TUHH / Institut für Mathematik / Forschungsgebiete / Linearisation of vector-valued functions

Linearisation of vector-valued functions

Description

It is a classical idea to represent vector-valued functions by continuous linear operators [13]. 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 [10] 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 [12], for instance power series representations of holomorphic functions [9], and the extension of $$E$$-valued functions via weak extensions [3,11], 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, 1955. 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 weighted spaces of 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. Vector-valued holomorphic functions in several variables. Funct. Approx. Comment. Math., 63(2):247?275, 2020. doi: 10.7169/facm/1861.

[10] 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.

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

[12] 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.

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