Integral linear operator

An integral bilinear form is a bilinear functional that belongs to the continuous dual space of $X{\widehat {\otimes }}_{\epsilon }Y$, the injective tensor product of the locally convex Topological Vector Spaces (TVSs) X and Y. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form.

These maps play an important role in the theory of nuclear spaces and nuclear maps.

Definition - Integral forms as the dual of the injective tensor product

See also: Injective tensor product and Projective tensor product

Let X and Y be locally convex TVSs, let $X\otimes _{\pi }Y$ denote the projective tensor product, $X{\widehat {\otimes }}_{\pi }Y$ denote its completion, let $X\otimes _{\epsilon }Y$ denote the injective tensor product, and $X{\widehat {\otimes }}_{\epsilon }Y$ denote its completion. Suppose that $\operatorname {In} :X\otimes _{\epsilon }Y\to X{\widehat {\otimes }}_{\epsilon }Y$ denotes the TVS-embedding of $X\otimes _{\epsilon }Y$ into its completion and let ${}^{t}\operatorname {In} :\left(X{\widehat {\otimes }}_{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }$ be its transpose, which is a vector space-isomorphism. This identifies the continuous dual space of $X\otimes _{\epsilon }Y$ as being identical to the continuous dual space of $X{\widehat {\otimes }}_{\epsilon }Y$.

Let $\operatorname {Id} :X\otimes _{\pi }Y\to X\otimes _{\epsilon }Y$ denote the identity map and ${}^{t}\operatorname {Id} :\left(X\otimes _{\epsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\pi }Y\right)_{b}^{\prime }$ denote its transpose, which is a continuous injection. Recall that $\left(X\otimes _{\pi }Y\right)^{\prime }$ is canonically identified with $B(X,Y)$, the space of continuous bilinear maps on $X\times Y$. In this way, the continuous dual space of $X\otimes _{\epsilon }Y$ can be canonically identified as a subvector space of $B(X,Y)$, denoted by $J(X,Y)$. The elements of $J(X,Y)$ are called integral (bilinear) forms on $X\times Y$. The following theorem justifies the word integral.

Theorem The dual J(X, Y) of $X{\widehat {\otimes }}_{\epsilon }Y$ consists of exactly those continuous bilinear forms v on $X\times Y$ that can be represented in the form of a map

$b\in B(X,Y)\mapsto v(b)=\int _{S\times T}b{\big \vert }_{S\times T}\left(x^{\prime },y^{\prime }\right)\operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)$

where S and T are some closed, equicontinuous subsets of $X_{\sigma }^{\prime }$ and $Y_{\sigma }^{\prime }$, respectively, and $\mu$ is a positive Radon measure on the compact set $S\times T$ with total mass $\leq 1$. Furthermore, if A is an equicontinuous subset of J(X, Y) then the elements $v\in A$ can be represented with $S\times T$ fixed and $\mu$ running through a norm bounded subset of the space of Radon measures on $S\times T$.

Integral linear maps

A continuous linear map $\kappa :X\to Y^{\prime }$ is called integral if its associated bilinear form is an integral bilinear form, where this form is defined by $(x,y)\in X\times Y\mapsto (\kappa x)(y)$. It follows that an integral map $\kappa :X\to Y^{\prime }$ is of the form:

$x\in X\mapsto \kappa (x)=\int _{S\times T}\left\langle x^{\prime },x\right\rangle y^{\prime }\operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)$

for suitable weakly closed and equicontinuous subsets S and T of $X^{\prime }$ and $Y^{\prime }$, respectively, and some positive Radon measure $\mu$ of total mass ≤ 1. The above integral is the weak integral, so the equality holds if and only if for every $y\in Y$, $\left\langle \kappa (x),y\right\rangle =\int _{S\times T}\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime },y\right\rangle \operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)$.

Given a linear map $\Lambda :X\to Y$, one can define a canonical bilinear form $B_{\Lambda }\in Bi\left(X,Y^{\prime }\right)$, called the associated bilinear form on $X\times Y^{\prime }$, by $B_{\Lambda }\left(x,y^{\prime }\right):=\left(y^{\prime }\circ \Lambda \right)\left(x\right)$. A continuous map $\Lambda :X\to Y$ is called integral if its associated bilinear form is an integral bilinear form. An integral map $\Lambda :X\to Y$ is of the form, for every $x\in X$ and $y^{\prime }\in Y^{\prime }$:

$\left\langle y^{\prime },\Lambda (x)\right\rangle =\int _{A^{\prime }\times B^{\prime \prime }}\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime \prime },y^{\prime }\right\rangle \operatorname {d} \mu \left(x^{\prime },y^{\prime \prime }\right)$

for suitable weakly closed and equicontinuous aubsets $A^{\prime }$ and $B^{\prime \prime }$ of $X^{\prime }$ and $Y^{\prime \prime }$, respectively, and some positive Radon measure $\mu$ of total mass $\leq 1$.

