| Popis: |
As in the preceding section we assume that E is a Riesz space having the principal projection property. Let 0 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 ∑ 1 m v k = e and the >e-step functions s = ∑ 1 m a k−1 v k and S = ∑ 1 m 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 $$ 0 \leq f - s \leq \in e{\text{ }}and{\text{ 0}} \leq S - f \leq \in e. $$ |
