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Algebra i Analiz, 1993, Volume 5, Issue 6, Pages 69–96 (Mi aa415)  

This article is cited in 1 scientific paper (total in 1 paper)

Research Papers

Classification of finite-dimensional algebras generated by the calkin image of a composition operator on $L^p$ with weight

A. Böttcher, H. Heidler

Technische Universität Chemnitz, Fakultät für Mathematik
Abstract: Given a countable infinite set $X$ and a weight $\mu\colon X\to(0,\infty)$, we denote by $l_{\mu}^p(X)$ the Banach space of all functions $f\colon X\to\mathbb C$ such that $\sum_{x\in X}|f(x)|^p\mu(x)<\infty$. The composition operator $C_a$ on $l_{\mu}^p(X)$ induced by a self-map $a\colon X\to X$ is defined by $(C_af)(x)=f(a(x))$. We establish a criterion for $C_a$ to be essentially algebraic, i.e., for the existence of a polynomial $q(z)$ such that $q(C_a)$ is compact. The polynomial $q(z)$ of minimal degree with this property is referred to as the essentially characteristic polynomial of $C_a$. We provide a list of all polynomials that may be the essentially characteristic polynomial of some composition operator on $l_{\mu}^p(X)$, which results in a complete classification of the finite-dimensional algebras generated by the Calkin image of a single composition operator on $l_{\mu}^p(X)$.
Keywords: composition operators, finite-dimensional algebras, algebraic operators, Calkin algebra.
Received: 13.04.1993
Bibliographic databases:
Document Type: Article
Language: English
Citation: A. Böttcher, H. Heidler, “Classification of finite-dimensional algebras generated by the calkin image of a composition operator on $L^p$ with weight”, Algebra i Analiz, 5:6 (1993), 69–96; St. Petersburg Math. J., 5:6 (1994), 1099–1119
Citation in format AMSBIB
\Bibitem{BotHei93}
\by A.~B\"ottcher, H.~Heidler
\paper Classification of finite-dimensional algebras generated by the calkin image of a~composition operator on~$L^p$ with weight
\jour Algebra i Analiz
\yr 1993
\vol 5
\issue 6
\pages 69--96
\mathnet{http://mi.mathnet.ru/aa415}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1270062}
\zmath{https://zbmath.org/?q=an:0814.47033|0824.47022}
\transl
\jour St. Petersburg Math. J.
\yr 1994
\vol 5
\issue 6
\pages 1099--1119
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  • This publication is cited in the following 1 articles:
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    Алгебра и анализ St. Petersburg Mathematical Journal
     
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