Relation to Hilbert spaces

The following result shows that integral maps "factor through" Hilbert spaces.

Proposition: Suppose that $u:X\to Y$ is an integral map between locally convex TVS with Y Hausdorff and complete. There exists a Hilbert space H and two continuous linear mappings $\alpha :X\to H$ and $\beta :H\to Y$ such that $u=\beta \circ \alpha$.

Furthermore, every integral operator between two Hilbert spaces is nuclear. Thus a continuous linear operator between two Hilbert spaces is nuclear if and only if it is integral.

Sufficient conditions

Suppose that $u:X\to Y$ is a continuous linear map between locally convex TVSs.

• Every nuclear map is integral.
  • An important partial converse is that every integral operator between two Hilbert spaces is nuclear.
• If $u:X\to Y$ is integral then so is its transpose ${}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }$.
  • Suppose that the transpose ${}^{t}u:Y_{b}^{\prime }\to X_{b}^{\prime }$ of the continuous linear map $u:X\to Y$ is integral. Then $u:X\to Y$ is integral if the canonical injections $\operatorname {In} _{X}:X\to X^{\prime \prime }$ (defined by $x\mapsto$ value at x) and $\operatorname {In} _{Y}:Y\to Y^{\prime \prime }$ are TVS-embeddings (which happens if, for instance, X and $Y_{b}^{\prime }$ are barreled or metrizable).
• If X and Y are normable spaces then $u:X\to Y$ is integral if and only if ${}^{t}u:Y^{\prime }\to X^{\prime }$ is integral.
• Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that $\alpha :A\to B$, $\beta :B\to C$, and $\gamma :C\to D$ are all continuous linear operators. If $\beta :B\to C$ is an integral operator then so is the composition $\gamma \circ \beta \circ \alpha :A\to D$.

Properties

• Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If $\alpha :A\to B$, $\beta :B\to C$, and $\gamma :C\to D$ are all integral linear maps then their composition $\gamma \circ \beta \circ \alpha :A\to D$ is nuclear.
  • Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection $u:X\to X$ cannot be an integral operator.

See also

• Auxiliary normed spaces
• Final topology
• Injective tensor product
• Nuclear operators
• Nuclear spaces
• Projective tensor product
• Topological tensor product

References

• Diestel, Joe (2008). The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society. ISBN 0-8218-4440-7. OCLC 185095773. {{cite book}}: Invalid |ref=harv (help)
• Dubinsky, Ed (1979). The structure of nuclear Fréchet spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09504-7. OCLC 5126156. {{cite book}}: Invalid |ref=harv (help)
• Grothendieck, Grothendieck (1966). Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society. ISBN 0-8218-1216-5. OCLC 1315788. {{cite book}}: Invalid |ref=harv (help)
• Husain, Taqdir (1978). Barrelledness in topological and ordered vector spaces. Berlin New York: Springer-Verlag. ISBN 3-540-09096-7. OCLC 4493665. {{cite book}}: Invalid |ref=harv (help)
• Khaleelulla, S. M. (1982). Counterexamples in topological vector spaces. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. {{cite book}}: Invalid |ref=harv (help)
• Nlend, H (1977). Bornologies and functional analysis : introductory course on the theory of duality topology-bornology and its use in functional analysis. Amsterdam New York New York: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier-North Holland. ISBN 0-7204-0712-5. OCLC 2798822. {{cite book}}: Invalid |ref=harv (help)
• Nlend, H (1981). Nuclear and conuclear spaces : introductory courses on nuclear and conuclear spaces in the light of the duality. Amsterdam New York New York, N.Y: North-Holland Pub. Co. Sole distributors for the U.S.A. and Canada, Elsevier North-Holland. ISBN 0-444-86207-2. OCLC 7553061. {{cite book}}: Invalid |ref=harv (help)
• Pietsch, Albrecht (1972). Nuclear locally convex spaces. Berlin,New York: Springer-Verlag. ISBN 0-387-05644-0. OCLC 539541. {{cite book}}: Invalid |ref=harv (help)
• Robertson, A. P. (1973). Topological vector spaces. Cambridge England: University Press. ISBN 0-521-29882-2. OCLC 589250. {{cite book}}: Invalid |ref=harv (help)
• Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC 48092184. {{cite book}}: Invalid |ref=harv (help)
• Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)
• Treves, François (2006). Topological vector spaces, distributions and kernels. Mineola, N.Y: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. {{cite book}}: Invalid |ref=harv (help)
• Wong (1979). Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag. ISBN 3-540-09513-6. OCLC 5126158. {{cite book}}: Invalid |ref=harv (help)

External links

• Nuclear space at ncatlab