On a necessary aspect for the Riesz basis property for indefinite Sturm-Liouville problems (Q2929430)

In [Proc. R. Soc. Edinb., Sect. A, Math. 126, No. 5, 1097--1112 (1996; Zbl 0865.34017)] \textit{H. Volkmer} observed that the inequality NEWLINE\[NEWLINE\left( \int_{-1}^{1} \frac{1}{\mid r\mid }\mid f^{\prime }\mid ^{2}dx\right) ^{2}\leq K^{2}\int_{-1}^{1}\mid f\mid ^{2}dx\int_{-1}^{1}\mid (\frac{1}{r}f^{\prime })^{\prime }\mid ^{2}dxNEWLINE\]NEWLINE is satisfied with some positive constant \(K>0\) for a certain class of functions \(f\) on \([-1,1]\) if the eigenfunctions of the problem NEWLINE\[NEWLINE-y^{\prime \prime }=\lambda r(x)y,~y(-1)=y(1)NEWLINE\]NEWLINE form a Riesz basis of the Hilbert space \(L_{\mid r\mid }^{2}(-1,1)\). Here, the weight \(r\in L^{1}(-1,1)\) is assumed to satisfy \(xr(x)>0\) a.e. on \((-1,1)\). The author present two criteria in terms of Weyl--Titchmarsh \(m\)-functions for the Volkmer inequality to be valid. Note that one of these criteria is new even for the classical Hardy-Littlewood-Polya-Everitt (HELP) inequality. Using these results the author improve the result of Volkmer by showing that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, the author shows that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if \(r\) is odd.