8. Schatten classes

The polar decomposition of operators has shown us that the absence of commutativity is not an insurmountable obstacle to establishing a parallel with the polar representation of complex numbers, even if in fact not all the properties are preserved, such as triangular inequality.

This parallel can be continued with the definition of a scale of subspaces of the set of compact operators possessing properties analogous to those of the spaces L ^p : the Schatten classes ${\mathfrak{L}}^p$ which form an intermediate category between operators of finite rank and compact operators. They play an important role in the study of the completeness of generalized eigenvectors...