Linearisation of vector-valued functions

Working Groups: Lehrstuhl Angewandte Analysis

Collaborators (MAT): Dr. Karsten Kruse

Description

It is a classical idea to represent vector-valued functions by continuous linear operators [15]. 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 [7,13,14] 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 for all complete (or Banach) \(E\), 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 complete (or Banach) \(E\), which we investigate in [2] for weighted spaces of \(\mathcal{C}^{k}\)-smooth functions. In [4] we study the stronger property that \(\mathcal{F}(\Omega)\) is nuclear in the case of weighted spaces of \(\mathcal{C}^{\infty}\)-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 [3,5,10,12] 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 [9,14], for instance power series representations of holomorphic functions [6], and the extension of \(E\)-valued functions via weak extensions [8,11,13,14], 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. 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.
[3] 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.
[4] 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.
[5] 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(3):1-24, 2020. doi: 10.1007/s13398-020-00863-x.
[6] K. Kruse. Vector-valued holomorphic functions in several variables. Funct. Approx. Comment. Math., 63(2):247-275, 2020. doi: 10.7169/facm/1861.
[7] 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.
[8] K. Kruse. Extension of vector-valued functions and sequence space representation, 2021. arXiv:1808.05182 (to appear in Bull. Belg. Math. Soc. Simon Stevin).
[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. Surjectivity of the \(\overline{\partial}\)-operator between weighted spaces of smooth vector-valued functions. Complex Var. Elliptic Equ., 67(11):2676-2707, 2022. doi: 10.1080/17476933.2021.1945587.
[11] K. Kruse. Extension of vector-valued functions and weak-strong principles for differentiable functions of finite order. Annals of Functional Analysis, 13(1):1-26, 2022. doi: 10.1007/s43034-021-00154-5.
[12] K. Kruse. The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes. Collect. Math., 74(1):81-112, 2023. doi: 10.1007/s13348-021-00337-2.
[13] K. Kruse. Linearisation of weak vector-valued functions, 2022. arXiv:2207.04681.
[14] K. Kruse. On vector-valued functions and the \(\varepsilon\)-product. Habilitation thesis, Technische Universität Hamburg (to appear).
[15] L. Schwartz. Espaces de fonctions différentiables à valeurs vectorielles. J. Analyse Math., 4:88-148, 1955. doi: 10.1007/BF02787718.