Functional Calculas and Multiplication

Abstract

As in the preceding section we assume that E is a Riesz space having the principal projection property. Let 0 < e ∈ E. Since we shall restrict ourselves to properties holding in the principal ideal A _e generated by e we may also assume that A _e = E, i.e., e is a strong unit in E. Given the element f ∈ E = A _e , there exist real numbers a, b (a <b) and a number δ > 0 such that ae ≤ f ≤ (b -δ) e. The interval [a, b] is then sometimes called a spectral interval of f. Let [a, b] be such a spectral interval of f and let

$$ \rho :a = {\alpha _0} < {\alpha _1} < \cdots < {\alpha _m} = b $$

be a partition of [a, b]. For any a ∈ [a, b] the band projection onto the band generated by (αe - f)⁺ will be denoted by P _α Note that P _α0 = 0 and P _αm = I (the identity operator). The equality P _αm = P _b = I holds because be − f ≤ δe, so the band generated by (be − f) ⁺ = be − f is the band generated by e i.e.,it is E. Writing

$$ {\upsilon _k} = \left( {{P_{{\alpha _k}}} - {P_{{\alpha _{k - 1}}}}} \right)e{\text{ }}for{\text{ }}k = 1, \ldots ,m, $$

the elements v _k are pairwise disjoint components of e such that ∑ ^m₁ v _k = e and the >e-step functions s = ∑ ^m₁ a _k−1 v _k and S = ∑ ^m₁ α _k v _k satisfy s ≤ f ≤ S (see the proof of Freudenthal’s spectral theorem in the preceding section). The elements s and S are called the lower sum and upper sum belonging to f and the partition P. If the partition points are sufficiently near to each other, then both s and S are near to f. Precisely stated, if α _k − α _k−1 ≤ ∈ for>k = 1,…, m, then 0 ≤ S − s ≤ ∈ e, so

$$ 0 \leq f - s \leq \in e{\text{ }}and{\text{ 0}} \leq S - f \leq \in e. $$