ABSTRACT
Chapter 5
Matrix Formulations
In Chapter 4 we saw how to approximate a bounded operator T on a
complex Banach space X by a sequence (T
n
) of bounded nite rank
operators on X as well as how to nd an approximate solution of the
eigenvalue problem for T by solving the eigenvalue problem for T
n
. In
the present section we show that the eigenvalue problem for a bounded
nite rank operator
e
T can be solved by reducing it to a matrix eigen-
value problem in a canonical way. For a bounded nite rank operator
e
T , solutions of the operator equation
e
Tx x = y (where y 2 X is given
and x 2 X is to be found) or of the eigenvalue problem
e
T' = ' (where
0 6= ' 2 X and 0 6= 2
j
C are to be found) have long been obtained with
the help of matrix computations. This is usually done in a variety of
ways, depending on the specic nature of the nite rank operator
e
T . A
unied treatment for solutions of operator equations involving nite rank
operators was given in [73]. It was extended to eigenvalue problems for
nite rank operators in [34] and [53]. Our treatment here is along those
lines. Although the operator T
K
n
which appears in the singularity sub-
traction technique discussed in Subsection 4.2.3 is not of nite rank, the
eigenvalue problem for it can still be reduced to matrix computations.
We discuss a related question about nding a basis for a nite dimen-
sional spectral subspace for T
