β Scribed by Ivan P. Gavrilyuk; Vladimir L. Makarov
No coin nor oath required. For personal study only.
We consider the operator equation SX -~)~t i U, XV, = Y where { U, }. { t~ } are some commutative sets of operators but in general { U, } need not commute with { 1,)}. Particular cases of this equation are the Syivester and Ljaptmov equations. We give a new representation and an approximation of the solution which is suitable to perform it algorithmically. Error estimates are given which show exponential covergence lbr bounded operators and polynomial convergence for unbounded ones. Based on these considerations we construct an iterative process and give an existence theorem for the operator equation Z-' + A ~Z + A., = O, arising for example when solving an abstract second order differential equation with non-commutative coefficients.
π SIMILAR VOLUMES
We consider the problem how big is the set of solutions of a given functional equation in the set of approximate solutions. It happens that in the cases of linear functional equations (like Cauchy, Jensen) or linear inequalities (like convex, Jensen convex) the sets of solutions are very small subse
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