The Weyl calculus and Clifford analysis

Abstract

The connection between Clifford analysis and the Weyl functional calculus for an n-tuple of bounded selfadjoint operators is noted. The operators do not necessarily commute with each other.

Exponential bounds for noncommuting systems of matrices

“…In the case that an exponential bound like (1) holds, γ(A, B) coincides with the support of the Weyl functional calculus for (A, B) (see [6]). As is usual in spectral theory, γ(A, B) is just the set of singularities of the Cauchy kernel ω → G ω (A, B) in the same way that the spectrum of a single operator is the set of singularities of its resolvent family.…”

“…Moreover, it is proved in [6] that the Weyl functional calculus and the functional calculus mentioned above agree on their common domain and the two related notions of "joint spectrum" γ(T ) for the system T coincide. It is well known that the bound (1) , it is not obvious how the imaginary parts of the eigenvalues will grow as | ζ| → ∞.…”

The Riesz-Dunford and Weyl Functional Calculi

“…According to [4,Proposition 4.19, Theorem 4.22], the linear mapping T is continuous from H M (γ(A)) into L(X). An appeal to the complex Cauchy integral formula and properties of the Bochner integral (see [5] for a similar argument) give…”

“…In the case that A 1 , A 2 are selfadjoint, γ(A) coincides with the support of the Weyl functional calculus for A [5]. As is usual in spectral theory, γ(A) is just the set of singularities of the Cauchy kernel ω −→ G ω (A) in the same way that the spectrum of a single operator is the set of singularities of its resolvent family.…”

Function theory in sectors

“…, A n do not commute with one another. If the n-tuple A has a Weyl functional calculus [1], that is, it satisfies suitable exponential growth conditions, then formula (2) is the restriction to real-analytic functions f of the Weyl functional calculus f → f (A) [4,Corollary 5.5].…”