# Nuclear operator

In mathematics, **nuclear operators** are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).

Throughout let *X*,*Y*, and *Z* be topological vector spaces (TVSs) and *L* : *X* → *Y* be a linear operator (no assumption of continuity is made unless otherwise stated).

Nuclear automorphisms of a Hilbert space are called **trace class** operators.

Let *X* and *Y* be Hilbert spaces and let *N* : *X* → *Y* be a continuous linear map whose absolute value is *R* : *X* → *X*. The following are equivalent:

When *X* and *Y* are Banach spaces, then this new definition of *nuclear mapping* is consistent with the original one given for the special case where *X* and *Y* are Banach spaces.

The following is a type of *Hahn-Banach theorem* for extending nuclear maps: