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On the spectrum of the operator which is a composition of integration and substitution

Volume 185 / 2008

Ignat Domanov Studia Mathematica 185 (2008), 49-65 MSC: 34l20, 45C05, 47A10, 47A75. DOI: 10.4064/sm185-1-3

Abstract

Let $\phi : [0,1]\rightarrow [0,1]$ be a nondecreasing continuous function such that $\phi(x)>x$ for all $x\in (0,1)$. Let the operator $V_{\phi} : f(x)\mapsto \int_0^{\phi(x)}f(t)\,dt$ be defined on $L_2[0,1]$. We prove that $V_{\phi}$ has a finite number of nonzero eigenvalues if and only if $\phi(0)>0$ and $\phi(1-\varepsilon)=1$ for some $0<\varepsilon<1$. Also, we show that the spectral trace of the operator $V_{\phi}$ always equals $1$.

Authors

  • Ignat DomanovInstitute of Applied Mathematics
    and Mechanics
    Ukrainian National Academy of Sciences
    R. Luxemburg St. 74
    83114 Donetsk, Ukraine
    and
    Mathematical Institute
    of the Academy of Sciences
    of the Czech Republic
    Žitná 25
    CZ-115 67 Praha 1, Czech Republic
    e-mail

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