Isometries of a function space (Q1096847)

\textit{R. E. Edwards} proves in his paper: Can. J. Math. 17, 839-846 (1965; Zbl 0128.347) that all bipositive isomorphisms of L p \((1\leq p<\infty)\) convolution algebras of a compact group are induced by bicontinuous isomorphisms of the group. By changing the algebra isomorphism from bipositive to an isometry, \textit{R. S. Strichartz} establishes in his paper: Proc. Am. Math. Soc. 17, 858-862 (1966; Zbl 0168.114), the same type of results with the exception of \(p=2.\) In this note the author of this paper gives a general form to these facts proving the isometry equivalent to a combination of a function h and a Borel measurable mapping \(\phi\) of real numbers R. The norm of a function in L \(1\cap L\) p(R), denoted by \(\| f\|_{\cap}\), is defined by \(\| f\|_{\cap}=\| f\|_ p+\| f\|_ 1\). The main theorem: Let \(1\leq p<\infty\), \(p\neq 2\) and U be a one-one onto linear transformation on positive functions of L \(1\cap L\) p(R) such that \(\| Uf\|_{\cap}=\| f\|_{\cap}\). Then there is a one-one Borel measurable mapping \(\phi\) of R onto itself and a function h such that \(Uf=h(f(\phi))\) for all positive \(f\in L\) \(1\cap L\) p(R).