Norm structure functions and extremeness criteria for operators on \(L_ p\) \((p\leq 1)\) or onto \(C(K)\) (Q1913490)

If \(E,F\) are Banach spaces, let \({\mathcal L}={\mathcal L}(E,F)\) denote the Banach space of bounded linear operators mapping \(E\) into \(F\), and let \({\mathcal U}={\mathcal U}(E,F)\) denote the unit ball \(\{T\in{\mathcal L}:|T|\leq 1\}\). The results of this paper state extremeness criteria for operators in \(\mathcal U\) for cases in which \(E=L_p(\mu)\), \(F=L_q(\nu)\), where \((X,{\mathcal F},\mu)\), \((X,{\mathcal F},\nu)\) are measure spaces and \(0<p\leq 1\leq q<\infty\), or \(E=F=C(K)\), the space of continuous functions on a compact Hausdorff space \(K\). For bounded operator \(T\), let \[ {\mathcal M}(T)=\{\theta\in L^+_\infty(\mu):|T(f)|\leq|\theta f|\text{ for }f\in E\}, \] where \(E=L_p(\mu)\), \(F\) has `an \(s\)-norm', \(p<s<1\). Then \(\delta_1(T)\) is defined to be \(\inf\{\theta:\theta\in{\mathcal M}(T)\}\), and if \({\mathcal W}(T)=\{|T(f)|:f\in E,\;|f|\leq 1\}\), then \(\delta_\infty(T)\) is defined to be \(\sup\{\phi:\phi\in{\mathcal W}(T)\}\). In the main results of the paper, it is shown that (I) the operator \(T\in{\mathcal U}(L_p(\mu),L_q(\nu))\) is extreme if and only if \(\delta_1(T)=1\) in case \(p=1\leq q\) or \(p=1=q\), or if and only if \(\delta_1(T)=\psi_{\@(\mu)}\), where \(\@(\mu)\) is the supremum of atoms of \(\mu\), in the case \(p<1<q\) or \(p<1=q\); and (II) if \(E\), \(F\) are AM-spaces, \(B\) is a subalgebra of \(F\), \({\mathcal U}_1=\{S\in{\mathcal U}:SE\subseteq B\}\) then \(T\in{\mathcal U}_1\) is extreme if and only if \(\delta_\infty(T)=|T|1\). References of the paper include some special cases of the general statements.