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On a Conjecture of Barry Simon on Trace Ideals

Cobos, Fernando and Kühn, Thomas (1989) On a Conjecture of Barry Simon on Trace Ideals. Duke mathematical journal , 59 (1). pp. 295-299. ISSN 0012-7094

Official URL: http://projecteuclid.org/euclid.dmj/1077307842

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Abstract

Let H denote a Hilbert space, T a compact operator on H, {sn(T)}1 n=1 the eigenvalues of |T|, and Sp (p > 0) the set of all such T for which {sn(T)}1 n=1 is in `p. If A and B are bounded linear operators on L2, say that B pointwise dominates A if |A(x)(t)| B(|x|)(t) a.e. for all x(t) in L2. It is known that if p = 2n for some positive integer n, B is in Sp, and B pointwise dominates A, then A is also in Sp. Simon has conjectured that this result fails for p < 2, and has given a counterexample for 0 < p 1. The authors provide a counterexample for the remaining cases where 1 < p < 2.


Item Type:Article
Uncontrolled Keywords:Trace ideals; compact operator; pointwise dominates
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:15609
Deposited On:13 Jun 2012 08:12
Last Modified:23 Oct 2013 16:12

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