Wold-type extension for N-tuples of commuting contractions

Marek Kosiek; Alfredo Octavio

Studia Mathematica (1999)

  • Volume: 137, Issue: 1, page 81-91
  • ISSN: 0039-3223

Abstract

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Let (T1,…,TN) be an N-tuple of commuting contractions on a separable, complex, infinite-dimensional Hilbert space ℋ. We obtain the existence of a commuting N-tuple (V1,…,VN) of contractions on a superspace K of ℋ such that each V j extends T j , j=1,…,N, and the N-tuple (V1,…,VN) has a decomposition similar to the Wold-von Neumann decomposition for coisometries (although the V j need not be coisometries). As an application, we obtain a new proof of a result of Słociński (see [9])

How to cite

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Kosiek, Marek, and Octavio, Alfredo. "Wold-type extension for N-tuples of commuting contractions." Studia Mathematica 137.1 (1999): 81-91. <http://eudml.org/doc/216676>.

@article{Kosiek1999,
abstract = {Let (T1,…,TN) be an N-tuple of commuting contractions on a separable, complex, infinite-dimensional Hilbert space ℋ. We obtain the existence of a commuting N-tuple (V1,…,VN) of contractions on a superspace K of ℋ such that each $V_j$ extends $T_j$, j=1,…,N, and the N-tuple (V1,…,VN) has a decomposition similar to the Wold-von Neumann decomposition for coisometries (although the $V_j$ need not be coisometries). As an application, we obtain a new proof of a result of Słociński (see [9])},
author = {Kosiek, Marek, Octavio, Alfredo},
journal = {Studia Mathematica},
keywords = {contractions; dilations; extensions; alternative dilation theory; Wold-type decomposition},
language = {eng},
number = {1},
pages = {81-91},
title = {Wold-type extension for N-tuples of commuting contractions},
url = {http://eudml.org/doc/216676},
volume = {137},
year = {1999},
}

TY - JOUR
AU - Kosiek, Marek
AU - Octavio, Alfredo
TI - Wold-type extension for N-tuples of commuting contractions
JO - Studia Mathematica
PY - 1999
VL - 137
IS - 1
SP - 81
EP - 91
AB - Let (T1,…,TN) be an N-tuple of commuting contractions on a separable, complex, infinite-dimensional Hilbert space ℋ. We obtain the existence of a commuting N-tuple (V1,…,VN) of contractions on a superspace K of ℋ such that each $V_j$ extends $T_j$, j=1,…,N, and the N-tuple (V1,…,VN) has a decomposition similar to the Wold-von Neumann decomposition for coisometries (although the $V_j$ need not be coisometries). As an application, we obtain a new proof of a result of Słociński (see [9])
LA - eng
KW - contractions; dilations; extensions; alternative dilation theory; Wold-type decomposition
UR - http://eudml.org/doc/216676
ER -

References

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  1. [1] H. Bercovici, Notes on invariant subspaces, Bull. Amer. Math. Soc. 23 (1990), 1-36. Zbl0727.47001
  2. [2] H. Bercovici, Commuting power-bounded operators, Acta Sci. Math. (Szeged) 57 (1993), 55-64. Zbl0819.47017
  3. [3] H. Bercovici, C. Foiaş, and C. Pearcy, Dual Algebras with Applications to Invariant Subspaces and Dilation Theory, CBMS Regional Conf. Ser. in Math. 56, Amer. Math. Soc., Providence, RI, 1985. Zbl0569.47007
  4. [4] L. Kérchy, Unitary asymptotes of Hilbert space operators, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 191-201. Zbl0807.47005
  5. [5] M. Kosiek and A. Octavio, On common invariant subspaces for N-tuples of commuting contractions with rich spectrum, to appear. Zbl1061.47007
  6. [6] M. Kosiek, A. Octavio, and M. Ptak, On the reflexivity of pairs of contractions, Proc. Amer. Math. Soc. 123 (1995), 1229-1236. Zbl0836.47006
  7. [7] M. Kosiek and M. Ptak, Reflexivity of n-tuples of contractions with rich joint left essential spectrum, Integral Equations Operator Theory 13 (1990), 395-420. Zbl0743.47032
  8. [8] A. Octavio, Coisometric extension and functional calculus for pairs of contractions, J. Operator Theory 31 (1994), 67-82. Zbl0872.47009
  9. [9] M. Słociński, On the Wold type decomposition of a pair of commuting isometries, Ann. Polon. Math. 37 (1980), 255-262. Zbl0485.47018
  10. [10] B. Sz.-Nagy and C. Foiaş, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970. Zbl0201.45003

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