%%@texfile{% %% filename="chap1.tex", %% version="1.1", %% date="21-JUN-1991", %% filetype="AMS-LaTeX: documentation", %% copyright="Copyright (C) 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.", %% author="American Mathematical Society", %% address="American Mathematical Society, %% Technical Support Department, %% P. O. Box 6248, %% Providence, RI 02940, %% USA", %% telephone="401-455-4080 or (in the USA) 800-321-4AMS", %% email="Internet: Tech-Support@Math.AMS.org", %% checksumtype="line count", %% checksum="70", %% codetable="ISO/ASCII", %% keywords="latex, amslatex, ams-latex", %% abstract="This file is part of the AMS-\LaTeX{} package, version 1.1. %% It is part of the monograph sample, testbook.tex (q.v.)." %%} %%% end of file header % \chapter[Operators with Compact Resolvent]% {Operators with Compact Resolvent\\ Which Are Close to Being Normal} \section{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 $\bold C$ and the function $\psi(z)$ is holomorphic in $U$, then let $M_\psi(U)=\sup\{|\psi (z)|\colon z\in U\}$, and denote by $n_\psi(U)$ the number of roots of $\psi(z)$ in $U$ (counting multiplicity). Also, let $D_r=\{z\colon |z|<r\}$, $M_\psi(r)=M _\psi(D_r)$, and $n_\psi(r)=n_\psi(D_r)$. \begin{lem}[Phragm\'en-Lindel\"of theorem] Suppose the function $f(z)$ is holomorphic inside the angle $\Omega=\{z\colon | \arg z|<\pi(2\alpha)^{-1}\}$ $(\alpha\geq1)$ and on its sides, and for some $\beta<\alpha$ \begin{equation} \varliminf_{r\to\infty}r^{-\beta}\log\sup_{|z|=r}|f(z)|<\infty \end{equation} If $|f(z)|\leq M(|\arg z|=\pi(2\alpha)^{-1})$, then $|f(z)|\leq M$ for all $z\in\Omega$. \end{lem} %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \endinput