# Journal of Operator Theory

Volume 32, Issue 2, Fall 1994 pp. 311-329.

Spectral invariance and the holomorphic functional calculus of J.L. Taylor in $\Psi*$-algebras**Authors**: Robert Lauter

**Author institution:**Fachbereich 17 - Mathematik, Johannes Gutenberg-UniversitÃ¤t Mainz, D-55099 Mainz, Germany

**Summary:**If X is a Hilbert space it is shown that very general subalgebras A of $\mathcal L (X)$ contain the holomorphic functional calculus in several variables in the sense of J.L. Taylor. In particular, Taylorâ€™s holomorphic functional calculus applies to $\Psi*$-algebras (cf. [12], Definition 5.1), and so gives a useful tool for the investigation of certain algebras of pseudo-differential operators and of FrÃ©chet operator algebras on singular spaces. Taylorâ€™s holomorphic functional calculus applies also to algebras of $n \times n$-matrices with elements in $\Psi*$-algebras and even more general algebras. Furthermore, an example shows that Taylorâ€™s holomorphic functional calculus for at least three commuting operators on a Hilbert space is, in general, richer than any other multidimensional holomorphic functional calculus in commutative subalgebras of $\mathcal L (X)$.

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