L$ is developed by expressing $L$ in the form $L = T + S$, where $T$ is the principal part determined by the $n$th order derivative and $S$ is the part determined by the lower-order derivatives. The spectral theory of $T$ is developed first using operator theory, and then the spectral theory of $L$ is developed by treating $L$ as a perturbation of $T$. Regular and irregular boundary values are allowed for $T$, and regular boundary values are considered for $L$. Special features of the spectral theory for $L$ and $T$ include the following: calculation of the eigenvalues, algebraic multiplicities and ascents; calculation of the associated family of projections which project onto the generalized eigenspaces; completeness of the generalized eigenfunctions; uniform bounds on the family of all finite sums of the associated projections; and expansions of functions in series of generalized eigenfunctions of $L$ and $T$.

Contents:

""Contents""; ""Preface""; ""Chapter 1. Unbounded Linear Operators""; ""1. Introduction""; ""2. Closed Linear Operators""; ""3. Analytic Vector-Valued Functions""; ""4. Spectral Theory""; ""5. Poles of the Resolvent""; ""Chapter 2. Fredholm Operators""; ""1. Basic Properties""; ""2. Spectral Theory for Fredholm Operators""; ""3. Spectral Theory for Index Zero""; ""4. Hilbert-Schmidt Operators""; ""5. Quasi-Nilpotent Hilbert-Schmidt Operators""; ""6. A Hilbert-Schmidt Completeness Theorem""; ""Chapter 3. Introduction to the Spectral Theory of Differential Operators""; ""1. An Overview""

""2. Sobolev Spaces""""3. The Characteristic Determinant and Eigenvalues""; ""4. Algebraic Multiplicities""; ""Chapter 4. Principal Part of a Differential Operator""; ""1. The Principal Part T""; ""2. The Characteristic Determinant of T""; ""3. The Green's Function of ...""; ""4. Alternate Representations""; ""5. The Boundary Values: Case n = 2v""; ""6. The Boundary Values: Case n = 2v � 1""; ""7. The Eigenvalues: Case n = 2v""; ""8. The Eigenvalues: Case n = 2v � 1""; ""9. Completeness of the Generalized Eigenfunctions""

""Chapter 5. Projections and Generalized Eigenfunction Expansions""""1. The Associated Projections: n � 2v""; ""2. The Associated Projections: n = 2v � 1""; ""3. Expansions in the Generalized Eigenfunctions""; ""Chapter 6. Spectral Theory for General Differential Operators""; ""1. The Resolvents of T and L""; ""2. The Operator SR[sub(...)](T) and Completeness""; ""3. Background Theory of Projections""; ""4. The Spectral Theory of L: n = 2v, Case 1""; ""5. The Spectral Theory of L: n = 2v, Case 2""; ""6. The Spectral Theory of L: n = 2v � 1, Case 1""; ""Bibliography""; ""Index""; ""A""

""B""""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 247-248) and index.

Description based on print version record.

ISBN:

1-4704-1300-0