Generalized Riesz basis property in the analysis of neutral type systems

Abstract

The functional differential equation of neutral type is studied. We consider the corresponding operator model in Hilbert space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:msub> <mml:mi mathvariant="normal">L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>;</mml:mo> <mml:msup> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and prove that there exists a sequence of invariant finite-dimensional subspaces which constitute a Riesz basis in M 2 . We also give an example emphasizing that the generalized eigenspaces do not form a Riesz basis.