Nuclear operators between Banach spaces

In mathematics, a nuclear operator or a trace-class operator is an oparator whose trace can in a certain sense said to be finite. Nuclear operators are usually defined on Hilbert spaces, but the definition can be extened to Banach spaces, the extension having been given by Grothendieck.

Compact operator

An operator ${\mathcal {L}}$ on a Hilbert space ${\mathcal {H}}$

${\mathcal {L}}:{\mathcal {H}}\to {\mathcal {H}}$

is said to be a compact operator if it can be written in the form

${\mathcal {L}}=\sum _{n=1}^{N}\lambda _{n}\langle f_{n},\cdot \rangle g_{n}$

where $1\leq N\leq \infty$ and $f_{1},\ldots ,f_{N}$ and $g_{1},\ldots ,g_{N}$ are (not necessarily complete) orthonormal sets. Here, $\lambda _{1},\ldots ,\lambda _{N}$ are a set of real numbers, the singular values of the operator, obeying $\lambda _{N}\to 0$ if $N=\infty$. The bracket $\langle \cdot ,\cdot \rangle$ is the scalar product on the Hilbert space; the sum on the right hand side must converge in norm.