$${\mathfrak L}$$-Resolvent Matrices for Hamiltonian Integral Systems

Abstract In this paper we consider a two–dimensional Hamiltonian integral system on an interval $$[ a,b )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We investigate the definiteness and surjectivity properties and define associated with this system the maximal $$S_{\max }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mo>max</mml:mo> </mml:msub> </mml:math> and minimal $$S_{\min }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mo>min</mml:mo> </mml:msub> </mml:math> linear relations. For an improper gauge $${\mathfrak L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> , we calculate the $${\mathfrak L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -resolvent matrix and describe the set of $${\mathfrak L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -resolvents of the linear relation $$S_{\min }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mo>min</mml:mo> </mml:msub> </mml:math> in the cases when the system is either quasiregular, limit circle, or limit point at b .