Infinite products of resolvents of accretive operators
This article deals with the space \({\mathcal M}_m\) of \(m\)-accretive operators (an operator \(A\) is \(m\)-accretive, if \(\|x-y\|\leq\|x-y+ r(u- v)\|\) (\(x,y\in D(A)\), \(u\in Ax\), \(v\in Ay\), \(r> 0\)) and \(R(I+ rA)= X\) \((r> 0)\)). The basic results are concerned with the behavior of iterations \(J^A_{r_n}\circ J^A_{r_{n-1}}\circ\cdots\circ J^A_{r_1}\) of resolvents \(J^A_r= (1+ rA)^{-1}\) of operators \(A\), where \(r_n\) is a sequence of positive numbers such that \(\overline r_n\leq r_n\leq\widetilde r\), \(\overline r_n\) is a fixed sequence for which \(\overline r_n< 1\) \((n= 1,2,\dots)\), \(\lim_{n\to\infty} \overline r_n= 0\), \(\sum^\infty_{n=1}\overline r_n= \infty\). The authors prove the existence of a set \({\mathcal F}\) which is a countable intersection of open everywhere dense sets in \({\mathcal M}_m\) and such that for each \(A\in{\mathcal F}\), each \(\delta> 0\) and each \(K>0\) there exist a neighbourhood \(U\) of \(A\) and an integer \(n_0\geq 1\) such that for each \(B\in U\) \[ \|J^B_{r_n}\circ J^B_{r_{n-1}}\circ\cdots\circ J^B_{r_1}x- J^B_{r_n}\circ J^B_{r_{n-1}}\circ\cdots\circ J^B_{r_1} y\|\leq\delta\;(x,y\in X,\;\|x\|,\|y\|\leq K,\;n\geq n_0). \] A similar result is proved for the space \({\mathcal M}^*_m\) of operators from \({\mathcal M}_m\) which have a zero; in this situation there exists a set \({\mathcal F}\) which is a countable intersection of open everywhere dense sets in \({\mathcal M}_m\) and such that for each \(A\in{\mathcal F}\) one can find a unique \(x_A\in X\) with \(0\in A(x_A)\) and, for each \(\delta> 0\) and each \(K>0\), there exist a neighbourhood \(U\) of \(A\) and an integer \(n_0\geq 1\) such that for each \(B\in U\) \[ \|J^B_{r_n}\circ J^B_{r_{n-1}}\circ\cdots\circ J^B_{r_1} x- x_A\|\leq\delta\quad (x\in X,\;\|x\|\leq K,\;n\geq n_0). \]
scientific article; zbMATH DE number 1545394