On Lambert Multipliers with kato property Between $L^p-$spaces

‎In this paper‎, ‎we investigate the structure of the space of multipliers arising from the range of a composition operator $c_{\varphi}$‎, ‎which is induced by the conditional expectation between two $L^p(\Sigma)$‎ ‎spaces‎. ‎After introducing the necessary preliminaries on measure spaces‎, ‎$L^p$-spaces‎, ‎and the conditional expectation operator‎, ‎we define weighted conditional multipliers and establish conditions under which they generate bounded operators between different $L^p(\Sigma)$‎ ‎and $L^q(\Sigma)$ spaces‎. ‎Using fundamental properties of conditional expectation‎, ‎we characterize the structural behavior of these multipliers and explore their relation to injectivity and surjectivity moduli of bounded linear operators‎. ‎The results provide a precise framework for describing conditional multipliers and open new perspectives for the study of composition operators in functional analysis‎.