On the extension problem for accretive differential operators (Q1066456)

Let \(T_ 0\) denote the minimal operator corresponding to the differential expression \[ \tau y(x)=(w(x))^{-1}\sum^{n}_{j=0}(- 1)^ j(P_{n-j}(x)y^{(j)}(x))^{\quad (j)},\quad x\in I, \] in the weighted space \(L^ 2_ w(I)\); the coefficients \(p_{_ j}\) are assumed to be real, I is a closed interval of the real line and \(\tau\) is assumed to be regular. Under the assumption that the Dirichlet integral of \(\tau\) exists and is an inner product on the maximal domain of \(\tau\) (defined in terms of quasi-derivatives), a complete description is given of all maximal accretive extensions of \(T_ 0\) in terms of explicit boundary conditions. The principal tool used is the work of R. S. Phillips on the abstract extension problem for accretive (and dissipative) operators.