Integral linear operator

Short description: Mathematical function

An integral bilinear form is a bilinear functional that belongs to the continuous dual space of [math]\displaystyle{ X \widehat{\otimes}_{\epsilon} Y }[/math], 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

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

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

Theorem — The dual J(X, Y) of [math]\displaystyle{ X \widehat{\otimes}_{\epsilon} Y }[/math] consists of exactly those continuous bilinear forms c on [math]\displaystyle{ X \times Y }[/math] that can be represented in the form of a map

[math]\displaystyle{ b \in B(X, Y) \mapsto v(b) = \int_{S \times T} b\big\vert_{S \times T} \left( x', y' \right) \mathrm{d} \mu\!\left( x', y' \right) }[/math]

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

Integral linear maps

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

[math]\displaystyle{ x \in X \mapsto \kappa(x) = \int_{S \times T} \left\langle x', x \right\rangle y' \mathrm{d} \mu\! \left( x', y' \right) }[/math]

for suitable weakly closed and equicontinuous subsets S and T of [math]\displaystyle{ X' }[/math] and [math]\displaystyle{ Y' }[/math], respectively, and some positive Radon measure [math]\displaystyle{ \mu }[/math] of total mass ≤ 1. The above integral is the weak integral, so the equality holds if and only if for every [math]\displaystyle{ y \in Y }[/math], [math]\displaystyle{ \left\langle \kappa(x), y \right\rangle = \int_{S \times T} \left\langle x', x \right\rangle \left\langle y', y \right\rangle \mathrm{d} \mu\! \left( x', y' \right) }[/math].

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

[math]\displaystyle{ \left\langle y', \Lambda(x) \right\rangle = \int_{A' \times B''} \left\langle x', x \right\rangle \left\langle y'', y' \right\rangle \mathrm{d} \mu\! \left( x', y'' \right) }[/math]

for suitable weakly closed and equicontinuous aubsets [math]\displaystyle{ A' }[/math] and [math]\displaystyle{ B'' }[/math] of [math]\displaystyle{ X' }[/math] and [math]\displaystyle{ Y'' }[/math], respectively, and some positive Radon measure [math]\displaystyle{ \mu }[/math] of total mass [math]\displaystyle{ \leq 1 }[/math].

Relation to Hilbert spaces

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

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

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

Every nuclear map is integral. An important partial converse is that every integral operator between two Hilbert spaces is nuclear.

Suppose that A, B, C, and D are Hausdorff locally convex TVSs and that [math]\displaystyle{ \alpha : A \to B }[/math], [math]\displaystyle{ \beta : B \to C }[/math], and [math]\displaystyle{ \gamma: C \to D }[/math] are all continuous linear operators. If [math]\displaystyle{ \beta : B \to C }[/math] is an integral operator then so is the composition [math]\displaystyle{ \gamma \circ \beta \circ \alpha : A \to D }[/math].

If [math]\displaystyle{ u : X \to Y }[/math] is a continuous linear operator between two normed space then [math]\displaystyle{ u : X \to Y }[/math] is integral if and only if [math]\displaystyle{ {}^{t}u : Y' \to X' }[/math] is integral.

Suppose that [math]\displaystyle{ u : X \to Y }[/math] is a continuous linear map between locally convex TVSs. If [math]\displaystyle{ u : X \to Y }[/math] is integral then so is its transpose [math]\displaystyle{ {}^{t}u : Y^{\prime}_b \to X^{\prime}_b }[/math]. Now suppose that the transpose [math]\displaystyle{ {}^{t}u : Y^{\prime}_b \to X^{\prime}_b }[/math] of the continuous linear map [math]\displaystyle{ u : X \to Y }[/math] is integral. Then [math]\displaystyle{ u : X \to Y }[/math] is integral if the canonical injections [math]\displaystyle{ \operatorname{In}_X : X \to X'' }[/math] (defined by [math]\displaystyle{ x \mapsto }[/math] value at x) and [math]\displaystyle{ \operatorname{In}_Y : Y \to Y'' }[/math] are TVS-embeddings (which happens if, for instance, [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y^{\prime}_b }[/math] are barreled or metrizable).

Properties

Suppose that A, B, C, and D are Hausdorff locally convex TVSs with B and D complete. If [math]\displaystyle{ \alpha : A \to B }[/math], [math]\displaystyle{ \beta : B \to C }[/math], and [math]\displaystyle{ \gamma: C \to D }[/math] are all integral linear maps then their composition [math]\displaystyle{ \gamma \circ \beta \circ \alpha : A \to D }[/math] is nuclear. Thus, in particular, if X is an infinite-dimensional Fréchet space then a continuous linear surjection [math]\displaystyle{ u : X \to X }[/math] 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

Bibliography

• Template:Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited
• Template:Dubinsky The Structure of Nuclear Fréchet Spaces
• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
• Husain, Taqdir; Khaleelulla, S. M. (1978). Barrelledness in Topological and Ordered Vector Spaces. Lecture Notes in Mathematics. 692. Berlin, New York, Heidelberg: Springer-Verlag. ISBN 978-3-540-09096-0. OCLC 4493665. 
• Khaleelulla, S. M. (July 1, 1982). written at Berlin Heidelberg. Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 936. Berlin New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. 
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Template:Hogbe-Nlend Bornologies and Functional Analysis
• Template:Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces
• Template:Pietsch Nuclear Locally Convex Spaces
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. 
• Rudin, Walter (January 1, 1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. https://archive.org/details/functionalanalys00rudi. 
• Template:Ryan Introduction to Tensor Products of Banach Spaces
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. 
• Template:Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products

External links

• Nuclear space at ncatlab