Startseite The role of the algebraic structure in Wold-type decomposition
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

The role of the algebraic structure in Wold-type decomposition

  • G. A. Bagheri Bardi ORCID logo , Zbigniew Burdak ORCID logo EMAIL logo und Akram Elyaspour
Veröffentlicht/Copyright: 30. Juni 2021
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 33 Heft 4

Abstract

In recent works [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗-rings, Linear Algebra Appl. 539 2018, 117–133] and [G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 2019, 102–118], the algebraic analogues of the three major decomposition theorems of Wold, Nagy–Foiaş–Langer and Halmos–Wallen were established in the larger category of Baer *-rings. The results have their versions for commuting pairs in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. In this work, ignoring topological structures, we give an algebraic approach to obtain them in Baer *-rings.

Keywords: compression; contraction; partial isometry; power partial isometry; shift; truncated shift
MSC 2010: 47A05; 47A15; 47A45

Communicated by Siegfried Echterhoff

Funding statement: The second author’s research was supported by the Ministry of Science and Higher Education of the Republic of Poland.

References

[1] G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, Wold-type decompositions in Baer ∗-rings, Linear Algebra Appl. 539 (2018), 117–133. 10.1016/j.laa.2017.10.024Suche in Google Scholar

[2] G. A. Bagheri-Bardi, A. Elyaspour and G. H. Esslamzadeh, The role of algebraic structure in the invariant subspace theory, Linear Algebra Appl. 583 (2019), 102–118. 10.1016/j.laa.2019.08.022Suche in Google Scholar

[3] S. K. Berberian, Baer *-Rings, Springer, Berlin, 1972. 10.1007/978-3-642-15071-5Suche in Google Scholar

[4] S. K. Berberian, Baer Rings and Baer *-Rings, Springer, Berlin, 2003. Suche in Google Scholar

[5] Z. Burdak, On a decomposition for pairs of commuting contractions, Studia Math. 181 (2007), no. 1, 33–45. 10.4064/sm181-1-3Suche in Google Scholar

[6] Z. Burdak, M. Kosiek, P. Pagacz and M. Sł ociński, A unified approach to the decomposition theorems in Baer *-rings, Results Math. 76 (2021), no. 3, Article ID 131. 10.1007/s00025-021-01415-4Suche in Google Scholar

[7] Z. Burdak, M. Kosiek and M. Sł ociński, The canonical Wold decomposition of commuting isometries with finite dimensional wandering spaces, Bull. Sci. Math. 137 (2013), no. 5, 653–658. 10.1016/j.bulsci.2012.12.007Suche in Google Scholar

[8] X. Catepillán, M. Ptak and W. Szymański, Multiple canonical decompositions of families of operators and a model of quasinormal families, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1165–1172. 10.1090/S0002-9939-1994-1189538-7Suche in Google Scholar

[9] P. R. Halmos and L. J. Wallen, Powers of partial isometries, J. Math. Mech. 19 (1969/1970), 657–663. 10.1512/iumj.1970.19.19054Suche in Google Scholar

[10] I. Kaplansky, Rings of Operators, W. A. Benjamin, New York, 1968. Suche in Google Scholar

[11] D. Popovici, A Wold-type decomposition for commuting isometric pairs, Proc. Amer. Math. Soc. 132 (2004), no. 8, 2303–2314. 10.1090/S0002-9939-04-07331-9Suche in Google Scholar

[12] M. Sł ociński, On the Wold-type decomposition of a pair of commuting isometries, Ann. Polon. Math. 37 (1980), no. 3, 255–262. 10.4064/ap-37-3-255-262Suche in Google Scholar

[13] B. Sz. -Nagy and C. Foiaş, Sur les contractions de l’espace de Hilbert. IV, Acta Sci. Math. (Szeged) 21 (1960), 251–259. Suche in Google Scholar

[14] H. Wold, A Study in the Analysis of Stationary Time Series, Almqvist and Wiksell, Stockholm, 1954. Suche in Google Scholar

Received: 2020-12-28
Revised: 2021-05-18
Published Online: 2021-06-30
Published in Print: 2021-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Higher depth mock theta functions and q-hypergeometric series
  3. Topological and algebraic properties of universal groups for right-angled buildings
  4. On the socles of characteristically inert subgroups of Abelian p-groups
  5. Priestley duality for MV-algebras and beyond
  6. The cardinality of μM,D‐orthogonal exponentials for the planar four digits
  7. Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
  8. Ordered fields dense in their real closure and definable convex valuations
  9. Third Hankel determinants for two classes of analytic functions with real coefficients
  10. A common range problem for model spaces
  11. Generalized Ricci flow on nilpotent Lie groups
  12. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
  13. The role of the algebraic structure in Wold-type decomposition
  14. Incidences between Euclidean spaces over finite fields
  15. Cancellation in algebraic twisted sums on GL_m
https://doi.org/10.1515/forum-2020-0362
Schlagwörter für diesen Artikel
compression; contraction; partial isometry; power partial isometry; shift; truncated shift

Artikel in diesem Heft

  1. Frontmatter
  2. Higher depth mock theta functions and q-hypergeometric series
  3. Topological and algebraic properties of universal groups for right-angled buildings
  4. On the socles of characteristically inert subgroups of Abelian p-groups
  5. Priestley duality for MV-algebras and beyond
  6. The cardinality of μM,D‐orthogonal exponentials for the planar four digits
  7. Associated prime ideals of equivariant coinvariant algebras, Steenrod operations, and Krull’s Going Down Theorem
  8. Ordered fields dense in their real closure and definable convex valuations
  9. Third Hankel determinants for two classes of analytic functions with real coefficients
  10. A common range problem for model spaces
  11. Generalized Ricci flow on nilpotent Lie groups
  12. Endpoint estimates for a trilinear pseudo-differential operator with flag symbols
  13. The role of the algebraic structure in Wold-type decomposition
  14. Incidences between Euclidean spaces over finite fields
  15. Cancellation in algebraic twisted sums on GL_m
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 21.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2020-0362/html?lang=de
Button zum nach oben scrollen