% \iffalse
%%% ====================================================================
%%% @LaTeX-file{
%%% author = "American Mathematical Society",
%%% version = "1.2beta",
%%% date = "11-Oct-1994",
%%% time = "15:08:35 EDT",
%%% filename = "app.tex",
%%% copyright = "Copyright (C) 1994 American Mathematical Society,
%%% all rights reserved. Copying of this file is
%%% authorized only if either:
%%% (1) you make absolutely no changes to your copy,
%%% including name; OR
%%% (2) if you do make changes, you first rename it
%%% to some other name.",
%%% address = "American Mathematical Society,
%%% Technical Support,
%%% Electronic Products and Services,
%%% P. O. Box 6248,
%%% Providence, RI 02940,
%%% USA",
%%% telephone = "401-455-4080 or (in the USA and Canada)
%%% 800-321-4AMS (321-4267)",
%%% FAX = "401-331-3842",
%%% checksum = "62749 81 386 3780",
%%% email = "tech-support@math.ams.org (Internet)",
%%% codetable = "ISO/ASCII",
%%% keywords = "latex, amslatex, ams-latex, amstex, documentation",
%%% supported = "yes",
%%% abstract = "This file is part of the AMS-\LaTeX{} package.
%%% It is part of the monograph sample,
%%% testbook.tex (q.v.).",
%%% docstring = "The checksum field above contains a CRC-16
%%% checksum as the first value, followed by the
%%% equivalent of the standard UNIX wc (word
%%% count) utility output of lines, words, and
%%% characters. This is produced by Robert
%%% Solovay's checksum utility.",
%%% }
%%% ====================================================================
% \fi
%-----------------------------------------------------------------------------
% Beginning of app.tex
%-----------------------------------------------------------------------------
\appendix
\chapter[Nonselfadjoint Equations]%
{On the Eigenvalues and Eigenfunctions\\
of Certain Classes of Nonselfadjoint Equations}
\section{Compact operators} In an appropriate Hilbert space, all
the equations considered below can be reduced to the
form
\begin{equation}
y=L(\lambda)y+f,\qquad L(\lambda)=K_0+\lambda K_1+\dots+\lambda^n
K_n,
\end{equation}
where $y$ and $f$ are elements of the Hilbert space,
$\lambda$ is a complex parameter, and the $K_i$ are
compact operators.
A compact operator $R(\lambda)$ is the resolvent of
$L(\lambda)$ if $(E+R)(E-L)=E$. If the resolvent exists
for some $\lambda=\lambda_0$, it is a meromorphic function
of $\lambda$ on the whole plane. We say that $y$ is
an eigenelement for the eigenvalue $\lambda=c$, and that
$y_1,\dots,y_k$ are elements associated with it (or
associated elements) if
\begin{equation}
y=L(c)y,\quad y_k=L(c)y_k+\frac{1}{1!}\,\frac{\partial L(c)}{\partial c}
y_{k-1}+\dots+\frac{1}{k!}\,\frac{\partial^kL(c)}{\partial c^k}y.
\end{equation}
Note that if $y$ is an eigenelement and $y_1,\dots,y_k$
are elements associated with it, then $y(t)=e^{ct}(y_k
+y_{k-1}t/1!+\dots+yt^k/k!)$ is a solution of the equation
$y=K_0y+K_1\partial y/\partial t+\dots+K_n\partial^ny/
\partial t^n$.
\endinput
%-----------------------------------------------------------------------------
% End of app.tex
%-----------------------------------------------------------------------------