On the range of monotone composite mappings (Q2764703)

Let \(X\) and \(U\) be reflexive Banach spaces, \(A: X\to U\) a linear map and \(T: U\to U^*\) a monotone set-valued map. Then the composite (set-valued) map \(A^* TA:X\to X^*\) is also monotone. The main purpose of the author is to show that for a large class of monotone maps \(T\) it holds: NEWLINE\[NEWLINE\text{range}(A^* TA)\approx A^*(\text{range }T)NEWLINE\]NEWLINE in the sense that the closures and the relative interiors of the two sets are equal. As a byproduct he obtains a simple proof of a maximality result for composite mappings.