The number of closed ideals in $L(L_p)$

We show that there are $2^{2^{\aleph_0}}$ different closed ideals in the Banach algebra $L(L_p(0,1)), 1 \lt p \neq 2 \lt \infty$. This solves a problem in A. Pietsch’s 1978 book “Operator Ideals”. The proof is quite different from other methods of producing closed ideals in the space of bounded operators on a Banach space; in particular, the ideals are not contained in the strictly singular operators and yet do not contain projections onto subspaces that are non-Hilbertian. We give a criterion for a space with an unconditional basis to have $2^{2^{\aleph_0}}$ closed ideals in terms of the existence of a single operator on the space with some special asymptotic properties. We then show that for $1 \lt q \lt 2$ the space $\mathfrak{X}_q$ of Rosenthal, which is isomorphic to a complemented subspace of $L_q(0,1)$, admits such an operator.

Keywords

ideals of operators, $L_p$ spaces

2010 Mathematics Subject Classification

46E30, 47L20

Full Text (PDF format)

Received 26 March 2020

Accepted 8 February 2021

Published 24 November 2021