Linearisation_of_vector-valued_functions.md

Linearisation of vector-valued functions

Working Groups: aa

Collaborators (MAT): kkruse

Description

It is a classical idea to represent vector-valued functions by continuous linear operators [3]. Let

$\mathcal{F}(\Omega)$

be a locally convex Hausdorff space (lcHs) of functions from a set

$\Omega$

to the field

$\mathbb{K}$

of real or complex numbers and

$E$

an lcHs over

$\mathbb{K}$

. Then Schwartz'

$\varepsilon$

-product of

$\mathcal{F}(\Omega)$

and

$E$

is defined as the space of continuous linear operators

$$\mathcal{F}(\Omega)\varepsilon E :=L_{e}(\mathcal{F}(\Omega)_{\kappa}',E).$$

Supposing that the point-evaluations

$\delta_{x}$

belong to the dual space

$\mathcal{F}(\Omega)'$

for all

$x\in\Omega$

and that there is an lcHs

$\mathcal{F}(\Omega,E)$

consisting of

$E$

-valued functions on

$\Omega$

which is the counterpart of

$\mathcal{F}(\Omega)$

, linearisation of

$\mathcal{F}(\Omega,E)$

means that the map

$$S\colon \mathcal{F}(\Omega)\varepsilon E \to \mathcal{F}(\Omega,E),\; u\longmapsto[x\mapsto u(\delta_{x})],$$

is a well-defined topological isomorphism.

In [11] we derive sufficient conditions on

$E$

and on the properties and structures of the functions and function spaces

$\mathcal{F}(\Omega)$

and

$\mathcal{F}(\Omega,E)$

such that the map

$S$

is a topological isomorphism. Once the isomorphism

$S$

is established, the famous approximation property of a space

$\mathcal{F}(\Omega)$

is equivalent to the property that every function in

$\mathcal{F}(\Omega,E)$

can be approximated by functions with values in finite dimensional subspaces of

$E$

for any lcHs

$E$

, which we investigate in [4] for weighted spaces of

$\mathcal{C}^{k}$

-smooth functions. In [7] we study the stronger property that

$\mathcal{F}(\Omega)$

is nuclear in the case of weighted

$\mathcal{C}^{\infty}$

-smooth functions.

Nuclearity can be used to transfer the surjectivity of a continuous linear map

$T\colon \mathcal{F}(\Omega)\to\mathcal{F}(\Omega)$

to the

$\varepsilon$

-product

$T\varepsilon \operatorname{id}_{E}\colon \mathcal{F}(\Omega)\varepsilon E\to\mathcal{F}(\Omega)\varepsilon E$

for Fréchet spaces

$\mathcal{F}(\Omega)$

and

$E$

by Grothendieck's classical tensor product theory [1]. In combination with the topological isomorphism

$S$

this implies that the surjectivity of a continuous linear partial differential operator can be transfered from the scalar-valued to the vector-valued case, which we study for the Cauchy-Riemann operator

$T=\overline{\partial}$

on weighted spaces of

$\mathcal{C}^{\infty}$

-smooth functions in [2,5,6,8] even for

$E$

beyond the class of Fréchet spaces.

Another application of the topological isomorphism

S

lies in lifting series representations from scalar-valued to

E

-valued functions [9], for instance power series representations of holomorphic functions [10], and the extension of

E

-valued functions via weak extensions [3,12], i.e. to answer the question:

Let

\Lambda

be a subset of

\Omega

and

G

a linear subspace of

E'

. Let

f\colon \Lambda\to E

be such that for every

e'\in G

, the function

e'\circ f\colon\Lambda\to \mathbb{K}

has an extension in

\mathcal{F}(\Omega)

. When is there an extension F\in\mathcal{F}(\Omega,E) of f, i.e. F_{\mid \Lambda} = f ?

References

[1] A. Grothendieck. Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16. AMS, Providence, RI, 1966. doi: 10.1090/memo/0016.

[2] K. Kruse. Surjectivity of the \overline{\partial}-operator between spaces of weighted smooth vector-valued functions, 2018. arXiv:1810.05069.

[3] K. Kruse. Extension of vector-valued functions and sequence space representation, 2019. arXiv:1808.05182.

[4] K. Kruse. The approximation property for weighted spaces of differentiable functions. In M. Kosek, editor, Function Spaces XII, volume 119 of Banach Center Publ., 233--258, Inst. Math., Polish Acad. Sci., Warszawa, 2019. doi: 10.4064/bc119-14.

[5] K. Kruse. The Cauchy-Riemann operator on smooth Fréchet-valued functions with exponential growth on rotated strips. PAMM, 19(1):1--2, 2019. doi: 10.1002/pamm.201900141.

[6] K. Kruse. The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes, 2019. arXiv:1901.02093.

[7] K. Kruse. On the nuclearity of weighted spaces of smooth functions. Ann. Polon. Math., 124(2):173--196, 2020. doi: 10.4064/ap190728-17-11.

[8] K. Kruse. Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 114(141):1--24, 2020. doi: 10.1007/s13398-020-00863-x.

[9] K. Kruse. Series representations in spaces of vector-valued functions via Schauder decompositions. Math. Nachr., 294(2):354--376, 2021. doi: 10.1002/mana.201900172.

[10] K. Kruse. Vector-valued holomorphic functions in several variables. Funct. Approx. Comment. Math., 63(2):247--275, 2020. doi: 10.7169/facm/1861.

[11] K. Kruse. Weighted spaces of vector-valued functions and the \varepsilon-product, Banach J. Math. Anal., 14(4):1509--1531, 2020. doi: 10.1007/s43037-020-00072-z.

[12] K. Kruse. Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order, 2021. arXiv:1910.01952.

[13] L. Schwartz. Espaces de fonctions différentiables à valeurs vectorielles. J. Analyse Math., 4:88--148, 1955. doi: 10.1007/BF02787718.