A constructive proof of the composition rule for Taylor's functional calculus (Q2703044)

For a commuting tuple \(a= (a_1,\dots, a_n)\) of bounded operators in a given Banach space, let us denote by \(\sigma(a)\) the (Taylor) joint spectrum of \(a\). For \(g= (g_1,\dots, g_m)\) a tuple of functions holomorphic in a neighborhood of \(\sigma(a)\) and \(f\) a function holomorphic in a neighborhood of \(g(\sigma(a))\), \textit{M. Putinar} showed [J. Oper. Theory 7, 149-155 (1982; Zbl 0483.46028)] that one has \(f(g(a))= f\circ g(a)\), where \(h(a)\) is given by the holomorphic functional calculus for each holomorphic function \(h\) in a neighborhood of \(\sigma(a)\). In the present paper, a new constructive proof of this result is proposed, which can be regarded as a continuation of a research started by the first named author.