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Monotone operator functions, gaps and power moment problem

Osaka, Hiroyuki; Silvestrov, Sergei LU and Tomiyama, Jun (2007) In Mathematica Scandinavica 100(1). p.161-183
Abstract
The article is devoted to investigation of the classes of functions belonging to the gaps between classes Pn+1 (I) and P, (I) of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps P-n(I) Pn+1 (I) for bounded intervals L We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from... (More)
The article is devoted to investigation of the classes of functions belonging to the gaps between classes Pn+1 (I) and P, (I) of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps P-n(I) Pn+1 (I) for bounded intervals L We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a proof of the Lowner theorem on integral representation of operator monotone functions. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematica Scandinavica
volume
100
issue
1
pages
161 - 183
publisher
Matematisk Institut
external identifiers
  • wos:000247570100006
  • scopus:34248548897
ISSN
0025-5521
language
English
LU publication?
yes
id
7f371a74-8e52-464a-be6e-8ec778f9b534 (old id 647671)
alternative location
http://www.mscand.dk/article.php?id=3009
date added to LUP
2007-12-17 08:50:17
date last changed
2017-03-12 04:16:07
@article{7f371a74-8e52-464a-be6e-8ec778f9b534,
  abstract     = {The article is devoted to investigation of the classes of functions belonging to the gaps between classes Pn+1 (I) and P, (I) of matrix monotone functions for full matrix algebras of successive dimensions. In this paper we address the problem of characterizing polynomials belonging to the gaps P-n(I) Pn+1 (I) for bounded intervals L We show that solution of this problem is closely linked to solution of truncated moment problems, Hankel matrices and Hankel extensions. Namely, we show that using the solutions to truncated moment problems we can construct continuum many polynomials in the gaps. We also provide via several examples some first insights into the further problem of description of polynomials in the gaps that are not coming from the truncated moment problem. Also, in this article, we deepen further in another way into the structure of the classes of matrix monotone functions and of the gaps between them by considering the problem of position in the gaps of certain interesting subclasses of matrix monotone functions that appeared in connection to interpolation of spaces and in a proof of the Lowner theorem on integral representation of operator monotone functions.},
  author       = {Osaka, Hiroyuki and Silvestrov, Sergei and Tomiyama, Jun},
  issn         = {0025-5521},
  language     = {eng},
  number       = {1},
  pages        = {161--183},
  publisher    = {Matematisk Institut},
  series       = {Mathematica Scandinavica},
  title        = {Monotone operator functions, gaps and power moment problem},
  volume       = {100},
  year         = {2007},
}