Quadratic inequalities for functionals in l∞

Abstract

For a class of operators $T$ on $l^{\infty}$ and $T$-invariant functionals $\varphi$ we prove inequalities between $\varphi(x)$, $\varphi(x^{2})$ and the upper density of the sets $P_r:=\{n \in \mathbb{N}_0: \varphi((T^{n}x)\cdot x) \gt r\}.$ Applications are given to Banach limits and integrals.