A new general condition for the existence as a closed operator of the sum of two operators
by
J-Ph. Labrousse
University of Nice, France

Let A, B be two closed linear operators defined on a domain contained in a Hilbert space H and with range in that same Hilbert space. A reasonably general condition is given under which the sum A + B of the two operators is itself a closed operator. As applications of this it can be shown that if A is Fredholm and if B is an operator which is compact equivalent with A (see LABROUSSE and MERCIER, Equivalences compactes entre deux operateurs fermes sur un espace de Hilbert, Math. Nachr. 133 (1987) 91-105 for definition of compact equivalence) then the intersection of the domain of B with the domain of A is dense in the domain of A, A + B is Fredholm and (A^* + B^*)^* = A + B. A similar result is obtained when B is close to A (in the gap topology). The above results can also be extended to linear relations.

Date received: June 13, 2000