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.

An operator L on a Hilbert space H

is compact if it can be written in the form

where 1 ≤ N ≤ ∞ and ${\displaystyle f_{1},\ldots ,f_{N}}$ and ${\displaystyle g_{1},\ldots ,g_{N}}$ 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 = ∞.

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