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On norm closed ideals in L(\ell_p\oplus\ell_q)

B. Sari·Th. Schlumprecht·N. Tomczak-Jaegermann·V. G. Troitsky
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Abstract

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.

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B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky. 2005-09-19. On norm closed ideals in L(\ell_p\oplus\ell_q). https://arxiv.org/abs/math/0509414

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