On generalized resolvents and characteristic matrices of differential operators

TL;DR

This paper extends the theory of generalized resolvents and characteristic matrices for differential operators of even order with operator coefficients, using boundary triplets to connect boundary value problems with spectral parameters.

Contribution

It develops a new parametrization of characteristic matrices via boundary triplets and Nevanlinna boundary parameters, generalizing known results for differential operators.

Findings

Provides a block-matrix representation of characteristic matrices.

Establishes a formula similar to Krein-Naimark for these matrices.

Derives integral operator representations of resolvents with Green functions.

Abstract

The main objects of our considerations are differential operators generated by a formally selfadjoint differential expression of an even order on the interval $[0,b> (b\leq \infty)$ with operator valued coefficients. We complement and develop the known Shtraus' results on generalized resolvents and characteristic matrices of the minimal operator $L_0$. Our approach is based on the concept of a decomposing boundary triplet which enables to establish a connection between the Straus' method and boundary value problems (for singular differential operators) with a spectral parameter in a boundary condition. In particular we provide a parametrization of all characteristic matrices $\Om (\l)$ of the operator $L_0$ immediately in terms of the Nevanlinna boundary parameter $\tau(\l)$. Such a parametrization is given in the form of the block-matrix representation of $\Om (\l)$ as well as by means of the formula for $\Om (\l)$ similar to the well known Krein-Naimark formula for generalized resolvents. In passing we obtain the representation of canonical and generalized resolvents in the form of integral operators with the operator kernel (the Green function) defined in terms of fundamental operator solutions of the equation $l[y]-{\l}y=0$.