ABSTRACT
Chapter 4
Finite Rank Approximations
Consider a complex Banach space X and a bounded linear operator T
on X , that is, T 2 BL(X). In Chapter 2, we saw how spectral values of
nite type and the associated spectral subspaces of T are approximated
if a sequence (T
n
) in BL(X) `converges' to T ; specically if T
n
n
! T , and
more generally if T
n
! T . We shall see in Chapter 5 that if the rank
of T
n
is nite, then the spectral computations for T
n
can be reduced
to solving a matrix eigenvalue problem in a canonical way. For this
reason, we now consider various situations in which it is possible to nd
a sequence (T
n
) of bounded nite rank operators such that T
n
n
! T ,
or at least, T
n
! T . First we treat some nite rank approximations
which employ bounded nite rank projections. Next we treat the case of
Fredholm integral operators, for which degenerate kernel approximations
as well as approximations based on quadrature rules are described. For a
weakly singular integral operator T , we present an approximation (T
K
n
)
such that T
K
n
= T
n
+U
n
, where T
n
is of nite rank, T
K
n
! T but T
K
n
cc
!
