The essential spectrum of linear operator pencils. (Das wesentliche Spektrum linearer Operatorpencils.) (German) Zbl 0947.47010
A theory about the essential spectrum of operator pencils is presented with application to the Orr-Sommerfeld eigenvalue problem for instability of incompressible viscous flow with the Blasius flow profile. Let \(A\), \(B\) be linear operators in Banach spaces \(X,Y,A:D(A)\to Y\) \((D(A)\subset X)\) is a closed operator, and \(B:X\to Y\) bounded. Consider the eigenvalue problem
\[
Au=\lambda Bu,\quad \lambda\in\mathbb{C},\quad u\in D(A),\quad u\neq 0.\tag{i}
\]
The resolvent set for (i) is
\[
\rho(A,B)= \{\lambda\in\mathbb{C}\mid A-\lambda B:D(A)\to Y\text{ is bijective}\}
\]
and the spectral set is \(\sigma(A,B)= \mathbb{C}\setminus \rho(A, B)\). The author identifies five subsets \(\sigma_{ek}(A, B)\), \(k= 1,2,\dots, 5\) of \(\sigma(A, B)\) as essential spectral sets and show that \(\sigma_{ek}(A, B)\subset\sigma_{ek+ 1}(A, B)\), \(k= 1,\dots, 4\). Eigenvalues of finite multiplicity are contained in \(\sigma(A,B)\setminus \sigma_{e5}(A, B)\). The Orr-Sommerfeld equation with Blasius flow profile is formulated abstractly in the form (i) with \(A= A_0+ K:D(A_0)\subset D(K)\to Y\), \(B: X\to Y\), \(B\) is bounded bijective, \(K\) is \(A_0\)-compact, \(A\), \(B\) are respectively fourth-order and second-order ordinary differential operators and \(A_0\) is a differential operator with constant coefficients. The author determines \(\sigma_{e4}(A, B)= M\) explicitly as a ray in \(\mathbb{C}\) and proves that \(\sigma_{ek}(A, B)= M\), \(k= 1,2,\dots, 5\).
Reviewer: J.B.Butler jun.(Portland)
MSC:
47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
47A75 | Eigenvalue problems for linear operators |
47A10 | Spectrum, resolvent |
76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
35Q35 | PDEs in connection with fluid mechanics |