Let ℬ(ℋ) denote the algebra of
all bounded linear operators on a complex separable Hilbert space. This paper is
concerned with reducing the norm of a product of operators by compact
perturbations of one or more of the factors. For any T in ℬ(ℋ), it is well known that
the infimum, ∥T∥e=inf{ ∥T + K∥ : K is a compact operator} is attained by some
compact perturbation T + K0. For T a noncompact product of n operators,
T = T1⋯Tn, it is proved that this infimum can be obtained by a compact
perturbation of any one of the factors. If T is a compact product, so that the infimum
is zero, it is shown that there are compact perturbations T1+ K1,⋯,Tn+ Kn of the
factors of T such that the product (T1+ K1)⋯(Tn+ Kn) is zero; furthermore, it
may be necessary to perturb every factor of T in order to obtain this zero
infimum. These results are applied to an arbitrary operator T to find a compact
perturbation T + K with ∥(T + K)2∥ = ∥T2∥e and ∥(T + K)3∥ = ∥T3∥e;
here the identical factors are perturbed in identical fashion to achieve both
infima. Stronger theorems of this latter sort are proved for special classes of
operators.
