An operator bound related to regular operators
The main theorem in this paper asserts, essentially, that if \(T\) is a bounded linear operator on \(L^p(\mu)\), \(f_j, g_j \in L^\infty(\mu)\) for \(1\leq j\leq k\) and for \(h\in L^\infty(\mu)\) we let \(Q(h)\) denote the corresponding multiplication operator on \(L^p(\mu)\) defined by \(Q(h)\varphi=h\varphi\) then there is a constant \(C\) such that \[ |\sum^k_{j=1} Q(g_j)TQ(f_j)|\leq C|\sum^k_{j=1} f_j\otimes g_j|_\infty\tag{*} \] if and only if \(T\) is regular. Proposition 1, which asserts the automatic validity of (*) when \(p=1\) or \(p=\infty\) merely reflects the fact that on \(L^1(\mu)\) or on \(L^\infty(\mu)\) all bounded linear operators are regular. Applications to various concrete classes of operators are given. Reviewer's remark. The reviewer has been recently informed by A. Wickstead that in fact the constant \(C\) may be taken to be the regular norm of \(T\), which in this setting is simply \(||T||\). The main result above is valid in a much broader setting. A simple version is to consider a Dedekind complete Banach lattice \(E\), let \(Z(E)\) denote its centre (i.e. all linear operators on \(E\) for which there is \(\lambda\in\mathbb{R}\) with \(|Tx|\leq\lambda|x|\) for all \(x\in E)\) on which the regular and operator norms coincide. The space of regular operators on \(E\), \({\mathcal L}^r(E)\) is again a Dedekind complete Banach lattice under the regular norm and has its own centre \(Z({\mathcal L}^r(E))\). The mapping from the algebraic tensor product of \(Z(E)\) with itself, equipped with the norm \(|\cdot|_\lambda\), into \(Z({\mathcal L}^r(E))\) is an isometry. Furthermore the inequality analogous to (*) holds for some \(C\) if and only if \(T\) is regular.
scientific article; zbMATH DE number 938529