Nuclear operators between Banach spaces

In mathematics, nuclear operators between Banach spaces are a linear operators between Banach spaces in infinite dimensions that share some of the properties of their counter-part in finite dimension. In Hilbert spaces such operators are usually called trace class operators and one can define such things as the trace. In Banach spaces this is no longer possible for general nuclear operators, it is however possible for ${\tfrac {2}{3}}$ -nuclear operator via the Grothendieck trace theorem.

The general definition for Banach spaces was given by Grothendieck. This article presents both cases but concentrates on the general case of nuclear operators on Banach spaces.

Nuclear operators on Hilbert spaces

An operator ${\mathcal {L}}$ on a Hilbert space ${\mathcal {H}}$ ${\mathcal {L}}:{\mathcal {H}}\to {\mathcal {H}}$ is compact if it can be written in the form^{[citation needed]} ${\mathcal {L}}=\sum _{n=1}^{N}\rho _{n}\langle f_{n},\cdot \rangle g_{n},$ 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 $\{\rho _{1},\ldots ,\rho _{N}\}$ is a set of real numbers, the set of singular values of the operator, obeying $\rho _{n}\to 0$ 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.

An operator that is compact as defined above is said to be nuclear or trace-class if $\sum _{n=1}^{\infty }|\rho _{n}|<\infty .$

Properties

A nuclear operator on a Hilbert space has the important property that a trace operation may be defined. Given an orthonormal basis $\{\psi _{n}\}$ for the Hilbert space, the trace is defined as $\operatorname {Tr} {\mathcal {L}}=\sum _{n}\langle \psi _{n},{\mathcal {L}}\psi _{n}\rangle .$

Obviously, the sum converges absolutely, and it can be proven that the result is independent of the basis^{[citation needed]}. It can be shown that this trace is identical to the sum of the eigenvalues of ${\mathcal {L}}$ (counted with multiplicity).

Nuclear operators on Banach spaces

The definition of trace-class operator was extended to Banach spaces by Alexander Grothendieck in 1955.

Let $A$ and $B$ be Banach spaces, and $A^{\prime }$ be the dual of $A,$ that is, the set of all continuous or (equivalently) bounded linear functionals on $A$ with the usual norm. There is a canonical evaluation map $A^{\prime }\otimes B\to \operatorname {Hom} (A,B)$ (from the projective tensor product of $A$ and $B$ to the Banach space of continuous linear maps from $A$ to $B$ ). It is determined by sending $f\in A^{\prime }$ and $b\in B$ to the linear map $a\mapsto f(a)\cdot b.$ An operator ${\mathcal {L}}\in \operatorname {Hom} (A,B)$ is called nuclear if it is in the image of this evaluation map.^[1]

$q$ -nuclear operators

An operator ${\mathcal {L}}:A\to B$ is said to be nuclear of order $q$ if there exist sequences of vectors $\{g_{n}\}\in B$ with $\Vert g_{n}\Vert \leq 1,$ functionals $\left\{f_{n}^{*}\right\}\in A^{\prime }$ with $\Vert f_{n}^{*}\Vert \leq 1$ and complex numbers $\{\rho _{n}\}$ with $\sum _{n}|\rho _{n}|^{q}<\infty ,$ such that the operator may be written as ${\mathcal {L}}=\sum _{n}\rho _{n}f_{n}^{*}(\cdot )g_{n}$ with the sum converging in the operator norm.

Operators that are nuclear of order 1 are called nuclear operators: these are the ones for which the series $\sum \rho _{n}$ is absolutely convergent. Nuclear operators of order 2 are called Hilbert–Schmidt operators.