Maximal Monotonicity for the Precomposition with a Linear Operator

We give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator A∗ ◦ T ◦A when A is a linear continuous mapping between two reflexive Banach spaces and T is a maximal monotone operator. As a special case we get the weakest constraint qualification that ensures the maximal monotonicity of the sum of two maximal monotone operators on a reflexive Banach space. Then we give a weak constraint qualification assuring the Brézis-Haraux-type approximation of the range of the subdifferential of the precomposition to A of a proper convex lower semicontinuous function in non-reflexive Banach spaces, extending and correcting in a special case an older result due to Riahi.