Chapter 37: Linear operators between function spaces
371
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).
372
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.
373
Decreasing rearrangements
The classes T, T×; the space M0,∞;
decreasing rearrangements u*; ∥u*∥p=∥u∥p;
∫|Tu×v|≤∫u*×v* if ∈T; the
very weak operator topology and compactness of T; v is
expressible as Tu, where ∈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.
374
Rearrangement-invariant spaces
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.
375
Kwapien's theorem
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.
376
Integral operators
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.
377
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.
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