A pseudo-inverse for integro substitution operators

In this paper, we characterize integro substitution operators that maintain closed ranges within \(L^{2}(X,\Sigma ,\mu )\) $L^2(X,\mathrm{\Sigma },\mu )$ , where \((X,\Sigma ,\mu )\) $(X,\mathrm{\Sigma },\mu )$ represents a \(\sigma \) $\sigma$ -finite measure space. Next, we give an explicit formula for the pseudo-inverse denoted as \((T_{u}^{\varphi })^{\dagger }\) ${(T_u^{\phi })}^{†}$ of a integro substitution operators \(T_{u}^{\varphi }\) $T_u^{\phi }$ on this space using conditional expectation operators.