ABSTRACT
Chapter 2
Spectral Approximation
Our main interest lies in determining clusters of spectral values of nite
type and the associated spectral subspaces of a given bounded linear
operator T . Since exact computations are almost always impossible, we
attempt to obtain numerical approximations. Usually these approxima-
tions are the exact results of spectral computations on a bounded linear
operator
e
T which is close to the given operator T . Often the operator
e
T
is chosen to be a member of a sequence (T
n
) of bounded linear operators
which converges to T pointwise, or with respect to the operator norm,
or in a `collectively compact' manner. Other than these three modes
of convergence, we study a new mode of convergence which is strong
enough to yield the desired spectral results and general enough to be
applicable in a number of situations. Under this mode of convergence,
properties similar to the upper semicontinuity and the lower semicon-
tinuity of the spectrum are proved. The chapter concludes with error
bounds rst for the approximation of a simple eigenvalue and the cor-
responding eigenvector, and then for the approximation of a cluster of
multiple eigenvalues and a basis for the associated spectral subspace.
