International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 1, Pages 25-32

Generalized dissipativeness in a Banach space

David R. Gurney

Southeastern Louisiana University, SLU 541, Hammond 70402, LA, USA

Received 12 December 1994

Copyright © 1996 David R. Gurney. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Suppose X is a real or complex Banach space with dual X* and a semiscalar product [,]. For k a real number, a subset B of X×X will be called k-dissipative if for each pair of elements (x1,y1), (x2,y2) in B, there existsh{fX*:[x,f]=|x|2=|f|2}such thatRe[y1y2,h]k|x1x2|2.This definition extends a notion of dissipativeness which is equivalent to having k equal zero here. A number of definitions and theorems related to this original dissipative notion are generalized in the present paper to fit the k-dissipative situation, and proofs are given for the new theorems.