Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
2.0
Journals
Axioms
Special Issues
Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications
share announcement

Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (20 November 2016) | Viewed by 15293

Special Issue Editor

Prof. Dr. Palle E.T. Jorgensen

E-Mail Website
Guest Editor
Department of Mathematics, 14 MLH, The University of Iowa, Iowa City, IA 52242-1419, USA
Interests: mathematical physics; Euclidean field theory; reflection positivity; representation theory; operators in Hilbert space; harmonic analysis; fractals; wavelets; stochastic processes; financial mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Special Issue aims to bring together a variety of topics in infinite-dimensional analysis, as they relate to operator theory in a general setting, encompassing both multivariable operator theory, and representation theory, representations of groups, and representations of algebras that arise in applications.

Infinite-dimensional analysis is a multi-faceted area that covers a host of mathematical topics, ranging from pure to applied, from harmonic to numerical, and from partial differential equations (PDE) to measure theory. Within the applied areas, the range certainly covers topics from physics, statistics, and engineering. What these themes have in common is that they all involve infinite-dimensional spaces, linear or non-linear. Within statistics and stochastic processes, we study measures of function spaces. In white noise analysis, we study measures of tempered distributions. An especially successful new direction in stochastic processes, which involves multivariable operator theory and non-commutativity, is free probability, as initiated by Dan Voiculescu.

In applications of non-commutative geometry, in physics, quantum mechanics, and quantum information theory (QIT), we study operators in Hilbert spaces. While the setting in QIT is often finite-dimensional, the framework for quantum theory is inherently infinite-dimensional Hilbert space.

The study of more than one operator at a time (systems of linear operators) is a more recent vintage, and it is motivated by geometry, spectral theory, operator algebras, representations of groups, and by applications. One trend in the multivariable case, starting with the work of Kadison-Singer, emphasizes analogues of analyticity. This has, since, been followed up vigorously with some remarkable successes in complex and algebraic geometry.

Early researchers considered either n-tuples of operators or representations of algebras. Many researchers have now adopted the language of Hilbert modules.

More recent, is the solution by Adam Marcus, Dan Spielman, and Nikhil Srivastava to the five-decade-old Kadison-Singer problem (KS). While the solution to KS entails deep tools from combinatorics, the framework is multi-variable operator theory.

Prof. Dr. Palle E.T. Jorgensen
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • infinite-dimensional analysis
  • operator theory
  • representations of groups
  • operator algebras
  • free probability
  • Kadison-Singer
  • white noise analysis
  • measures on function spaces
  • non-commutative harmonic analysis

Published Papers (3 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

455 KiB  
Article
Fundamental Results for Pseudo-Differential Operators of Type 1, 1
by Jon Johnsen
Axioms 2016, 5(2), 13; https://doi.org/10.3390/axioms5020013 - 19 May 2016
Cited by 1 | Viewed by 3919
Abstract
This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type 1 , 1 in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. [...] Read more.
This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type 1 , 1 in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type 1 , 1 -operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander’s twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type 1 , 1 -operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type 1 , 1 -operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type 1 , 1 -context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions. Full article
(This article belongs to the Special Issue Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications)
491 KiB  
Article
Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus
by Palle Jorgensen and Feng Tian
Axioms 2016, 5(2), 12; https://doi.org/10.3390/axioms5020012 - 17 May 2016
Cited by 4 | Viewed by 5789
Abstract
We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations [...] Read more.
We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest. Full article
(This article belongs to the Special Issue Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications)
►▼ Show Figures

Figure 1

431 KiB  
Article
Modular Nuclearity: A Generally Covariant Perspective
by Gandalf Lechner and Ko Sanders
Axioms 2016, 5(1), 5; https://doi.org/10.3390/axioms5010005 - 29 Jan 2016
Cited by 6 | Viewed by 4853
Abstract
A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use a [...] Read more.
A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use a modular ℓp -condition, which is an extension of a strengthened modular nuclearity condition to generally covariant theories. The modular nuclearity condition was previously introduced in Minkowski space, where it played an important role in constructive two dimensional algebraic QFT’s. We show that our generally covariant extension of this condition makes sense for a vast range of theories, and that it behaves well under causal propagation and taking mixtures. In addition we show that our modular ℓp -condition holds for every quasi-free Hadamard state of a free scalar quantum field (regardless of mass or scalar curvature coupling). However, our condition is not equivalent to the Hadamard condition. Full article
(This article belongs to the Special Issue Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-3
Back to TopTop