Similarity of operators associated with the Volterra relation
If \(Vf(x)=\int_0^x f(t)\,dt\) is the classical Volterra operator in \(L^p(0, 1)\) and \(S\) is the multiplication operator \(Sf(x)=xf(x)\), then the commutation relation \(SV-VS=V^2\) holds. It was proved by the author [cf.\,Theorem 11.17 in Lect.\ Notes Math.\ 1012 (1983; Zbl 0527.47001)] that \(S+\alpha V\) is similar to \(S\) if and only if \(\Re\alpha=0\), the result extended later to bounded operators \(S,V\) satisfying the commutation relation (and additional conditions) [\textit{S.\,Kantorovitz}, Trans.\ Am.\ Math.\ Soc.\ 156, 193--218 (1971; Zbl 0215.21002)] and then to \(S\) unbounded with \(V\)-invariant domain and \(iS\) generating a bounded strongly continuous group \(G(t)\) [\textit{S.\,Kantorovitz} and \textit{K.\,J.\thinspace Pei}, Indiana Univ.\ Math.\ J.\ 23, 1177--1197 (1974; Zbl 0287.47027)]. In particular, \(S - V\) is not similar to \(S\). The main result of the present paper is that \((S - V) + P\) is similar to \(S\) for all \(P\) in the similarity sub-orbit \({\mathcal V}=\{S(a)VS(-a)\mid a\in{\mathbb R}\}\) of \(V\) (the group \(G(t)\) need not be bounded). There is a sharper result for bounded \(S\). The results are applied to scalar-type spectral operators.
scientific article; zbMATH DE number 5547489
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