Conditional multiplication operators (Q2447320)

Let \((X,{\mathcal X},\mu)\) be a \(\sigma\)-finite measure space with set of \({\mathcal X}\)-measurable functions denoted by \(L^0({\mathcal X})\), and for \(1\leq p\leq\infty\), the \(L^p\)-space be defined by \[ L^p(X)= \{[f]:\|[f]\|_p= \| f\|_p< \infty\}, \] where \([f]\) is the equivalence class of functions which differ from \(f\) on sets of measure \(0\), and \[ \| f\|_p= \Biggl(\int_X|f(x)|^p d\mu(x)\Biggr)^{1/p},\quad 1\leq p<\infty. \] Let \({\mathcal A}\) be a subalgebra of \({\mathcal X}\) and, if \(f\in L^0({\mathcal X})\bigcup_{r\geq 1} L^r(X)\), let \(E(f)\) denote the \({\mathcal A}\)-measurable functions such that \(E(f)\in{\mathcal A}\) and \(\int_F f\,d\mu= \int_F E(f)\,d\mu\), \(F\in{\mathcal A}\). If \(w\) and \(u\) are functions in \(L^0({\mathcal A})\cap \text{Dom}(E)\), then the multiplication transformation \(M_w\) and the conditional multiplication operator, \(T_u\), induced by the weight function \(u\), are defined by \(M_w(f)= wf\), \(T_u(f)= uE(f)+ fE(u)- E(u)E(f)\). For \(1\leq p<\infty\), the class of operators \({\mathcal K}_p\) with norm \(\|\cdot\|_{p\to p}\), and the class of weight functions \({\mathcal L}_p\) with norm \(\|\cdot\|_{{\mathcal L},p}\) are defined by \({\mathcal K}_p= \{T_u\mid T_u: L^p({\mathcal X})\to L^p({\mathcal X})\text{ is bounded}\}\), \({\mathcal L}_p= \{u: T_u\in{\mathcal K}_p\}\), \(\| u\|_{{\mathcal L},p}= (\| E(|u|^p)\|_\infty)^{1/p}\). The main theorems of this paper include{\parindent=0.6cm\begin{itemize}\item[(1)] if \(T_u: L^p({\mathcal X})\to L^p({\mathcal X})\) is a bounded linear operator, then \(T_u\) has closed range if and only if a number \(\delta> 0\) may be determined such that \(E(|u|^p)\geq\delta\) a.e. on \(\text{supp}(E(u))\);\item[(2)] \(({\mathcal L}_p,\|\cdot\|_{{\mathcal L},p})\) is a Banach space and \(\| u\|_{{\mathcal L},p}\leq\| T_u\|_{p\to p}\leq 3\| u\|_{{\mathcal L},p}\). \end{itemize}}