Common Fixed Point Theorems and Defect Indices with Failure of Rational Dilation

Abstract

We generalize results on common fixed points in ordered cone metric spaces by weakening the respective contractive condition. Then, the notions of quasicontraction and g-quasicontraction are introduced in the setting of ordered cone metric spaces and respective (common) fixed point Theorems are shown. We show that the equality holds for unitary or the eigen values are all in the open unit disk. We also consider the defect index for a finite Blaschke product. We study common fixed points for the self and non-self type maps in cone metric spaces. For particular class of E-contractions, we show it necessary for the existence of rational dilation that the corresponding fundamental operators satisfy certain conditions. Then we construct an E-contraction from that particular class which fails to satisfy the certain condition. We produce a concrete functional model for pure E-isometries and a class of E-contractions analogous to the pure isometries in one variable.