Deviations of Riesz projections of Hill operators with singular potentials

Abstract

It is shown that the deviations $P_n-P_n^0$ of Riesz projections $$P_n=\frac{1}{2\pi i}{\int }_{C_n}(z-L{)}^{-1}dz,C_n=\{|z-n^2|=n\},$$ of Hill operators $Ly=-y^{\prime \prime }+v(x)y,x\in [0,\pi ],$ 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 $Ran{P}_n.$