Binary products commutator

The commutator binary product introduced by Goscinski and coworkers was chosen so that the special propagators in the Linderberg-Ohrn theory17 could be written explicitly in the form (1.56). Using the connection formula (3.19), one hence obtains... 

It should be observed that the GNS space A contains the Hilbert-Schmidt space as a subspace, but also that it is considerably larger, since it is now sufficient to require that the product AT1/2 should be a Hilbert-Schmidt operator. As an example, we note that the identity operator 1 belongs to the GNS space but not to the HS space. By using (3.17) and (3.48), one may now write the commutator binary product (A T) in the form ... 

The commutator binary product defined through (1.54) or (3.17) and the GNS binary product (3.48) have played an important role in revealing the mathematical structure of the special propagator methods and their connection with the EOM approach. In the opinion of the author, however, it is of great advantage to work with a binary product with respect to which the Liouvillian superoperator L from the very beginning is self-adjoint, and the HS binary product (2.2) would then be the most natural starting point. 

In principle, it should be simpler to use the Hilbert-Schmidt binary product than the commutator binary product (1.54), since the former is positive definite and nondegenerate and makes the Liouvillian L a self-... 

We remind the reader that all permutations are imitary operators. Since binary permutations are equal to dieir own inverses, they are also Hermitian. Products of commuting binaries are also Hermitian. 

Binary composition in a set of abstract elements g,, whatever its nature, is always written as a multiplication and is usually referred to as multiplication whatever it actually may be. For example, if g, and g, are operators then the product g,- gy means carry out the operation implied by gy and then that implied by g,. If g, and gy are both -dimensional square matrices then g, gy is the matrix product of the two matrices g, and gy evaluated using the usual row x column law of matrix multiplication. (The properties of matrices that are made use of in this book are reviewed in Appendix Al.) Binary composition is unique but is not necessarily commutative g, g, may or may not be equal to gy gt. In order for a set of abstract elements g, to be a G, the law of binary composition must be defined and the set must possess the following four properties. 

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