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On isometric domains of positive operators on Orlicz spaces. (English) Zbl 0536.46019
If \(\phi\) is a non-negative convex strictly increasing function on \(R=(- \infty,\infty)\), then the Orlicz space \(L_{\phi}\) consists of functions of the form f on R such that \(\int_{R}\phi(| f(x)| r)\quad dx<\infty\) for some number \(r>0\). The ’Luxemburg norm’ \(\| \cdot \|\) on \(L_{\phi}\) is defined by \(\| f\| =\inf \{\beta>0:\int_{R}\phi(| f(x)| /\beta)\quad dx\leq 1.\) [Ref.: G. Weiss, Port. Math. 15, 35-47 (1956; Zbl 0071.330)]. If \(T:L_{\phi}\to L_{\phi},\) then the space M(T) is defined to be \(\{f\in L_{\phi}:\| T(f)\| =\| T\|\| f\| \}.\) In the main theorem of this paper, the author states that if M(T) is a linear subspace of \(L_{\phi}\) for every positive linear operator \(T:L_{\phi}\to L_{\phi},\) then \(\phi(t)=Ct^ p,\) where \(C>0\), and \(1<p<\infty.\)
(Reviewer’s comments): (i) A statement in the paper’s introduction that M(T) is a linear lattice when \(\phi(t)=t^ p\), \(1<p<\infty\), appears doubtful. (It is not obvious that f and g in M(T) implies \(f+g\) in M(T)). In the special case \(\phi(t)=t^ p\), representations of functions in M(T) are indicated in the reviewer’s paper: Bull. Math. No.6, 53-67 (1982; Zbl 0507.47014).
(ii) Although \(\| U_{a,b,c,d}(f)\| \leq(\| \chi_{(c,d)}\| \| \chi_{(a,b)}\|^{-1})f,\) is derived from Jensen’s inequality, the result \(\| \xi U_{0,b,0,c}\| =1\) indicated in the proof of the main theorem is not obvious.
Reviewer: G.O.Okikioulu

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47B38 Linear operators on function spaces (general)