Regularizers of Closed Operators

Abstract

Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.

On the continuous linear image of a Banach space

“…Theorems 4.8 and 4.10 together show that the results of Lin (1974) are also valid for bounded strictly cosingular operators in place of (bounded) strictly singular operators.…”

“…The result can be deduced from the proof of Goldberg (1966) PROOF, (a) The stability of the classes <p, under bounded strictly singular perturbation for i = 1,2,3,4 follows from known results. Indeed, <p t and <p + are stable (Goldberg (1966), V.2.1), the stability of q> t u p 2 follows from Lin (1974), Theorem 1, and taking the complement of q> t u q> 2 in q> + shows that <p A is stable. The stability of (p, u cp 3 (and hence of q> 3 ) follows again from Lin (1974), Theorem 2.…”

“…Indeed, <p t and <p + are stable (Goldberg (1966), V.2.1), the stability of q> t u p 2 follows from Lin (1974), Theorem 1, and taking the complement of q> t u q> 2 in q> + shows that <p A is stable. The stability of (p, u cp 3 (and hence of q> 3 ) follows again from Lin (1974), Theorem 2.…”

“…If there exists t / e [ X * ] such that TU = I + F, where F is a bounded operator of finite dimensional range, then U is called a bounded rightregularizer of T(cf. Lin (1974)).…”

“…PROOF. The strictly singular case is treated in Lin (1974). Here we deal with the strictly cosingular case.…”