Ufa Mathematical Journal
    Volume 9, Number 1, pp. 18-28

    Analogue of the Orlov's theorem about deficiency numbers for matrix differential operators of the second order

    Braeutigam I.N., Mirzoev K.A., Safonova T.A.


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