Compact inverses of first-order normal differential operators (Q2492974)

Let \(H\) be a separable Hilbert space and let \(L^{2} = L^{2} (H, (a,b))\) be the Hilbert space of all \(H\)-valued strongly measurable functions \(u\) defined a.e.\ on \((a,b)\) and satisfying \(\int_{a}^{b} | u (t) |^{2} dt < \infty.\) In the space \(L^{2}\), a linear differential operator expression of the first-order \(l(u)=u'(t)+Au(t)\) is considered, where \(A\) is a linear normal operator in \(H\) the real part \(A_{R}\) of which is a linear positive definite self-adjoint operator in \(H\) (it is assumed that the greatest lower bound of \(A_{R}\) is equal to 1). Denote by \(L_{0}\) (resp., \(L_{0}^{+}\)) the minimal operator generated in \(L^{2}\) by the differential expression \(l(u)\) (resp., by the adjoint expression \(l^{*}(v)=-v'(t)+A^{*}v(t)\)). The maximal operators generated by \(l(u)\) (resp., \(l^{*}(v)\)) are denoted by \(L\) (resp., \(L^{+}\)). Let \(U(t,s)\), \(t,s\in [a,b]\), be the family of evolution operators corresponding to the homogeneous differential equation \(U_{t}'(t,s)f+iA_{I}U(t,s)f=0\) (here, \(A_{I} = \text{Im}\,A\)) with \(U(s,s)f=f\), \(f\in D(A)\) (here, \(D(A)\) stands for the domain of the operator \(A\)). The author states that under the assumption \(A_{R}^{1/2} [D(L)\cap D(L^{+})]\subset W_{2}^{1} (H,(a,b))\) (here, \(W_{2}^{1}(H,(a,b))\) designates the Sobolev space), each normal extension \(L_{n}\) of the minimal operator \(L_{0}\) in \(L^{2}\) is generated by the expression \(l(u)\) with the boundary condition \(u(b)=U(b,a)Wu(a)\), where \(W\) and \(A_{R}^{1/2}WA_{R}^{1/2}\) are unitary operators in \(H\). The unitary operator \(W\) is determined uniquely by the extension \(L_{n}\), i.e., \(L_{n} = L_{W}\), and, conversely, the restriction of the maximal operator \(L\) to the manifold of vector functions \(u(t)\in D(L)\cap D(L^{+})\) that satisfy the above mentioned boundary condition for some unitary operator \(W\), where \(A_{R}^{1/2}WA_{R}^{-1/2}\) is also a unitary operator in \(H\), constitutes a normal extension of the minimal operator \(L_{0}\) in the space \(L^{2}\). Results concerning the structure of the spectrum of a normal extension of \(L_{0}\) are also formulated. It is proved that the spectrum of a normal extension \(L\) coincides with the set of all \(\lambda \in \mathbb C\) such that \(\lambda = \lambda_{0} + \frac{2k\pi i}{b - a}\), where \(\lambda_{0}\) is a set of solutions to the equation \(e^{-\lambda_{0}(b-a)} - \mu = 0\), \(\mu \in \sigma (W^{*} e^{-A_{R}(b - a)})\), \(k \in \mathbb Z\). In the particular case of \(\dim H < \infty\), the operator \(L_{W}\) has only a discrete spectrum and for the \(s\)-numbers of the inverse operator \(L_{W}^{-1}\), there holds \(S_{n} (L_{W}^{-1}) \sim \frac{b - a}{2\pi n}\) as \(n \longrightarrow \infty\). It is also stated that the compactness of the resolvent of \(A_{R}\) implies the compactness for \(L_{W}\) and if, in addition, \(\lambda_{n}(A_{R})\sim cn^{\alpha}\) (\(c > 0\), \(0<\alpha<\infty\)), then \(S_{n} (L_{W}^{-1}) \sim d n^{-\beta}\) with \(0 < d < \infty\) and \(\beta = \frac{\alpha}{1 + \alpha}\).