A numerical investigation of the quadratic numerical range of Hain-Lüst operators

Abstract

The quadratic numerical range (QNR) was introduced by Langer and Tretter [Spectral decomposition of some non-self-adjoint block operator matrices, J. Oper. Theory 39 (1998), pp. 339–359] as a tool for estimating the spectra of operators which admit a natural block matrix representation with respect to a decomposition of the underlying Hilbert space. We consider the problem of computing the QNR numerically for block differential operators, particularly these of Hain-Lüst type and Stokes type.

Keywords:

• block operator matrices
• estimates for eigenvalues
• quadratic numerical range
• finite difference
• Hain-Lüst operators
• spectral pollution

2010 AMS Subject Classifications:

• 65L15
• 65L12
• 34L15
• 34L16