Approximation theorems for positive operators on \(L^ p\)-spaces
For \(T\in {\mathcal L}(L^ p(m),L^ r(n))\), where (X,\({\mathcal A},m)\) and (Y,\({\mathcal B},n)\) are \(\sigma\)-finite mesure spaces, let \(M(T)=\{f\in L^ p(m):\| Tf\| =\| T\| \| f\| \}\) and \(J(T)=\{\sup p f: f\in M(T)\}\). A linear bounded operator T is said to be positive (T\(\geq 0)\) if Tf\(\geq 0\) whenever \(f\geq 0\). The set of all positive operators is denoted by \({\mathcal L}_+(L^ p(m),L^ r(n))\) and the positive part of the unit ball of \({\mathcal L}(L^ p(m),L^ r(n))\) is denoted by \({\mathcal P}\). Let \(1<r\leq p<\infty\) and \({\mathcal N}=\{T\in {\mathcal L}_+(L^ p(m),L^ r(n):\) \(X\in J(T),Y\in J(T^*)\}\), then the author proves that \({\mathcal N}\) is norm dense in \({\mathcal L}_+(L^ p(m),L^ r(n))\) and the convex hull conv(\({\mathcal N}\cap ext {\mathcal P}\}\) is strong operator dense in the positive part of the unit sphere of \({\mathcal L}(L^ p(m),L^ r(n))\). The author also considers extreme points of the positive part of the unit ball of \({\mathcal L}(\ell^ p,\ell^ r)\), and proves that, if \(0\neq T\in {\mathcal L}_+(\ell^ p,\ell^ r)\cap {\mathcal N}\), and \(1<r\leq p<\infty\), \(T\in {\mathcal N}\cap ext{\mathcal P}\) if and only if \(\| T\| =1\) and the connected components of the graph G(T) are trees. Furthermore, he proves that, when \(1<r\leq p<\infty\) and \(T\in {\mathcal L}(\ell^ p,\ell^ r)\) if \(T\in {\mathcal N}\cap ext{\mathcal P}\), then T is exposed but T is not a strongly exposed point of \({\mathcal P}\).
scientific article; zbMATH DE number 4200876
scientific article