On the Riesz Basisness of Systems Composed of Root Functions of Periodic Boundary Value Problems
On the Riesz Basisness of Systems Composed of Root Functions of Periodic Boundary Value Problems
We consider the nonself-adjoint Sturm-Liouville operator with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>and either periodic or antiperiodic boundary conditions. We obtain necessary and sufficient conditions for systems of root functions of these operators to be a Riesz basis in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mn>0,1</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:math>in terms of the Fourier coefficients of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math>.