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Functional Calculus for Semi-Bounded Operators

We build on the work by Davies, extending the Helffer-Sj\"ostrand Functional Calculus domain for semi-bounded operators on Banach spaces given a priori controlled growth of the resolvents. We employ Seeley's Extension Theorem to extend smooth functions on the half line to the whole line and thus indirectly define functions of these operators.


Introduction
Helffer and Sjöstrand [3] introduced a new formula into the field of spectral theory for self-adjoint operators. It was later shown by Davies [1] that the Helffer-Sjöstrand formula had greater implications in spectral theory than it was originally intended for. He constructed a functional calculus for a much wider class of operators on spaces other than Hilbert under certain assumptions on the norms of the resolvents. He in addition showed that in the special case of self-adjoint operators this coincided with the classical functional calculus. We show that we can widen the class of functions when we consider operators with spectrums which are bounded below, since positive operators take special treatment in the analysis of partial differential equations. The results enable explicit tractable definitions of certain functions of partial differential operators in particular the Heat Semigroup.
Our hypothesis in the pending analysis is H is a closed densely defined operator on a Banach space B, with spectrum σ (H) ⊂ R. It has resolvent operators (z − H) −1 defined and bounded for all z ∈ C satisfying: for some α ≥ 0 and all z ∈ R, where z := (1 + |z| 2 ) 1 2

Preliminaries
We introduce our main concepts. | d n f dx n | ≤ c n x β−n Definition 1. 3 We define the Algebra A as,

Algebra of Slow Decreasing functions
The strict inequality β < 0 is of importance and we conjecture that it cannot be extended to β ≤ 0.

Lemma 1.5 (Davies [2])
A is an algebra under pointwise multiplication. If f ∈ A then the expression defines a norm on A for each n moreover C ∞ c (R) is dense in A with this norm. The completion A n is a Banach space.

Lemma 1.6
The function x β is in A for each β < 0

Proof
The statement follows from the observation that if β < 0 and m ≥ n then The statement follows from the inequality

The Helffer Sjöstrand formula
We introduce the concept of almost analytic extensions due to Hörmander [4].
Definition 1.9 Given f ∈ A we define an almost analytic extensionf to the complex planef and define Definition 1.10 Given f ∈ A and H satisfying our initial hypothesis we define We recall some important results from [1] showing that we do indeed have a functional calculus ii. The operator f (H) is independent of n and the cut-off function σ, subject to n > α iii. If f is a smooth function of compact support disjoint from the spectrum of H then f (H) = 0 iv. If f and g are in A then

Semi-bounded Operators
We modify our main hypothesis by assuming the spectrum of H is bounded below and without loss of generality σ (H) ⊆ [0, ∞).
We introduce a new ring of functions A + Definition 2.1 S β + is the set of smooth functions on R + ∪ {0} with the same decaying property as S β that is for every n there is positive constant c n such that | d n f dx n | ≤ c n x β−n Then A + is defined appropriately and similarly we define the Banach space A + n with norm

Seeley's Extension Theorem
We present a theorem due to Seeley [5] which gives a linear extension operator for smooth functions from the half space to the whole space. This extension operator is continuous for many topologies including uniform convergence of each derivative. We demonstrate a brief proof as it contributes to the proof of continuity for our topology A + n for each n. Definition 2.2 Given f ∈ A + , φ ∈ A and real a we define two operators on

Theorem 2.3 Seeley's Extension Theorem.
There is a linear extension operator The proof of the theorem centres on the following lemma  See [5]. We recall from the proof b k = − 2 k and |a k | < e 4 The series is convergent since for all negative x the sum has only finite nonzero terms. It is evident that (Ef ) (0) = f (0) and when N ∈ N then for all and we deduce that for all n lim dx n (x) to complete the proof.
and the inequality follows.
Lemma 2.6 If φ ∈ A then S φ is a bounded operator with respect to each norm A simple application of Leibnitz gives and so we integrate to give and hence we have our estimate and hence the extension operator is continuous.