Nuclear operator

In mathematics, a nuclear operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis (at least on well behaved spaces; there are some spaces on which nuclear operators do not have a trace). Nuclear operators are essentially the same as trace class operators, though most authors reserve the term "trace class operator" for the special case of nuclear operators on Hilbert spaces.

The general definition for Banach spaces was given by Grothendieck. This article presents both cases concentrates on the general case of nuclear operators on Banach spaces; for more details about the important special case of nuclear (=trace class) operators on Hilbert space see the article on trace class operators.

Compact operator

An operator L on a Hilbert space H

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

is compact if it can be written in the form

$\mathcal{L} = \sum_{n=1}^N \rho_n \langle f_n, \cdot \rangle g_n$

where 1 ≤ N ≤ ∞ and $f_1,\ldots,f_N$ and $g_1,\ldots,g_N$ are (not necessarily complete) orthonormal sets. Here, ρ₁, ... ,ρ_N are a set of real numbers, the singular values of the operator, obeying ρ_n → 0 if N = ∞.

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

$\mbox{Tr} \mathcal {L} = \sum_n \langle \psi_n , \mathcal{L} \psi_n \rangle$.

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

On Banach spaces

See main article Fredholm kernel.

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' be the dual of A, that is, the set of all continuous or (equivalently) bounded linear functionals on A with the usual norm. Then 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 $\Vert g_n \Vert \le 1$, functionals $\{f^*_n\} \in A'$ with $\Vert f^*_n \Vert \le 1$ 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.

With additional steps, a trace may be defined for such operators when A = B.

Operators that are nuclear of order 1 are called nuclear operators: these are the ones for which the series ∑ρ_n is absolutely convergent. Nuclear operators of order 2 are called Hilbert–Schmidt operators.

More generally, an operator from a locally convex topological vector space A to a Banach space B is called nuclear if it satisfies the condition above with all f_n^* bounded by 1 on some fixed neighborhood of 0.

References

• A. Grothendieck (1955), Produits tensoriels topologiques et espace nucléaires,Mem. Am. Math.Soc. 16. MR 0075539
• A. Grothendieck (1956), La theorie de Fredholm, Bull. Soc. Math. France, 84:319–384. MR 0088665
• A. Hinrichs and A. Pietsch (2010), p-nuclear operators in the sense of Grothendieck, Mathematische Nachrichen 283: 232–261. doi:10.1002/mana.200910128 MR 2604120
• G. L. Litvinov (2001), "Nuclear operator", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4