Perturbation of m-accretive operators in banach spaces

Abstract

THERE ARE not many results on multiplicative perturbation theory dealing with development of criteria for an m-accretive operator to remain m-accretive after being composed with other operators. In this connection we mention the work of Gustafson and Lumer [l 11, who gave a criterion for m-accretivity of the composition KA on a Banach space X, where K is assumed to be an everywhere defined strongly accretive bounded linear operator and KA is accretive. The main purpose of this note is to weaken the requirement of strong accretivity of K to simple accretivity to conclude that the closure of densely defined operator KA is m-accretive. An example is given to illustrate applicability of this result even though the Gustafson and Lumer result is not applicable. Further a nonlinear analogue of Lumer’s result is proved. We suggest two unsolved problems in this direction. It is well known that the class of m-accretive operators arise in initial boundary value problems of differential equations of evolution type. Multiplicative perturbations of such operators have been studied previously in [l, 5, 6, 7, 8, 9, 10, 11, 131. For a description of m-accretive operators and their properties one can refer to [2, 3, 6, 12, 151. We state the above mentioned result of Gustafson and Lumer [l l] in the following form which can be compared more easily with our results in this work.