The deficiency indices of \(T_ 0(\tau^ +) T_ 0(\tau)\), where \(\tau\) is a linear ordinary differential expression (Q1188218)

Let \(\tau\) be a linear ordinary differential expression of order \(n\) with \(C^ \infty\) coefficients on a real interval \(I\) and let \(\tau^ +\) be its formal adjoint. For any differential expression \(\zeta\) let \(T_ 0(\zeta)\) denote its closed minimal and \(T_ 1(\zeta)\) its maximal operator in \(L^ 2(I)\). It is proved that \(T_ 0(\tau^ +)T_ 0(\tau)\) is a closed symmetric finite dimensional extension of \(T_ 0(\tau^ +\tau)\) and its adjoint is \(T_ 1(\tau^ +)T_ 1(\tau)\). The deficiency indices of \(T_ 0(\tau)T_ 0(\tau^ +)\) and \(T_ 0(\tau^ +)T_ 0(\tau)\) coincide and, hence, the deficiency indices of \(T_ 0(\tau^ +)T_ 0(\tau)\) are a lower bound for the deficiency indices of \(T_ 0(\tau\tau^ +)\). A Kodaira-like formula connecting the deficiency indices of \(T_ 0(\tau^ +)T_ 0(\tau)\) on \(I\) and on the intervals of a decomposition of \(I\) is established.