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Conditional multipliers and essential norm of $uC_\varphi$ between $L^p$ spaces. (English) Zbl 1200.47029
Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. The space of $\Sigma$-measurable functions on $X$ is denoted by $L^0(\Sigma)$, and if $\le p<\infty$, then $L^p(\Sigma)$ denotes $\{[f]:\Vert [f]\Vert_p=\Vert p\Vert_p< \infty\}$, where $[f]$ is the equivalence class of functions which differ from $f$ on sets of measure $0$, and for $1\le p\le\infty$, $\Vert\cdot\Vert_p$ is the $L^p$-norm. In the paper, ${\Cal A}\subseteq\Sigma$ denotes a complete $\sigma$-finite sub-algebra of $\Sigma$, and the operator $E^{\Cal A}=E$ is said to be a conditional expectation operator if for $f\ge 0$, $f\in L^p(\Sigma)$, $E(f)$ is ${\Cal A}$-measurable and $\int_{\Cal A}E(f)\,d\mu= \int_{\Cal A}f\,d\mu$, whenever the integrals are finite. If $1\le p,q<\infty$, then the set of conditional multipliers $K_{p,q}= K_{p,q}({\Cal A},\Sigma)$ is defined to be $\{u\in L^0(\Sigma): uL^p({\Cal A})\subseteq L^q(\Sigma)\}$, $M_u: L^p({\Cal A})\to L^q(\Sigma)$ denotes the multiplication operator $M_u(f)= uf$, and if $\varphi: X\to X$ is a non-singular measurable transformation with measure $u\circ\varphi^{-1}$ which is absolutely continuous with respect to $\mu$, the derivative $d(p\circ\varphi^{-1})/d\mu$ is denoted by $h$. If $C_\varphi$ denotes the composition operator, $C_\varphi(f)(x)= f\circ\varphi(x)= f(\varphi(x))$, then $K^\varphi_{p,q}$ is defined to be $\{u\in L^0(\Sigma): u\text{ Range}(C_\varphi)\subseteq L^q(\Sigma)\}$, and $$\Vert uC_\varphi f\Vert^q_q= \int_X S|f|^q\,d\mu,$$ $1\le q<\infty$, where $S= hE^{\varphi^{-1}}(\Sigma)/(E^{\Cal A}(|u|^q))\circ\varphi^{-1}$. The results of this paper include: {\parindent=7mm \item{(1)} $(K_{p,q}, \Vert\cdot\Vert_{p, q})$ is a Banach space, where $\Vert u\Vert_{p,q}=\Vert E(|u|^q)^{1/q}\Vert_r$, if $1\le q\le p$ and $1/r= (1/q)-(1/p)$, and $\Vert u\Vert_{p,q}= \{\sup_n E(|u|^q)(A_n)/\mu(A_n)^{1/r}\}$, if $1\le p\le q$, $1/r= (1/p)- (1/q)$, and $\{A_n\}$ is a countable set of pairwise disjoint ${\Cal A}$-atoms such that $X=B\cup(\bigcup A_n)$; \item{(2)} characterizations indicating that $u\in K^\varphi_{p,q}$ if and only if $S\in L^{1/(1-(q(p))}(\Sigma)$, where $1\le q\le p$; $u\in K^\varphi_{p,q}$ if and only $S= 0$ a.e on $B$ and $\sup_n\{S(A_n)/\mu(A_n)^{(q(p))-1}\}< \infty$, where $1\le p\le q$. \par}

47B20Subnormal operators, hyponormal operators, etc.
47B38Operators on function spaces (general)
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