On the spectrum of the operator which is a composition of integration and substitution
Ignat Domanov
Studia Mathematica, Tome 187 (2008), p. 49-65 / Harvested from The Polish Digital Mathematics Library
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Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator Vϕ:f(x)↦∫0ϕ(x)f(t)dt be defined on L₂[0,1]. We prove that Vϕ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator Vϕ always equals 1.

Publié le : 2008-01-01
EUDML-ID : urn:eudml:doc:285287 
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     year = {2008},
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Ignat Domanov. On the spectrum of the operator which is a composition of integration and substitution. Studia Mathematica, Tome 187 (2008) pp. 49-65. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm185-1-3/