Deviations of Riesz projections of Hill operators with singular potentials

2008/02/15 by Plamen Djakov, Djakov, Plamen, Boris Mityagin +1
Mathematics · #34L40 #47B06 #47E05 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #math.SP #msc:34L40 #msc:47B06 #msc:47E05

paper · pdf · doi:10.48550/arxiv.0802.2197

arxiv created 2008/02/15 · openalex publication_date 2008/02/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is shown that the deviations P_n -P_n⁰ of Riesz projections P_n = (1)/(2πi) ∫_Cn (z-L)^-1 dz, C_n=\|z-n²|= n\, of Hill operators L y = - y^{′ ′} + v(x) y, x ∈ [0,π], with zero and H^-1 periodic potentials go to zero as n → ∞ even if we consider P_n -P_n⁰ as operators from L¹ to L^∞. This implies that all L^p-norms are uniformly equivalent on the Riesz subspaces Ran P_n.

Related

• Bari-Markus property for Riesz projections of Hill operators with singular potentials
• Characterization of potential smoothness and Riesz basis property of Hill-Scrödinger operators with singular periodic potentials in terms of periodic, antiperiodic and Neumann spectra
• Riesz basis property of Hill operators with potentials in weighted\n spaces
• Bari-Markus property for Riesz projections of 1D periodic Dirac operators
• Characterization of potential smoothness and Riesz basis property of the Hill-Scrödinger operator in terms of periodic, antiperiodic and Neumann spectra