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.