Representation of Bilinear Forms in Hilbert Space by Linear Operators
Alan G.R. McIntosh · Transactions of the American Mathematical Society · 1968
Introduction.This paper is concerned with representing accretive bilinear forms in a Hubert space by maximal accretive operators in the same space.The operator A} associated with a bilinear form J is defined to be the operator with largest domain satisfying J[u, v] = (A}u, v) for all v e 3>(J), the domain of J.If J is accretive (ReJ[u, u]^0, ue!3(J)), then A} is accretive (Re (A3u, u)^0, u e 3i(Aj)).An accretive form J is defined to be representable if A, is maximal accretive (i.e. has no proper accretive extension).This implies that the spectrum of Aj is contained in the right half of the complex plane.Our aim is to give conditions on J under which J is representable.It is clear that a bounded accretive form on J? is representable (cf.Appendix).The first result of this kind for an unbounded form was derived by Friedrichs (1934) when he proved that the operator associated with a closed semibounded hermitian form is selfadjoint (cf.[2, Chapter 6]).This result was extended to a more general case by T. Kato who showed that a closed regularly accretive form is representable (see §4). (Recall that an accretive form is regularly accretive if \lm J[u, u]\ ^y Re J[u, u] for all u e S>(J) and some y ^0.)In this paper we consider accretive forms that are not necessarily regularly accretive.A representation theorem for such forms is presented in §3.Then in §4 it is shown that the known result for regularly accretive forms is a corollary of this theorem.In §5 we turn to the theory of partial differential equations, and investigate the form with domain ■ (where Í2 is an open subset of Rm, m£l, andfjk are C1 functions on Q.).Suppose