Extensions of<i>d∕dx</i>that generate uniformly bounded semigroups

Abstract

A = -d/dx on BC 0 (R + ), *D(Λ) = {/ G BC 0 (R + ) | /' e BC 0 (JR + )}, is an example of a maximal accretive operator that does not generate a contraction semigroup.It does, however, have extensions that generate uniformly bounded semigroups.A large class of such extensions are presented.The same is done with d/dx and -d/dx on C o [0,1].Introduction.The theory of accretive operators generalizes the theory of symmetric operators on a Hubert space.An operator, T, on a Hubert space is accretive if Re(7jc, JC)>0 for all x in the domain of T. A maximal accretive operator is one that has no proper accretive extensions.An m-accretive operator is one that generates a strongly-continuous contraction semigroup.The "m" suggests "maximal", because in a Hubert space, every maximal accretive operator is m-accretive.(See Theorem 0.) This need not be true in a general Banach space.Lumer and Phillips, in 1961, ([1], p. 688) first gave an example of a maximal accretive operator that is not m-accretive.The space is C 0 [0,1], and the accretive operator is d/dx, with domain {f\f exists,/' G C o [0,1]}.Lumer and Phillips show that any proper extension of this operator fails to be accretive.To verify that this operator is not m-accretive, one uses the well-known ([2], p. 240) fact, that, if T is a closed accretive operator, then (1 + T) is one-to-one, and Γis m-accretive if and only if the range of (1 + T) is the entire space.The range of (1 + d/dx), with the domain given above, can be explicitly calculated to be {g E C 0 [0,1] | /J e r g(r) dr -0), which is not dense in C 0 [0,1].Intuitively, this operator fails to be m-accretive because d/dx should generate the translation semigroup {Γ 5 } 5 > 0 > defined by (T s f)(ί) = f{t -s), but this semigroup does not take C o [0,1] into itself.

Locations

• Pacific Journal of Mathematics - View - PDF