The Stieltjes convolution and a functional calculus for non-negative ...

Another very powerful tool based on path integrals in the complex plane
and the Cauchy integral theorem rather than transform methods is the more
recently developed H ∞ -calculus (see for example the papers by McIntosh [13],
Cowling et al. [14] or the more recent one by Kalton and Weis [15], just to
name a few). One of the main advantages of this approach is that it naturally
leads to a spectral theorem, but it does not allow for non-commuting operators
and purely real Banach spaces.
We will now use our approach to the distributional Stieltjes transform to
define a functional calculus. Since we use the properties of the resolvents of
A only, the calculus can be applied to all M-bounded resolvent families. A
family {R(λ) : λ > 0} of bounded operators on a Banach space X is called
an M-bounded resolvent family if it fulfills the resolvent equation
R(λ 1 ) − R(λ 2 ) = (λ 2 − λ 1 )R(λ 1 )R(λ 2 ) ∀λ 1 ,λ 2 > 0
and has the property sup λ>0 ||λR(λ)|| ≤ M. It is easy to check that R(·)
is infinitely often differentiable with ∂ k R(·) = (−1) k k!R(·) k+1 . Note that the
resolvent families {(λ + A) −1 ; λ > 0} of every non-negative operator is M-
bounded.
We will restrict our functional calculus to those distributions in D S ∗(R n +) ′
and expect it to return only bounded operators. An extension to the whole
class D S (R + ) ′ for n = 1 would probably also return unbounded operators and
could provide an approach to fractional powers of operators with negative
exponents. This conjecture is based on the observation that for every α > 0
and an arbitrary m ∈ N with m > α the distribution S α := Cα,mD −1 m−1 λ m−α−1 ,
where C α,m := (m − 1)! −1 ∫ ∞
0 λ m−α−1 ( 1
λ+1 )m dλ, is in D S (R + ) ′ \ D S ∗(R + ) ′ with
∫ ∞
S(S α )(a) = Cα,m(m −1 − 1)! λ m−α−1( 1 ) mdλ = a −α .
0 λ + a
Definition 9 Let {R i (λ); λ > 0} for every i = 1,...,n be an M-bounded
resolvent family on the Banach space X. We define the linear mapping R R :
D S ∗(R n +) ′ → L(X) by
R R (S) := ∑ ∫ n∏
⃗ k! R i (λ i ) ki+1 dµ ⃗k ( ⃗ λ),
| ⃗ k|≤K
if S has the representation (8).
R n + i=1
It is not clear yet that our definition is independent of the particular representation
(8) of S. To prove this, we will first show Lemma 6 that tells us that
the values 〈R R (S)x,x ∗ 〉 do not depend on the representation from which the
operator R R (S) was derived.
Lemma 6 If we define ϕ x,x ∗ ∈ H(R n +) by ϕ x,x ∗( ⃗ λ) := 〈 ( ∏ n
i=1 R i (λ i ))x,x ∗〉
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