Functional Calculus for Infinitesimal Generators of Holomorphic Semigroups

Abstract

We give a functional calculus formula for infinitesimal generators of holomorphic semigroups of operators on Banach spaces, which involves the Bochner Riesz kernels of such generators. The rate of smoothness of operating functions is related to the exponent of the growth on vertical lines of the operator norm of the semigroup. The strength of the formula is tested on Poisson and Gauss semigroups in L 1 (R n ) and L 1 (G), for a stratified Lie group G. We give also a self-contained theory of smooth absolutely continuous functions on the half line [0, ). Academic Press

Some applications of fractional calculus to operator semigroups and functional calculus

“…(It is possible to prove a similar result to Theorem 4.1 when the operator A is assumed to have property (G α ) [GP,p. 329], [GMP,p.…”

“…As said before, the algebra AC (ν+ 1 2 ) 2,1 is invariant under the action of the isomorphisms u → u θ , θ > 0, on (0 , ∞) (thus, in the sequel, the symbol f (A) includes general expressions of the form g(A θ )). In particular, the fractional powers A θ with θ > 0 can be defined in a fairly simple way for an operator A with property (HG α ) [GP,. We can choose, for instance, to introduce −A θ as the infinitesimal generator of the holomorphic semigroup e z,θ (A) where e z,θ (u) = exp(−zu θ ), u > 0, ℜz > 0.…”

“…Sectorial operators. The calculus Φ can be approximated by standard holomorphic functional calculus, see [GP,p. 328].…”

“…It allows us to establish an explicit functional calculus for A in the following terms, see [GP,Theorem 4.1 and Theorem 6.3].…”

Algebra Structure of Operator-Valued Riesz Means

“…Given ∈ N, we consider the norm ‖ ⋅ ‖ ( ) defined by In fact, the space ( ) is a Banach algebra under pointwise multiplication for ∈ N (see [4,Proposition 3.4]). Applying Proposition 3, a second proof may be given.…”