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Quasicompact endomorphisms of commutative semiprime Banach algebras

100%
Feinstein J., Kamowitz H.
Banach Center Publications
|
2010
|
tom 91
|
nr 1
159-167
EN
This paper is a continuation of our study of compact, power compact, Riesz, and quasicompact endomorphisms of commutative Banach algebras. Previously it has been shown that if B is a unital commutative semisimple Banach algebra with connected character space, and T is a unital endomorphism of B, then T is quasicompact if and only if the operators Tⁿ converge in operator norm to a rank-one unital endomorphism of B. In this note the discussion is extended in two ways: we discuss endomorphisms of commutative Banach algebras which are semiprime and not necessarily semisimple; we also discuss commutative Banach algebras with character spaces which are not necessarily connected. In previous papers we have given examples of commutative semisimple Banach algebras B and endomorphisms T of B showing that T may be quasicompact but not Riesz, T may be Riesz but not power compact, and T may be power compact but not compact. In this note we give examples of commutative, semiprime Banach algebras, some radical and some semisimple, for which every quasicompact endomorphism is actually compact.
2
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Compact endomorphisms of $H^∞(D)$

100%
Feinstein J. F., Kamowitz H.
Studia Mathematica
|
1999
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tom 136
|
nr 1
87-90
EN
Compact composition operators on $H^∞(G)$, where G is a region in the complex plane, and the spectra of these operators were described by D. Swanton ( Compact composition operators on B(D), Proc. Amer. Math. Soc. 56 (1976), 152-156). In this short note we characterize all compact endomorphisms, not necessarily those induced by composition operators, on $H^∞(D)$, where D is the unit disc, and determine their spectra.
3
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Some properties of endomorphisms of Lipschitz algebras

80%
Kamowitz H., Scheinberg S.
Studia Mathematica
|
1990
|
tom 96
|
nr 3
255-261
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