On the spectrum of the operator which is a composition of integration and substitution
Volume 185 / 2008
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 $1$.
