ON NORM CLOSED IDEALS IN L(lp, lq)

Given two Banach spaces X and Y , we write L(X, Y ) for the space of all continuous linear operators from X to Y . A linear subspace J of L(X, Y ) is said to be an ideal if ATB ∈ J whenever A ∈ L(Y ), T ∈ J , and B ∈ L(X). It is known (see, e.g., Caradus:74 [CPY74]) that the only norm closed ideal in L(lp), 1 6 p < ∞ is the ideal of compact operators. The structure of closed ideals in L(lp ⊕ lq) for 1 6 p < q 6 ∞ is not completely clear, but it is known (see Pietsch:78 [Pie78]) that it can be reduced to describing the closed ideals in L(lp, lq). In this paper we describe More details on the reduction? Or just refer to Pietsch:78 [Pie78]? some new closed ideals in L(lp, lq). Throughout this paper, p and q always satisfy 1 6 p < q < ∞. We denote by p′ the conjugate of p, that is, 1 p + 1 p′ = 1. By a closed ideal of L(lp, lq) we mean an ideal closed in operator norm topology. We denote by K the closed ideal of all compact operators. It is known (see, e.g., Caradus:74 [CPY74]) that K is contained in every closed ideal of L(lp, lq). Is it well known? Or should we outline this? If Z is a Banach space, we denote J Z the closure of the set of all the operators in L(lp, lq) that factor through Z. It can be easily verified that if Z ∼= Z ⊕ Z then J Z is an ideal. For S ∈ L(lp, lq) we denote J S the closed ideal in L(lp, lq) generated by S, that is, the smallest closed ideal containing S. It is easy to see that J S consists exactly of the operators that can be approximated in norm by operators of the form ∑n i=1 AiSBi, where Ai ∈ L(lq) and Bi ∈ L(lp) as i = 1, . . . , n. If Do we indeed need linear combinations? A is an n × n scalar matrix, we write ‖A‖p,q for the norm of A as an operator from lp to l n q .