A class of extreme \(L_ p\) contractions, \(p\neq 1,2,\infty\), and real \(2\times 2\) extreme matrices (Q1072090)

Starting with a simple inequality that is sharper than the classical Clarkson-Lamperti inequality, we establish a sufficient condition for the extremeness of a contraction \(T: L_ p(X)\to L_ q(Y)\), \(2<p\leq q<\infty\). The condition: for some \(A\subset X\), the linear spans of \(\{1_ Af:\) \(\| Tf\| =\| f\| \}\) and of \(\{1_{Y\setminus B}g:\) \(\| T^*g\| =\| g\| \}\) are dense in \(L_ p(A)\) and \(L_{q/(q-1)}(Y\setminus B)\) respectively, where \(B=\sup \{\sup p Tf:\) \(f\in L_ p(A)\) and \(\| Tf\| =\| f\| \}\). Furthermore, the condition implies \(TL_ p(X\setminus A)\subset L_ q(Y\setminus B)\). From this, we obtain a characterization of extreme contractions from \(L_ p(X)\) to \(L_ p(Y)\), \(1<p<\infty\), \(p\neq 2\), that are direct sums of disjunctive and codisjunctive operators. (A disjunctive operator is one that maps functions with disjoint supports to those also with disjoint supports; its adjoint is a codisjunctive operator.) This generalizes a result of \textit{R. Grzaślewicz} [Port. Math. 40, 413-419 (1981; review above)], in which are considered only \(\ell_ p\) spaces. We prove that operators in the weak* closed convex hull of these particular extreme contractions have contractive linear moduli in the sense of \textit{R. V. Chacon} and \textit{U. Krengel} [Proc. Am. Math. Soc. 15, 553-559 (1964; Zbl 0168.117)]. We also study \(2\times 2\) extreme \(\ell_ p(2)\) matrices, \(1<p<\infty\), \(p\neq 2\), and use the results to extend our characterization to a larger class of contractions and to construct some new types of \(3\times 3\) and \(2\times 2\) extreme contractions on \(\ell_ p(3)\) and \(\ell_ p(2)\) respectively.