One-sided invertibility of discrete operators and their applications

Abstract

For $$p\in [1,\infty ]$$ , we establish criteria for the one-sided invertibility of binomial discrete difference operators $${{\mathcal {A}}}=aI-bV$$ on the space $$l^p=l^p(\mathbb {Z})$$ , where $$a,b\in l^\infty $$ , I is the identity operator and the isometric shift operator V is given on functions $$f\in l^p$$ by $$(Vf)(n)=f(n+1)$$ for all $$n\in \mathbb {Z}$$ . Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators $$A=aI-bU_\alpha $$ on the Lebesgue space $$L^p(\mathbb {R}_+)$$ for every $$p\in [1,\infty ]$$ , where $$a,b\in L^\infty (\mathbb {R}_+)$$ , $$\alpha $$ is an orientation-preserving bi-Lipschitz homeomorphism of $$[0,+\infty ]$$ onto itself with only two fixed points 0 and $$\infty $$ , and $$U_\alpha $$ is the isometric weighted shift operator on $$L^p(\mathbb {R}_+)$$ given by $$U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )$$ . Applications of binomial discrete operators to interpolation theory are given.