Nuclear operator

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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 concentrates on the general case of nuclear operators on Banach spaces; for the important special case of nuclear (=trace class) operators on Hilbert space see the article on trace class operators.

Compact operator[edit]

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[citation needed]

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

where 1 \le N \le \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 are a set of real numbers, the 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[edit]

A nuclear operator on a Hilbert space has the important property that its trace may be defined so that it is finite and is independent of the basis. Given any orthonormal basis \{\psi_n\} for the Hilbert space, one may define the trace as

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

since the sum converges absolutely and is independent of the basis[citation needed]. Furthermore, this trace is identical to the sum over the eigenvalues of \mathcal{L} (counted with multiplicity).

On Banach spaces[edit]

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

\inf \left\{ p\ge 1 : \sum_n |\rho_n|^p < \infty \right\} = q,

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 ∑ρn 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 fn* bounded by 1 on some fixed neighborhood of 0 and all gn bounded by 1 on some fixed neighborhood of 0.

References[edit]