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Compact Endomorphisms of Banach Algebras of Infinitely Differentiable Functions
Author(s) -
Feinstein Joel F.,
Kamowitz Herbert
Publication year - 2004
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610703005131
Subject(s) - endomorphism , pointwise , mathematics , commutative property , unit interval , banach algebra , ideal (ethics) , zero (linguistics) , interval (graph theory) , banach space , unit disk , discrete mathematics , pure mathematics , combinatorics , mathematical analysis , philosophy , linguistics , epistemology
Let ( M n ) be a sequence of positive numbers satisfying M 0 = 1 andM n + mM n M m⩾ (n + mm)for all non‐negative integers m, n . LetD ( [ 0 , 1 ] , M ) = { f ∈ C ∞ ( [ 0 , 1 ] ) :‖ f ‖ D = ∑ n = 0 ∞‖f ( n )‖ ∞M n< ∞} .With pointwise addition and multiplication, D ([0,1], M ) is a unital commutative semisimple Banach algebra. If lim n →∞ ( n !/ M n ) 1/n =0, then the maximal ideal space of the algebra is [0,1], and every non‐zero endomorphism T has the form Tf ( x )= f (ϕ( x )) for some selfmap ϕ of the unit interval. The authors have previously shown for a wide class of ϕ mapping the unit interval to itself that if ‖ϕ′‖ ∞ <1, then ϕ induces a compact endomorphism. The paper investigates the extent to which this condition is necessary, and the spectra of all compact endomorphisms of D ([0,1], M ) are determined. Some of the authors' earlier results on general endomorphisms of D ([0,1], M ) are simplified and strengthened.