Chapter 37: Linear operators between function spaces

371
The Chacon-Krengel theorem

*L*^{~}(*U*,*V*)=*L*^{×}(*U*;*V*)=** B**(

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**,

374
Rearrangement-invariant spaces

** T**-invariant subspaces of

375
Kwapien's theorem

Linear operators on *L*^{0} spaces; if **B** is measurable, a
positive linear operator from *L*^{0}(**A**) to *L*^{0}(**B**) can
be assembled from Riesz homomorphisms.

376
Integral operators

Kernel operators; free products of measure algebras and tensor
products of *L*^{0} spaces; tensor products of *L*^{1} spaces; abstract
integral operators (i) as a band in *L*^{×}(*U*,*V*) (ii)
represented by kernels belonging to *L*^{0}(**A**)⊗*L*^{0}(**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∈I}*L*^{0}(**A**_{i}) and
∏_{i∈I}*L*^{p}(**A**_{i}); reduced powers; universal mapping
theorems for function spaces on projective and inductive limits of
probability algebras.

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Revised 10.4.10