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What is the motivation for trace class operators? Can any body suggest the most general and standard reference that includes Schatten p class operators as well. I have following references

  1. Operator theory by Conway
  2. Operator theory in function spaces by Kehe Zhu

Can anybody suggest even better?

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  • $\begingroup$ Welcome to the site. I took the liberty of correcting a couple of typos and adding the reference-request tag. $\endgroup$
    – user25199
    Commented Dec 5, 2013 at 14:27
  • $\begingroup$ Generalizing Schatten classes one step further, we get noncommutative $L^p$ spaces: so maybe you could find some motivation in the survey by Pisier & Xu, in the Handbook II (p. 1459 et seq.): click here. $\endgroup$
    – Julien
    Commented Dec 5, 2013 at 21:14

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Compact operators have a countable set of singular values $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the Schatten $p$-class. So Schatten $p$-class operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making Schatten $2-$class a Hilbert space. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).

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    $\begingroup$ If I am not mistaken, you should have written "singular values" instead of "eigenvalues" in your second sentence. $\endgroup$
    – Mateusz Wasilewski
    Commented Dec 5, 2013 at 13:42
  • $\begingroup$ No problem, Marc. By the way, I was referring to the second sentence, the one concerning $\ell_p$-norms. $\endgroup$
    – Mateusz Wasilewski
    Commented Dec 5, 2013 at 16:07
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The justification for studying these operators is displayed in their name---they are the infinite dimensional operators for which a trace can be defined. In functional analysis, they are important since they have a natural Banach space structure for which they are the predual of the space of bounded linear operators. Comprehensive monographs on them have been penned by Pietsch, Retherford, Simon and König.

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As a further reference you can also consult the locally convex analysis monograph of Jarchow. A very comprehensive book, I like it a lot: it has some sections on $p$-summable operators also beyond the Hilbert space case. There a lot of new phenomena appear. So this might be interesting for you to put things into a slightly bigger context.

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You might look at Barry Simon's book "Trace Ideals and Their Applications", especially for applications in mathematical physics.

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  • MR0582655 Pietsch, Albrecht Operator ideals. Translated from German by the author. North-Holland Mathematical Library, 20. North-Holland Publishing Co., Amsterdam-New York, 1980. 451 pp. (MathSciNet)
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