MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested in the Banach space $\mathcal{K}=\mathcal{K}(\ell^2)$ of compact operators on $\ell^2$, however my questions can be stated for any $\mathcal{K}(E)$, where $E$ is an arbitrary Banach space. I think that everyone who tries to study "classical" operator spaces like $\mathcal{K}$, Schatten $p$-class operators etc. immediately discovers the similarity with "commutative" counterparts, i.e. $c_0$ and $\ell^p$. This phenomenon is visible when one uses (generalised) singular numbers for certain classes of operators. Again, I have got plenty of questions concerning this stuff, let me list at least two of them:

1) what are the complemented subspaces of $\mathcal{K}$? Is $\mathcal{K}$ complemented in $\mathcal{B}(\ell^2)$? Recently, Haydon and Argyros constructed an HI-space $E$ such that $\mathcal{K}(E)$ has codimension 1 $\mathcal{B}(E)$, thus complemented.

2) is every bounded operator from $p$-Schatten class to $\mathcal{K}$ compact?

What other properties $\mathcal{K}$ shares with $c_0$?

share|cite|improve this question
I've removed the "operator spaces" tag, since nowadays this usually refers to operator spaces in the sense of Effros, Ruan et al. – Yemon Choi Jun 8 '11 at 19:24
What if you search MathSciNet for "operator spaces" to see what it commonly refers to in recent years? – Gerald Edgar Jun 8 '11 at 20:17
@Gerald: Well, of the top 40 hits, all but 2 (maybe 3) seem to use the term in Yemon's sense. – Matthew Daws Jun 8 '11 at 20:33
up vote 8 down vote accepted

It is easy to see that whenever a space has an unconditional basis then the space of diagonal operators of the basis is equivalent to $\ell_\infty$. If $c_0$ embeds in $K(X,Y)$ then $K(X,Y)$ is not complemented in $B(X,Y)$. One reference for this is: M. FEDER. On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 34 (1980), 196-205.

It is also a direct consequence of a result from a Studia paper of Tong and Wilken from 1971. Here they prove that if $Y$ has an unconditional basis then $K(X,Y)$ is uncomplemented in $B(X,Y)$ (assuming the spaces are not equal).

As far as I know the Argyros-Haydon space is the first example of a space for which it is known that $K(X)$ is complemented in $B(X)$.

share|cite|improve this answer

Probably in (2) you meant to ask whether every bounded operator from $K$ into a Schatten $p$ class is compact, since every operator from $c_0$ into $\ell_p$, $p<\infty$, is compact. But no either way: $K$ and any Schatten $p$ class contain complemented subpspaces isometrically isomorphic to $\ell_2$ (e.g. operators whose matrix representation has zeroes except in the first column).

share|cite|improve this answer
You might look at Rosenthal's expository paper which discusses properties of $K$ and other $C^*$ algebras as Banach spaces and as operator spaces. – Bill Johnson Jun 9 '11 at 2:54
That is very useful. Thank you! – Tomek Kania Jun 9 '11 at 16:41

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.