% \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
%-----------------------------------------------------------------------------
