Deviations of Riesz projections of Hill operators with singular potentials

Details Deviations of Riesz projections of Hill operators with singular potentials Deviations of Riesz projections of Hill operators with singular potentials

2008-02-15

arxiv.org/abs/0802.2197v1

Mathematics

Spectral Theory

Scientific paper

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. $

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