The defect index of singular symmetric linear difference equations with real coefficients

Abstract

This paper is concerned with the defect index of singular symmetric linear difference equations of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 n"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with real coefficients and one singular endpoint. We show that their defect index <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies the inequalities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to d less-than-or-equal-to 2 n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\leq d \leq 2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that all values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in this range are realized. This parallels the well known result of Glazman for differential equations established about 1950. In addition, several criteria of the limit point and strong limit point cases are established.