A question on the product of two real linear operators

[Excimer]

I am reading The Principles of Quantum Mechanics 4th Ed by Paul Dirac, specifically where he introduces his own Bra-Ket notation. You can view this book as a google book.

http://books.google.com.au/books?id...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

On page 28 (on linear operators acting on kets and bras), after showing that "the conjugate complex of the product of two linear operators equals the product of the conjugate complexes of the factors in reverse order",

He then says that "if ξ and η are real, in general ξη is not real".

Can someone please explain or post a link to a reference showing how it can be that the product of two real linear operators is not necessarily real?

OR, could it be a misprint?

 

[Bill_K]

When he says "real", he means self-adjoint. So (ab)^† = b^†a^† = ba, which is not necessarily the same as ab. Thus the product of two self-adjoint operators is not self-adjoint if they don't commute.

 

[Excimer]

Thanks BIll,

You are referring to the previous page (27) where he says that when a linear operator is self adjoint it is called a 'real linear operator' (only because when the operator is a number and self adjoint then the conjuagte complex is real).

Is that correct?

So the term 'real' in this context now just means an operator that is self adjoint, rather than the normal meaning of the word real?

So one could replace his sentence using the word real to:

if ξ and η are both self adjoint, ξη is not necessarily also self adjoint.

Correct?