Nuclear Operator - On Banach Spaces

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

is said to be nuclear of order q if there exist sequences of vectors with, functionals with and complex numbers with

such that the operator may be written as

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 and all g_n bounded by 1 on some fixed neighborhood of 0.