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This paper studies the existence of bounded projections from the space B(X, Y) of bounded linear operators onto the subspace K(X, Y) of compact linear operators, where X and Y are normed spaces. For the cases where X and Y are either the I, or co spaces, the problem was first answered by E. Thorp [3 ]. For the cases where X is an infinite dimensional L-space or where Y is the space of all continuous functions on a topological space, various theorems were given by Arterburn and Whitley [1]. Owing to a recent result in [4], we will show that a more decisive answer can be given for the case where X, Y are Banach sequence spaces when the problem is considered from the approach of Thorp [3]. (See (1.2), (2.1), and (2.4).)

Let H \mathfrak {H} be a separable Hilbert space, B \mathfrak {B} the set of bounded linear operators on H \mathfrak {H} , and P P an orthogonal projection on H \mathfrak {H} . Denote the range of P P by R ( P ) R(P) . Let A A belong to B \mathfrak {B} . The general Wiener-Hopf operator associated with A A and P P is defined by T P ( A ) = P A | R ( P ) {T_P}(A) = PA|R(P) , the vertical bar denoting restriction. Let Q = I − P Q = I - P . The purpose of this paper is to disprove the general conjecture that if A A is an invertible element of B \mathfrak {B} , then the invertibility of T P ( A ) {T_P}(A) implies the invertibility of T Q ( A ) {T_Q}(A) . We also disprove the conjecture in an interesting special case.