Vol. 31, No. 1, 1969

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ISSN: 0030-8730
Isometries of certain function spaces

Kwok-Wai Tam

Vol. 31 (1969), No. 1, 233–246

Let X be a discrete symmetric Banach function space with absolutely continuous norm. We prove by the method of generalized hermitian operator that an operator U on X is an onto isometry if and only if it is of the form: Uf(.) = u(.)f(T.) all f X, where u is a unimodular function and T is a set isomorphism of the underlying measure space. That other types of isometries occur if the symmetry condition is not present is illustrated by an example. We completely describe the isometries of a reflexive Orlicz space L(L2) provided the atoms have equal mass (the atom-free case has been treated by G. Lumer); similarly for the case that no Hilbert subspace occurs.

Mathematical Subject Classification 2000
Primary: 46E30
Secondary: 46A45, 47B37
Received: 9 October 1967
Revised: 8 May 1969
Published: 1 October 1969
Kwok-Wai Tam