PATRICK FITZPATRICK AND JACOBO PEJSACHOWICZ
compact perturbation of a Fredholm operator is again Fredholm of the same
The preceding observations may be summarized by asserting that if T €
L(X,Y), then the following three assertions are equivalent:
(i) T € *Q(X,Y)
(ii) there exists K e K(X,Y) with T + K € GL(X,Y)
(iii) there exists R € GL(Y,X) with RT - Id € K(X).
An operator R as above is called a (strong) parametrix for T. In
the Soviet literature on singular integral equations, R is called an
equivalent regularizer for T (cf. [Z-K-K-P]).
Suppose now we have a (continuous) family a:A — $n(X,Y), where A
is a topological space. If £: A —-» GL(Y,X) is continuous and
0(A)a(A) - Id € K(X) for all A € A,
then we call 0: A —» GL(Y,X) a (strong) parametrix for a. In general,
a family a:A — $n(X,Y) does not have a parametrix. For instance, when
A = S , the non-existence of parametrices for certain x:S — $ (X,Y)
corresponds to the nontriviality of the Poincare group of $ (X,Y)
(cf.[Fi-Pe,4]). However, we have the following fundamental result.
Theorem 2.1 Let A be a paracompact, contractible, Hausdorff space and
suppose that a: A —- » $ (X,Y) is continuous. Then there exists a
parametrix 0: A — GL(Y,X) for a.
Proof: (See also [Z-K-K-P].) Since $ (X,Y) * 0 we may, without loss of
generality, assume that X = Y. Let G denote the orbit space of GL(X)
by the action of GLp(X). It is known (see [Pa]) that