Joint spectrum of subnormal \(n\)-tuples of composition operators (Q2782631)

For a selfmap \(\psi\) of the unit disk \(\mathbb{D}\) let \(C_\psi: f \mapsto f\circ\psi\) be the associated composition operator on the Hardy space \(H^2\). For the special case \(\psi(z)=\varphi_a(z)=z/(az+(a+1))\), \(a>0\), the author defines for \(\lambda>0\) the powers \(C_\psi^\lambda\) of \(C_\psi\) by \(C_\psi^\lambda=C_{\varphi_{(1+a)^\lambda-1}}\). NEWLINENEWLINENEWLINE\textit{C. C. Cowen} [Integral Equations Oper. Theory 11, No. 2, 151-160 (1988; Zbl 0638.47027)] showed that these operators form a strongly continuous, one-parameter semigroup of subnormal (hyponormal) composition operators.NEWLINENEWLINENEWLINEIn [ibid. 43, 385-396 (2002)], the author of the paper under review began his series of papers on studying \(n\)-tuples of subnormal (hyponormal) composition operators on \(H^2\) induced by fractional self-maps of \(\mathbb{D}\). In the present paper, the author now computes for \(\psi(z)=z/(z+2)\) the Taylor spectrum of the \(n\)-tuple \((C_{\psi}^{\lambda_1}, C_{\psi}^{\lambda_2},\dots, C_{\psi}^{\lambda_n})\). His tools come from the general theory of strongly continuous semigroups of operators and multiparameter spectral theory, in particular a theorem of \textit{R. E. Curto} and \textit{L. A. Fialkow} [Integral Equations Oper. Theory 10, 707-720 (1987; Zbl 0639.47004)].