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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

MSC:
 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)
Citations:
Zbl 0071.330; Zbl 0507.47014