About the trace class operators and their motivation 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 

  
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*Operator theory by Conway
  
*Operator theory in function spaces by Kehe Zhu
  

Can anybody suggest even better?
 A: 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).
A: 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.
A: 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.
A: You might look at Barry Simon's book "Trace Ideals and Their Applications", especially for applications in mathematical physics.
A: *

*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)

