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The role of the algebraic structure in Wold-type decomposition

  • G. A. Bagheri Bardi ORCID logo , Zbigniew Burdak ORCID logo EMAIL logo and Akram Elyaspour
From the journal Forum Mathematicum

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.

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.2017.10.024Search 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.2019.08.022Search in Google Scholar

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

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

[5] Z. Burdak, On a decomposition for pairs of commuting contractions, Studia Math. 181 (2007), no. 1, 33–45. 10.4064/sm181-1-3Search 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/s00025-021-01415-4Search 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.2012.12.007Search 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/S0002-9939-1994-1189538-7Search 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.1970.19.19054Search in Google Scholar

[10] I. Kaplansky, Rings of Operators, W. A. Benjamin, New York, 1968. Search 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/S0002-9939-04-07331-9Search 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-37-3-255-262Search 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. Search in Google Scholar

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

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