ON PERIODIC SOLUTIONS OF EQUATIONS WITH RIGHT INVERTIBLE OPERATORS INDUCED BY FUNCTIONAL SHIFTS

Zbigniew Binderman
1993 Demonstratio Mathematica  
Dedicated to the memory of Professor WITOLD POGORZELSKI Real shifts induced by right invertible operators were studied by D. Przeworska-Rolewicz [1]- [4] . Complex and functional extensions of these shifts were considered by the author [5]-[10]. In the present paper periodic solutions of equations and initial value problems, induced by functional shifts are studied. Periodic solutions of an equation with an operator of complex differentation are considered. 0. Denote by L(X) the set of all
more » ... r operators with domains and ranges in a linear space X over the field C of the complex numbers and by L 0 (X) the set of all operators A € L(X) with dom A = X. The set of all right invertible operators belonging to L(X) will be denoted by R(X). If D € R(X) then we denote by Hp the set of all right inverses of D. In the sequel we shall assume that dim ker D / 0 and that right inverses belong to L 0 (X). An operator F G L 0 (X) is said to be an initial operator for D corresponding to an R G 11 D if F 2 = F, FX = ker D and FR = 0. The set of all initial operators for a given D 6 R{X) is denoted by To- Here and in the sequel we admit that 0° := 1. We also write N for the set of all positive integers and No := {0} U N.
doi:10.1515/dema-1993-3-402 fatcat:6fwjfbdta5bjhbkw7zzohznche