Degenerate differential-operator equations of higher order and arbitrary weight (Q2920395)

The author considers a separable Hilbert space \({\mathcal H}\), \(f\in L_2((0,b),{\mathcal H})\), a linear operator \(A:{\mathcal H}\rightarrow {\mathcal H}\), a positive measurable function \(\rho\) on \((0,b]\) and the Dirichlet problem for the operator equation NEWLINE\[NEWLINE Pu \equiv (-1)^m(\rho(t)u^{(m)}(t))^{(m)}+Au=f,\quad t\in (0,b). NEWLINE\]NEWLINE The author defines the weighted Sobolev spaces {W}\(_p^m\), describes the behavior of the functions from these spaces close to \(t=0\) and proves some embedding and compactness theorems. Moreover, he proves under some conditions given in spectral terms, the existence and uniqueness of the generalized solution of the Dirichlet problem for every \(f\in L_2(0,b)\). Under some conditions on the spectrum of the operator \(A\), the author proves the unique solvability of the above operator equation for every \(f\in L_2((0,b),{\mathcal H})\) and gives the description of the spectrum for the corresponding operator \(P\).