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Weighted Hardy Spaces Over the Unit Ball: The Freely Noncommutative and Commutative Settings Read chapter Read first chapter

2024 | OriginalPaper | Chapter

Composition in Reproducing Kernel Hilbert Spaces à Rebours

Author : Franciszek Hugon Szafraniec

Published in: Operator and Matrix Theory, Function Spaces, and Applications

Publisher: Springer Nature Switzerland

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Abstract

This is a recapitulation of the talk (under the same title) designated for IWOTA2022 Session 21 “Truncated and Full Moment Problems, and Applications”.
Roughly, the content of this paper provides us with building up the univocal treatment of extending positive definiteness forward and backward. The paper is completed with exposing vital facts (in two Appendices) concerning reproducing kernel Hilbert spaces, the main ingredient of it.

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previous chapter Geometric Structures Related to a -Algebra
next chapter Kippenhahn’s Construction Revisited
Appendix
Available only for authorised users
Footnotes
1
For equivalent definitions of the boundedness condition look at [12] or [11].
 
2
Here the aforesaid composition is encoded.
 
3
This is a RKHS due to the criterion involving evaluation functionals. Here \( \operatorname {\mathrm {clolin}}\) stands for the “closed linear span of ...”. Though it is not commonly in use it is tempting to take advantage of it due to its suggestiveness.
 
4
In addition to [14] let us mention one more reference [22] related to that.
 
5
This is the educational material, just for reader’s convenience.
 
6
In other words, \(\mathcal D=\mathcal D_{K}\).
 
7
Notice completeness of \((\varPhi _\alpha )_{{\alpha }\in \mathrm {A}}\) appears a posteriori.
 
8
Putting
$$\displaystyle \begin{aligned} {} V\colon \mathcal E\ni f\to K_{1,f}\in\mathcal H, \end{aligned} $$
(14)
we obtain an anti-linear mapping, which in case of non-degenerate kernel is an injection and in case of isometric kernel is just an isometry.
 
9
Cf. (10) for the general set-up. Index u on the left hand side refers to the operation described in (10).
 
10
If \(\mathcal D\) is a dense subspace \(\mathcal H\) and \(\mathcal D\) is contained in the domain \(\mathcal D({A})\) of an operator A, then \(A^{\scriptscriptstyle {\#}}\)\(A^{\scriptscriptstyle {\#}}{\stackrel {{\scriptscriptstyle {\textsf {def}}}}{=}} A^{*}|{ }_{\mathcal D}\), as long \(\mathcal D\subset \mathcal D({A^{*}})\). \(\mathcal L^{\scriptscriptstyle {\#}}(\mathcal D)\)\(\mathcal L^{\scriptscriptstyle {\#}}(\mathcal D)\) is a set of all operators A for which \(A^{\scriptscriptstyle {\#}}\) exists; it is an algebra, and operation \({ }^{\scriptscriptstyle {\#}}\) is an involution of this algebra.
 
11
cf. [17, Lemma 2], [9], [12, Proposition 1], [11] and other references therein.
 
12
Here is the right moment to refer to [5, Chapter 3], [3, p. 155] or [8, p. 59].
 
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Metadata
Title
Composition in Reproducing Kernel Hilbert Spaces à Rebours
Author
Franciszek Hugon Szafraniec
Copyright Year
2024
Book
Operator and Matrix Theory, Function Spaces, and Applications

Print ISBN: 978-3-031-50612-3

Electronic ISBN: 978-3-031-50613-0

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-50613-0_17

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