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It is well known that the only proper non-trivial norm-closed ideal in the algebra L(X) for X=\ell_p (1 \le p < \infty) or X=c_0 is the ideal of compact operators. The next natural question is to describe all closed ideals of L(\ell_p\oplus\ell_q) for 1 \le p,q < \infty, p \neq q, or, equivalently, the closed ideals in L(\ell_p,\ell_q) for p < q. This paper shows that for 1 < p < 2 < q < \infty there are at least four distinct proper closed ideals in L(\ell_p,\ell_q), including one that has not been studied before. The proofs use various methods from Banach space theory., Comment: 24 pages

A 1972 duality conjecture due to Pietsch asserts that the entropy numbers of a compact operator acting between two Banach spaces and those of its adjoint are (in an appropriate sense) equivalent. This is equivalent to a dimension free inequality relating covering (or packing) numbers for convex bodies to those of their polars. The duality conjecture has been recently proved (see math.FA/0407236) in the central case when one of the Banach spaces is Hilbertian, which - in the geometric setting - corresponds to a duality result for symmetric convex bodies in Euclidean spaces. In the present paper we define a new notion of "convexified packing," show a duality theorem for that notion, and use it to prove the duality conjecture under much milder conditions on the spaces involved (namely, that one of them is K-convex)., Comment: 6 p., LATEX

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain $\ell_1$ but none of them is isomorphic to $\ell_1$. We also prove that for any countable set $C$ of spreading models generated by weakly null sequences there is a spreading model generated by a weakly null sequence which dominates each element of $C$. In certain cases this ensures that $X$ admits, for each $\alpha < \omega_1$, a spreading model $(\tilde x_i^\alpha)_i$ such that if $\alpha < \beta$ then $(\tilde x_i^\alpha)_i$ is dominated by (and not equivalent to) $(\tilde x_i^\beta)_i$. Some applications of these ideas are used to give sufficient conditions on a Banach space for the existence of a subspace and an operator defined on the subspace, which is not a compact perturbation of a multiple of the inclusion map.