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In Barry Simons book "Trace ideals and Their Applications" in "1.4 Rearrangement Inequalities and All That" he says:

"Let $a_n$ be an infinite sequence of numbers with $a_n\rightarrow 0$ as $n\rightarrow\infty$. $a_n^*$ is the sequence defined by $a_1^*=max |a_i|$,$a_1^*+a_2^*=max_{i\neq j}(|a_i|+|a_j|)$, etc. Thus $a_1^*\geq a_2^*\geq ...$ and the sets of $a_i^*$ and $|a_i|$ are identical counting multiplicities."

Now I don´t have the newest edition of the book. I just thought if You take the sequence $a=\{1,0,\frac{1}{2},0,\frac{1}{3},...\}$, then $a^*=\{1,\frac{1}{2},\frac{1}{3},...\} $ does not contain $0$. But the sets $\{|a_j|\}-\{0\}$ and $\{a_j^*\}-\{0\}$ are identical counting multiplicities.

Am I wrong, or is there a little inaccuracy in Simons book? Thanks for any help in advance.

Peter Melech
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1 Answers1

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You are right, and there a little inaccuracy in Simons book.
The correct statement should be:

Let $a_n$ be an infinite sequence of numbers with $a_n\rightarrow0$ as $n\rightarrow\infty$. $a_n^*$ is the sequence defined by $a_1^*=\max |a_i|$, $a_1^*+a_2^*=\max_{i\neq j}(|a_i|+|a_j|)$, etc. Thus $a_1^*\geq a_2^*\ge\ldots$ and the sets of $a_i^*$ and $|a_i|$ are identical counting multiplicities, with the sole exception of occurrences of $0$.

Since $a_n\rightarrow0$, every $a_n\ne0$ will eventually be picked up by the rearrangement process. On the other hand, the only possibility that the process will ever pick up some $a_n=0$ is that the sequence contains only finitely many non-zero elements.

Gamow
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