Cohomology of sheaves of Fréchet algebras and spectral theory (Q883693)

Let \(A\) be a Banach algebra. A finite-dimensional nilpotent Lie subalgebra \(\mathfrak a\) of \(A\) is called supernilpotent if every element \(u\in [{\mathfrak a},{\mathfrak a}]\) is nilpotent. The paper contains a construction of a holomorphic functional calculus for a noncommutative family of operators generating a supernilpotent Lie subalgebra. The main result of the paper is the following Theorem. Let \(X\) be a Banach space, let \(T\subset {\mathcal L}(X)\) be a finite family of operators generating a supernilpotent Lie subalgebra \({\mathfrak g}_T\) in \({\mathcal L}(X)\), and let \(U\) be an open neighbourhood of the Taylor spectrum \(\sigma(T)\) of the family \(T\). Then \({\mathfrak g}_T=\alpha({\mathfrak g})\) and \(T=\alpha(e)\), where \(\mathfrak g\) is a finite-dimensional nilpotent Lie algebra admitting a positive grading with Lie family \(e\) of generators and \(\alpha:{\mathfrak g}\to{\mathcal L}(X)\) is a Lie representation. Moreover, there exists a continuous unital algebra homomorphism \(\widetilde{\alpha}:{\mathfrak T}_{\mathfrak g}(U)\to{\mathcal L}(X)\) from the Fréchet algebra \({\mathfrak T}_{\mathfrak g}(U)\) of all formally radical functions on \(U\) in elements of the Lie algebra \(\mathfrak g\) into the Banach algebra \({\mathcal L}(X)\), and the homomorphism \(\widetilde{\alpha}\) extends the representation \(\alpha\). This result generalizes \textit{J.\,L.\thinspace Taylor}'s holomorphic functional calculus developed in [Adv.\ Math.\ 9, 183--252 (1972; Zbl 0271.46041)].