On deficiency index for some second order vector differential operators

Abstract

In this paper we consider the operators generated by the second order matrix linear symmetric quasi-differential expression $$ l[y]=-(P(y'-Ry))'-R^*P(y'-Ry)+Qy $$ on the set $[1,+\infty)$, where $P^{-1}(x)$, $Q(x)$ are Hermitian matrix functions and $R(x)$ is a complex matrix function of order $n$ with entries $p_{ij}(x),q_{ij}(x),r_{ij}(x)\in L^1_{loc}[1,+\infty)$ ($i,j=1,2,\dots,n$). We describe the minimal closed symmetric operator $L_0$ generated by this expression in the Hilbert space $L^2_n[1,+\infty)$. For this operator we prove an analogue of the Orlov's theorem on the deficiency index of linear scalar differential operators.