ONE-DIMENSIONAL STABLE DISTRIBUTIONS V. M. Zolotarev

Translator: H. H. McFaden 1980 Mathematics Subject Classification (1985 Revision). Primary 47A56,

47A10; Secondary 47A55, 47A70, 47A68, 47B15. Author addresses:

Faculty of Mathematics and Statistics, Moscow State University, Moscow, USSR Abstract. The file testbook.tex and its included files demonstrate a L^{A}TEX book setup using the documentstyle amsbook. The abstract will appear on the obverse of the title page.

Contents

Preface 1 Chapter I. Operators with Compact Resolvent 3 1. Auxiliary propositions from function theory 3 Chapter II. Keldysh Pencils 5 1. Holomorphic operator-valued functions 5 Appendix A. Nonselfadjoint Equations 7 1. Compact operators 7 Bibliography 9

Preface

This book is a study of polynomial operator pencils, i.e., operator polynomials of the form

A(λ) = A_{0} + λA_{1} + ⋯ + λ^{n}A_{n},

where λ is a spectral parameter and A_{0}, …, A_{n} are linear operators acting in a Hilbert space H. In the simplest cases A(λ) = A − λI and A(λ) = I − λA we come to the usual (linear) spectral problems.

Spectral problems for polynomial pencils arise naturally in diverse areas of mathematical physics (differential equations and boundary value problems, controllable systems, the theory of oscillations and waves, elasticity theory, and hydromechanics). This explains the steady interest in these problems over the last 35 years.

A consideration of the simplest model—a matrix pencil—enables us to see the essential differences between nonlinear and linear spectral problems. If the coefficients of the pencil A_{k} (k = 0, …, n) are matrices of order m, det A_{n} ≠ 0, and all the roots {λ_{k}}_{1}^{nm} of the characteristic equation det A(λ) = 0 are distinct, then the pencil A(λ) has nm eigenvectors {φ_{k}}_{1}^{nm} (i.e., A(λ_{k})φ_{k} = 0). One possible approach is to single out in the system {φ_{k}}_{1}^{nm} various subsystems {φ_{k_{j}}}_{j=1}^{m} forming bases in C^{m}. Another approach leads to the consideration of the system of vectors {φ_{k}, λ_{k}φ_{k}, …, λ_{k}^{n−1}φ_{k}) (k = 1, …, nm), which forms a basis in the space C^{nm}. Both these approaches are fruitful and admit farreaching generalizations.

2 PREFACE

CHAPTER I Operators with Compact Resolvent Which Are Close to Being Normal

1. Auxiliary propositions from function theory

Here we give statements of known results from function theory which are needed in what follows.

If U is a domain in the complex plane C and the function ψ(z) is holomorphic in U, then let M_{ψ}(U) = sup{|ψ(z)|: z ∈ U}, and denote by n_{ψ}(U) the number of roots of ψ(z) in U (counting multiplicity). Also, let D_{r} = {z: |z| < r}, M_{ψ}(r) = M_{ψ}(D_{r}), and n_{ψ}(r) = n_{ψ}(D_{r}).

Lemma 1.1 (Phragmén-Lindelöf theorem). Suppose the function f(z) is holomorphic inside the angle Ω = {z: | arg z| < π(2α)^{−1}} (α ≥ 1) and on its sides, and for some β < α

(1) \underline{lim}_{r→∞}r^{−β} log sup_{|z|=r} |f(z)| < ∞

If |f(z)| ≤ M(| arg z| = π(2α)^{−1}), then |f(z)| ≤ M for all z ∈ Ω.

4 I. OPERATORS WITH COMPACT RESOLVENT

CHAPTER II Keldysh Pencils

1. Holomorphic operator-valued functions

The main object of study in this book is polynomial operator pencils (operator polynomials). However, it is more convenient for us to give certain definitions and results for the more general case of holomorphic operator-valued functions.

Let U be a domain in C, B be a complex Banach space, and f(λ) a B-valued function defined in U. Such a function f(λ) is said to be strongly (weakly) holomorphic in U if for any λ_{0} ∈ U the strong (weak) limit

lim_{λ→λ_{0}} \frac{f(λ) − f(λ_{0})}{λ − λ_{0}} (= f′(λ_{0})) (2)

exists.

Obviously, f(λ) is weakly holomorphic if and only if any function ψ(f(λ)) is holomorphic, where ψ ∈ B^{∗}.

An operator-valued function A(λ) (λ ∈ U) with values in L(H) is said to be uniformly (strongly, weakly) holomorphic if for any λ_{0} ∈ U the uniform (strong, weak) limit

lim_{λ→λ_{0}} \frac{A(λ) − A(λ_{0})}{λ − λ_{0}} (= A′(λ_{0})) (3)

exists.

Test of citations: [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].

6 II. KELDYSH PENCILS

APPENDIX A On the Eigenvalues and Eigenfunctions of Certain Classes of Nonselfadjoint Equations

1. Compact operators

In an appropriate Hilbert space, all the equations considered below can be reduced to the form

y = L(λ)y + f, L(λ) = K_{0} + λK_{1} + ⋯ + λ^{n}K_{n}, (4)

where y and f are elements of the Hilbert space, λ is a complex parameter, and the K_{i} are compact operators.

A compact operator R(λ) is the resolvent of L(λ) if (E + R)(E − L) = E. If the resolvent exists for some λ = λ_{0}, it is a meromorphic function of λ on the whole plane. We say that y is an eigenelement for the eigenvalue λ = c, and that y_{1}, …, y_{k} are elements associated with it (or associated elements) if

y = L(c)y, y_{k} = L(c)y_{k} + \frac{1}{1!} \frac{∂L(c)}{∂c}y_{k−1} + ⋯ + \frac{1}{k!} \frac{∂^{k}L(c)}{∂c^{k}}y. (5)

Note that if y is an eigenelement and y_{1}, …, y_{k} are elements associated with it, then y(t) = e^{ct}(y_{k} + y_{k−1}t/1! + ⋯ + yt^{k}/k!) is a solution of the equation y = K_{0}y + K_{1}∂y/∂t + ⋯ + K_{n}∂^{n}y/∂t^{n}.

8 A. NONSELFADJOINT EQUATIONS

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