Volterra systems of operators (Q1110060)

Put \(T_{\alpha}=S+\alpha V\) (\(\alpha\in {\mathbb{C}})\), where the multiplication S: \(f(x)\to xf(x)\) and the classical Volterra operator \(V:f(x)\to \int^{x}_{0}f(t)dt\) acting on \(L^ p(0,1)\) \((1<p<\infty)\). The operators \(T_{\alpha}\) are having some interesting backgrounds. See, for instance, \textit{L. D. Faddeev} [Trudy Math. Inst. Steklov 73, 314-336 (1964; Zbl 0145.467)]. In the previous paper [Trans. Am. Math. Soc. 146, 61-67 (1970; Zbl 0189.135)], the athor obtained (1) \(| Re \alpha | \leq m\Leftrightarrow T_{\alpha}\in [C^ m]\), i.e. \(\exists\) constant K: \(\| p(T_{\alpha})\| =K\| p\|_{m,[0,1]}\) \((=K\sum^{m}_{j=1}(1/j!)\sup_{[0,1]}| p^{(j)}|\) (\(\forall\) polynomial p), (2) Re \(\alpha\) \(=Re \beta \Leftrightarrow T_{\alpha}\) is similar to \(T_{\beta}\), and (3) Re \(\alpha\) \(=0\Leftrightarrow T_{\alpha}\) is spectral. The present paper devotes to prove natural ``N-dimensional'' versions of the above results.