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 on the set [1 , + ∞ ), 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 ) ∈ 𝐿 1 𝑙𝑜𝑐 [1 , + ∞ ) ( 𝑖, 𝑗 = 1 , 2 , . . . , 𝑛 ). We describe the minimal closed symmetric operator 𝐿 0 generated by this expression in the Hilbert space 𝐿 2 𝑛 [1 , + ∞ ). For this operator we prove an analogue of the Orlov’s theorem on the deficiency index of linear scalar differential operators.