On the spectral properties of degenerate non-selfadjoint elliptic systems of differential operators (Q1769386)

Set \(H_\ell=\big(L^2(0,1)\big)^\ell\) and consider the non-selfadjoint differential operator \[ (Pu)(t)=-{{d}\over {dt}}\left( t^\alpha(1-t)^\alpha R(t){{d u(t)}\over {dt}}\right) \] on \(H_\ell\) with two-point (Dirichlet) boundary conditions. It is assumed that \(0\leq\alpha<1\) and \(R(t)\in C^2([0,1], \text{End}\,\mathbb C^\ell)\) is a matrix-valued function for each \(t\in[0,1].\) Suppose further that, for each \(t\in[0,1],\) the matrix \(R(t)\) has \(\ell\)-simple nonzero eigenvalues \(\mu_j(t)\in C^2[1,0],\) \(j=1,\ldots,\ell,\) such that \(\mu_j(t)\in \mathbb C\setminus \Phi,\) where \(\Phi=\{z\in \mathbb C\mid\text{arg\,}z\leq \varphi\}\) with \(\varphi\in(0,\pi).\) The authors investigate some spectral properties of the degenerate operator \(P\) acting on \(H_\ell.\) In particular, resolvent estimates are obtained for \(P\) under two-point boundary conditions both in \(H_1\) and \(H_\ell.\)