Scope

Infinite dimensional block operator matrices are matrices the entries of which are linear operators between Banach or Hilbert spaces. They arise in various areas of mathematics and its applications, e.g., in

system theory as Hamiltonians
the discretization of partial differential equations as large partitioned matrices due to sparsity patterns
non-linear analysis and saddle point problems
evolution problems as linearizations of second order Cauchy problems
linear operators describing coupled systems of partial differential equations with applications to fluid mechanics, magnetohydrodynamics, and quantum mechanics.

In all these applications, the spectral properties of the corresponding block operator matrices are of vital importance as they govern, for instance, the time evolution and hence the stability of the underlying physical or mechanical systems. It is the aim is to present a wide panorama of methods to investigate the spectral properties of block operator matrices. Particular emphasis is placed on classes of block operator matrices to which standard operator theoretical methods taught at universities do not readily apply: non-self-adjoint block operator matrices, block operator matrices with unbounded entries, non-semi-bounded block operator matrices, and classes of block operator matrices arising in mathematical physics. Particular interest is given to carefully presented applications ranging from hydrodynamics to quantum mechanics.

The course will be designed for students with various backgrounds and interests; it contains new results and methods ranging from the finite dimensional matrix case to unbounded operators in infinite dimensional spaces. New applications to problems from physics and mechanics will be given. In addition, the lectures will lead the students to various open problems in an active research area, thus challenging them to active participation.