Deviations of Riesz projections of Hill operators with singular potentials

TL;DR

This paper demonstrates that the deviations of Riesz projections for Hill operators with singular potentials diminish as the spectral parameter increases, ensuring uniform equivalence of all L^p norms on the associated subspaces.

Contribution

It establishes the asymptotic behavior of Riesz projection deviations for Hill operators with singular potentials, extending understanding to L^1 and L^∞ operator norms.

Findings

Deviations P_n - P_n^0 tend to zero as n increases.

L^p norms are uniformly equivalent on Riesz subspaces.

Results hold even for singular potentials in H^{-1}.

Abstract

It is shown that the deviations $P_n -P_n^0$ of Riesz projections $$ P_n = \frac{1}{2\pi i} \int_{C_n} (z-L)^{-1} dz, \quad C_n=\{|z-n^2|= n\}, $$ of Hill operators $L y = - y^{\prime \prime} + v(x) y, x \in [0,\pi],$ with zero and $H^{-1}$ periodic potentials go to zero as $n \to \infty $ even if we consider $P_n -P_n^0$ as operators from $L^1$ to $L^\infty. $ This implies that all $L^p$-norms are uniformly equivalent on the Riesz subspaces $Ran P_n. $