This article discusses the so-called monogenic functional calculus for finite tuples of bounded (possibly noncommuting) selfadjoint operators acting on a common Hilbert space or, more generally, for tuples \(\pmb{T} = (T_1,\dots,T_n)\) of bounded linear operators (acting on a common Banach space) such that the spectrum of \(\sum_{j=1}^n \alpha_j T_j\) is a subset of the real line for any real scalars \(\alpha_1,\dots,\alpha_n\). Using Clifford algebras and (a counterpart of) the Cauchy kernel for them, to every such a tuple \(\pmb{T}\) one assigns a compact subset \(\gamma(\pmb{T})\) of \(\mathbb{R}^n\) and defines an assignment \(f \mapsto f(\pmb{T})\) (called the monogenic functional calculus) where \(f\) runs over all complex-valued real-analytic functions (in \(n\) variables) defined on open neighbourhoods of \(\gamma(\pmb{T})\) and \(f(\pmb{T})\) is a bounded linear operator on the underlying Banach space \(X\). In this calculus, monomials \(p(x_1,\dots,x_n) = x_1^{k_1} \cdot \dots \cdot x_n^{k_n}\) act as follows: \[ p(\pmb{T}) = \frac{k_1! \cdot \dots \cdot k_n!}{(k_1+\dots+k_n)!} \sum_{\sigma} T_{\sigma(1)} \cdot \dots \cdot T_{\sigma(n)}, \] where \(\sigma\) runs over all functions from \(\{1,\dots,k_1+\dots+k_n\}\) onto \(\{1,\dots,n\}\) which assume the value \(j\) exactly \(k_j\) times (\(j=1,\dots,n\)).
It is worth noting that, if the operators \(T_j\) do not commute, the monogenic functional calculus for \(\pmb{T}\) is not multiplicative; that is, \((fg)(\pmb{T})\) differs from \(f(\pmb{T}) g(\pmb{T})\) for some \(f\) and \(g\). On the other hand, if \(T_j\) pairwise commute, then \(\gamma(\pmb{T})\) coincides with the Taylor spectrum of \(\pmb{T}\) and the monogenic calculus for \(\pmb{T}\) coincides with Taylor’s.
The author discusses (in detail) Clifford algebras, the Cauchy kernel for them and the Weyl calculus. He also gives some applications to harmonic analysis.
For the entire collection see [Zbl 1325.47001].