On non-holomorphic functional calculus for commuting operators

Abstract

We provide a general scheme to extend Taylor's holomorphic functional calculus for several commuting operators to classes of non-holomorphic functions. These classes of functions will depend on the growth of the operator valued forms that define the resolvent cohomology class. The proofs are based on a generalization of the so-called resolvent identity to several commuting operators.

Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula

“…In special cases, for instance when the spectrum is real, such a representation was known earlier, and was used by Droste [10], following Dynkin's approach (1.3), to obtain a smooth functional calculus in the multivariable case for operators with real spectra. This approach is extended to more general spectra in [18].…”

(Ultra)differentiable functional calculus and current extension of the resolvent mapping

“…If ,f is holomorphic in a neighborhood of a (a) and has compact support, then J However, the same formula may have meaning even for an f that is not necessarily holomorphic in a full neighborhood of a( a), provided that 8j(z) has enough decay when approaching a(a) to balance the growth of (some representative of) the resolvent. In one variable this approach was first exploited by Dynkin, [8]; for several commuting operators a similar approach is used by Droste, [7], and recently by Sandberg, [13]; for the case when is real, see [4]. Notice that such an approach will always require that a f -0 on a( a) which is a very strong restriction if a (a) contains some complex structure.…”

On the Taylor functional calculus

“…, A n ) be an n-tuple of mutually commuting operators acting on a Banach space X. The existence of the Taylor functional calculus [18], [19] (for simpler versions see [10], [8], [3]- [5] and [15]) is one of the most important results of spectral theory. However, the formula defining f (A) for a function f analytic on a neighbourhood of the Taylor spectrum has some drawbacks.…”