Nuclear operator


Nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces.

Preliminaries and notation

Throughout let X,Y, and Z be topological vector spaces and L : X → Y be a linear operator.
Let X and Y be vector spaces and let Bi be the space of all bilinear maps defined on and going into the underlying scalar field.
For every, let be the canonical bilinear form on Bi defined by for every u ∈ Bi.
This induces a canonical map defined by, where denotes the algebraic dual of Bi.
If we denote the span of the range of ? by X ⊗ Y then it can be shown that X ⊗ Y together with ? forms a tensor product of X and Y.
This gives us a canonical tensor product of X and Y.
If Z is any other vector space then the mapping Li → Bi given by u ↦ u ∘ ? is an isomorphism of vector spaces.
In particular, this allows us to identify the algebraic dual of X ⊗ Y with the space of bilinear forms on X × Y.
Moreover, if X and Y are locally convex topological vector spaces and if X ⊗ Y is given the ?-topology then for every locally convex TVS Z, this map restricts to a vector space isomorphism from the space of continuous linear mappings onto the space of continuous bilinear mappings.
In particular, the continuous dual of X ⊗ Y can be canonically identified with the space B of continuous bilinear forms on X × Y;
furthermore, under this identification the equicontinuous subsets of B are the same as the equicontinuous subsets of ''.

Nuclear operators between Banach spaces

There is a canonical vector space embedding defined by sending to the map
Assuming that X and Y are Banach spaces, then the map has norm . Thus it has a continuous extension to a map, where it is known that this map is not necessarily injective.. The range of this map is denoted by and its elements are called nuclear operators. is TVS-isomorphic to and the norm on this quotient space, when transferred to elements of via the induced map, is called the trace-norm and is denoted by. Explicitely, if is a nuclear operator then.

Characterization

Suppose that X and Y are Banach spaces and that is a continuous linear operator.
  1. is nuclear.
  2. There exists an sequence in the closed unit ball of, a sequence in the closed unit ball of, and a complex sequence such that and is equal to the mapping: for all. Furthermore, the trace-norm is equal to the infimum of the numbers over the set of all representations of as such a series.
Let X and Y be Banach spaces and let be a continuous linear operator.
Nuclear automorphisms of a Hilbert space are called trace class operators.
Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map.
Suppose that where R : X → X is the square-root of and U : X → Y is such that is a surjective isometry and.
Then N is a nuclear map if and only if R is a nuclear map;
hence, to study nuclear maps between Hilbert spaces it suffices to restrict one's attention to positive linear operators.

Characterizations

Let X and Y be Hilbert spaces and let N : X → Y be a continuous linear map whose absolute value is R : X → X.
The following are equivalent:

  1. N : X → Y is nuclear.
  2. R : X → X is nuclear.
  3. R : X → X is compact and is finite, in which case.
    • Here, is the trace of R and it is defined as follows: Since R is a continuous compact positive operator, there exists a sequence of positive numbers with corresponding non-trivial finite-dimensional and mutually orthogonal vector spaces such that the orthogonal of is equal to and for all k, for all ; the trace is defined as.
  4. is nuclear, in which case.
  5. There are two orthogonal sequences in X and in Y, and a sequence in such that for all,.
  6. N : X → ''Y' is an integral map.

Nuclear operators between locally convex spaces

Suppose that U is a convex balanced closed neighborhood of the origin in X and B is a convex balanced bounded Banach disk in Y with both X and Y locally convex spaces. Let and let be the canonical projection. One can define the auxiliary Banach space with the canonical map whose image,, is dense in as well as the auxiliary space normed by and with a canonical map being the canonical injection.
Given any continuous linear map one obtains through composition the continuous linear map ; thus we have an injection and we henceforth use this map to identify as a subspace of.
Definition: Let X and Y be Hausdorff locally convex spaces. The union of all as U ranges over all closed convex balanced neighborhoods of the origin in X and B ranges over all bounded Banach disks in Y, is denoted by and its elements are call nuclear mappings of X into Y.
When X and Y are Banach spaces, then this new definition of nuclear mapping is consistent with the original one given for the special case where X and Y are Banach spaces.

Sufficient conditions for nuclearity

Let X and Y be Hausdorff locally convex spaces and let be a continuous linear operator.
  1. is nuclear.
  2. There exists a convex balanced neighborhood U of the origin in X and a bounded Banach disk B in Y such that and the induced map is nuclear, where is the unique continuous extension of, which is the unique map satisfying where is the natural inclusion and is the canonical projection.
  3. There exist Banach spaces and and continuous linear maps,, and such that is nuclear and.
  4. There exists an equicontinuous sequence in, a bounded Banach disk, a sequence in B, and a complex sequence such that and is equal to the mapping: for all.
The following is a type of Hahn-Banach theorem for extending nuclear maps:
Let X and Y be Hausdorff locally convex spaces and let be a continuous linear operator.