Über Maximale Monotonie von Operatoren des Typs L*ϕ·L

Abstract

In a Hilbert space H, we consider operators of type A=L*ϕ·L, where L is a closed, linear operator and ϕ is a maximal cyclically monotone, coercive operator. The operators ϕ, L, L* and their inverses are not necessarily everywhere defined. Our principle result is a nonlinear extension of an earlier theorem of v. Neumann for A=L*L.Theorem: Suppose that either (L*)−1 is bounded or that both L−1 is bounded and, D(ϕ) υ N (L*). The L*ϕ·L, is maximal cyclically monotone. Maximality of sums\(\mathop \Sigma \limits_{i = O}^n (L*)^{i_\Phi } _i {}^ \circ L^i \) is also considered, and the theory is applied to concrete differential operators of the form\(\mathop \Sigma \limits_{i = 0}^n ( - 1)^i f_i (u^{[i]} )^{[i]} \), with monotone functions f1 and various boundary conditions.