Geodesic
On the spectrum of the operator which is a composition of integration and substitution
Ignat Domanov 1
1 Institute 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
Studia Mathematica, Tome 185 (2008) no. 1, pp. 49-65 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $0\varepsilon1$. Also, we show that the spectral trace of the operator $V_{\phi}$ always equals $1$.
DOI : 10.4064/sm185-1-3
Keywords: phi rightarrow nondecreasing continuous function phi operator phi mapsto int phi defined prove phi has finite number nonzero eigenvalues only phi phi varepsilon varepsilon spectral trace operator phi always equals

Ignat Domanov  1

1 Institute 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 
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Ignat Domanov. On the spectrum of the operator which is a composition
 of integration and substitution. Studia Mathematica, Tome 185 (2008) no. 1, pp. 49-65. doi: 10.4064/sm185-1-3

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