Chapter 37: Linear operators between function spaces
The Chacon-Krengel theorem
L~(U,V)=L×(U;V)=B(U;V) for L-spaces U and V; the class T(0)μν of ∥ ∥1-decreasing, ∥ ∥∞-decreasing linear operators from M1,0(A,μ) to M1,0(B,μ).
The ergodic theorem
The Maximal Ergodic Theorem and the Ergodic Theorem for operators in T(0)μμ; for inverse-measure-preserving functions φ:X→X; limit operators as conditional expectations; applications to continued fractions; mixing and ergodic transformations.
The classes T, T×; the space M0,∞; decreasing rearrangements u*; ∥u*∥p=∥u∥p; ∫|Tu×v|≤∫u*×v* if T∈T; the very weak operator topology and compactness of T; v is expressible as Tu, where T∈T, iff ∫0tv*≤∫0tu* for every t; finding T such that ∫Tu×v=∫u*×v*; the adjoint operator from T(0)μν to T(0)νμ.
T-invariant subspaces of M1,∞, and T-invariant extended Fatou norms; relating T-invariant norms on different spaces; rearrangement-invariant sets and norms; when rearrangement-invariance implies T-invariance.
Linear operators on L0 spaces; if B is measurable, a positive linear operator from L0(A) to L0(B) can be assembled from Riesz homomorphisms.
Kernel operators; free products of measure algebras and tensor products of L0 spaces; tensor products of L1 spaces; abstract integral operators (i) as a band in L×(U,V) (ii) represented by kernels belonging to L0(A)⊗L0(B) (iii) as operators converting weakly convergent sequences into order*-convergent sequences; operators into M-spaces or out of L-spaces.
Function spaces of reduced products
Measure-bounded Boolean homomorphisms on products of probability algebras; associated maps on subspaces of ∏i∈IL0(Ai) and ∏i∈ILp(Ai); reduced powers; universal mapping theorems for function spaces on projective and inductive limits of probability algebras.
ro-PDF (results-only version).
Return to contents page.