  This paper is primarily concerned with the problem of maximality for the sum
$A+B$ and composition $L^{*}ML$ in non-reflexive Banach space settings under
qualifications constraints involving the domains of $A,B,M$. Here $X$, $Y$ are
Banach spaces with duals $X^{*}$, $Y^{*}$, $A,B:X\rightrightarrows X^{*}$,
$M:Y\rightrightarrows Y^{*}$ are multi-valued maximal monotone operators, and
$L:X\to Y$ is linear bounded. Based on the Fitzpatrick function, new
characterizations for the maximality of an operator as well as simpler proofs,
improvements of previously known results, and several new results on the topic
are presented.
