Vol. 68, No. 1, 1977

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Norms of compact perturbations of operators

Catherine Louise Olsen

Vol. 68 (1977), No. 1, 209–228

Let () 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 (), it is well known that the infimum, Te = 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 = T1Tn, 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= T2e and (T + K)3= T3e; 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.

Mathematical Subject Classification 2000
Primary: 47A55
Received: 4 August 1975
Published: 1 January 1977
Catherine Louise Olsen