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Chapter

Chapter

# Finite Rank Approximations

DOI link for Finite Rank Approximations

Finite Rank Approximations book

# Finite Rank Approximations

DOI link for Finite Rank Approximations

Finite Rank Approximations book

## 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

!