Nuclear operator

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In mathematics, 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 (TVSs).

Preliminaries and notation[]

Throughout let X,Y, and Z be topological vector spaces (TVSs) and L : X → Y be a linear operator (no assumption of continuity is made unless otherwise stated).

• The projective tensor product of two locally convex TVSs X and Y is denoted by ${\displaystyle X\otimes _{\pi }Y}$ and the completion of this space will be denoted by ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$.
• L : X → Y is a topological homomorphism or homomorphism, if it is linear, continuous, and ${\displaystyle L:X\to \operatorname {Im} L}$ is an open map, where ${\displaystyle \operatorname {Im} L}$, the image of L, has the subspace topology induced by Y.
  • If S is a subspace of X then both the quotient map X → X/S and the canonical injection S → X are homomorphisms.
• The set of continuous linear maps X → Z (resp. continuous bilinear maps ${\displaystyle X\times Y\to Z}$) will be denoted by L(X, Z) (resp. B(X, Y; Z)) where if Z is the underlying scalar field then we may instead write L(X) (resp. B(X, Y)).
• Any linear map ${\displaystyle L:X\to Y}$ can be canonically decomposed as follows: ${\displaystyle X\to X/\ker L\;\xrightarrow {L_{0}} \;\operatorname {Im} L\to Y}$ where ${\displaystyle L_{0}\left(x+\ker L\right):=L(x)}$ defines a bijection called the canonical bijection associated with L.
• X* or ${\displaystyle X'}$ will denote the continuous dual space of X.
  • To increase the clarity of the exposition, we use the common convention of writing elements of ${\displaystyle X'}$ with a prime following the symbol (e.g. ${\displaystyle x'}$ denotes an element of ${\displaystyle X'}$ and not, say, a derivative and the variables x and ${\displaystyle x'}$ need not be related in any way).
• ${\displaystyle X^{\#}}$ will denote the algebraic dual space of X (which is the vector space of all linear functionals on X, whether continuous or not).
• A linear map L : H → H from a Hilbert space into itself is called positive if ${\displaystyle \langle L(x),x\rangle \geq 0}$ for every ${\displaystyle x\in H}$. In this case, there is a unique positive map r : H → H, called the square-root of L, such that ${\displaystyle L=r\circ r}$.^[1]
  • If ${\displaystyle L:H_{1}\to H_{2}}$ is any continuous linear map between Hilbert spaces, then ${\displaystyle L^{*}\circ L}$ is always positive. Now let R : H → H denote its positive square-root, which is called the absolute value of L. Define ${\displaystyle U:H_{1}\to H_{2}}$ first on ${\displaystyle \operatorname {Im} R}$ by setting ${\displaystyle U(x)=L(x)}$ for ${\displaystyle x=R\left(x_{1}\right)\in \operatorname {Im} R}$ and extending ${\displaystyle U}$ continuously to ${\displaystyle {\overline {\operatorname {Im} R}}}$, and then define U on ${\displaystyle \ker R}$ by setting ${\displaystyle U(x)=0}$ for ${\displaystyle x\in \ker R}$ and extend this map linearly to all of ${\displaystyle H_{1}}$. The map ${\displaystyle U{\big \vert }_{\operatorname {Im} R}:\operatorname {Im} R\to \operatorname {Im} L}$ is a surjective isometry and ${\displaystyle L=U\circ R}$.
• A linear map ${\displaystyle \Lambda :X\to Y}$ is called compact or completely continuous if there is a neighborhood U of the origin in X such that ${\displaystyle \Lambda (U)}$ is precompact in Y.^[2]

  • In a Hilbert space, positive compact linear operators, say L : H → H have a simple spectral decomposition discovered at the beginning of the 20th century by Fredholm and F. Riesz:^[3]

    There is a sequence of positive numbers, decreasing and either finite or else converging to 0, ${\displaystyle r_{1}>r_{2}>\cdots >r_{k}>\cdots }$ and a sequence of nonzero finite dimensional subspaces ${\displaystyle V_{i}}$ of H (i = 1, 2, ${\displaystyle \ldots }$) with the following properties: (1) the subspaces ${\displaystyle V_{i}}$ are pairwise orthogonal; (2) for every i and every ${\displaystyle x\in V_{i}}$, ${\displaystyle L(x)=r_{i}x}$; and (3) the orthogonal of the subspace spanned by ${\textstyle \bigcup _{i}V_{i}}$ is equal to the kernel of L.^[3]

Notation for topologies[]

• σ(X, X′) denotes the coarsest topology on X making every map in X′ continuous and ${\displaystyle X_{\sigma \left(X,X'\right)}}$ or ${\displaystyle X_{\sigma }}$ denotes X endowed with this topology.
• σ(X′, X) denotes weak-* topology on X* and ${\displaystyle X_{\sigma \left(X',X\right)}}$ or ${\displaystyle X'_{\sigma }}$ denotes X′ endowed with this topology.
  • Note that every ${\displaystyle x_{0}\in X}$ induces a map ${\displaystyle X'\to \mathbb {R} }$ defined by ${\displaystyle \lambda \mapsto \lambda \left(x_{0}\right)}$. σ(X′, X) is the coarsest topology on X′ making all such maps continuous.
• b(X, X′) denotes the topology of bounded convergence on X and ${\displaystyle X_{b\left(X,X'\right)}}$ or ${\displaystyle X_{b}}$ denotes X endowed with this topology.
• b(X′, X) denotes the topology of bounded convergence on X′ or the strong dual topology on X′ and ${\displaystyle X_{b\left(X',X\right)}}$ or ${\displaystyle X'_{b}}$ denotes X′ endowed with this topology.
  • As usual, if X* is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be b(X′, X).

A canonical tensor product as a subspace of the dual of Bi(X, Y)[]

Let X and Y be vector spaces (no topology is needed yet) and let Bi(X, Y) be the space of all bilinear maps defined on ${\displaystyle X\times Y}$ and going into the underlying scalar field.

For every ${\displaystyle (x,y)\in X\times Y}$, let ${\displaystyle \chi _{(x,y)}}$ be the canonical linear form on Bi(X, Y) defined by ${\displaystyle \chi _{(x,y)}(u):=u(x,y)}$ for every u ∈ Bi(X, Y). This induces a canonical map ${\displaystyle \chi :X\times Y\to \mathrm {Bi} (X,Y)^{\#}}$ defined by ${\displaystyle \chi (x,y):=\chi _{(x,y)}}$, where ${\displaystyle \mathrm {Bi} (X,Y)^{\#}}$ denotes the algebraic dual of Bi(X, Y). If we denote the span of the range of