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

In 1996, H. Volkmer observed that the inequality (integral(1)(-1) 1/vertical bar r vertical bar f'vertical bar(2)dx)(2) <= K-2 integral(1)(-1) vertical bar f vertical bar(2) dx integral(1)(-1)vertical bar(1/r f')'vertical bar(2) dx is satisfied with some positive constant K > 0 for a certain class of functions f on [-1, 1] if the eigenfunctions of the problem -y '' = lambda r(x)y, y(-1) = y(1) = 0 form a Riesz basis of the Hilbert space L-vertical bar r vertical bar(2)(-1, 1). Here the weight r is an element of L-1(-1, 1) is assumed to satisfy xr( x) > 0 a.e. on (-1, 1). We 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 HELP inequality. Using these results we 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, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if r is odd. (C) 2014 WILEY- VCH Verlag GmbH & Co. KGaA, Weinheim