Opuscula Math. 36, no. 6 (2016), 799-806
http://dx.doi.org/10.7494/OpMath.2016.36.6.799

 
Opuscula Mathematica

2-hyperreflexivity and hyporeflexivity of power partial isometries

Kamila Piwowarczyk
Marek Ptak

Abstract. Power partial isometries are not always hyperreflexive neither reflexive. In the present paper it will be shown that power partial isometries are always hyporeflexive and \(2\)-hyperreflexive.

Keywords: power partial isometry, reflexive subspace, hyperreflexive subspace, hyperreflexive operator, hyporeflexive algebra.

Mathematics Subject Classification: 47L80, 47L45, 47L05.

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  1. W.T. Arveson, Interpolation problems in nest algebras, J. Funct. Anal. 20, (1975), 208-233.
  2. W.T. Arveson, Ten lectures on operator algebras, CBMS Regional Conference Series 55, Amer. Math. Soc., Providence (1984).
  3. E.A. Azoff, \(k\)-reflexivity in finite dimensional subspaces, Duke Math. J. 40 (1973), 821-830.
  4. E.A. Azoff, C.K. Fong, F. Gilfeather, A reduction theory for non-self-adjoint operator algebras, Trans. Amer. Math. Soc. 224 (1976), 351-366.
  5. E.A. Azoff, W.S. Li, M. Mbekhta, M. Ptak, On consistent operators and reflexivity, Integr. Equ. Oper. Theory 71 (2011), 1-12.
  6. H. Bercovici, Operator theory and arytmetic in \(H^{\infty}\), Mathematical Surveys and Monographs, vol. 26, Amer. Math. Soc., 1988.
  7. H. Bercovici, C. Foias, C. Pearcy, Dual Algebras with Applications to Invariant Subspaces and Dilation Theory, CBMS Regional Conference Series 56, Amer. Math. Soc., Providence, 1985.
  8. L. Brickman, P.A. Fillmore, The invariant subspace lattice of a linear transformation, Canad. J. Math. 19 (1967), 810-822.
  9. J.B. Conway, A Course in Operator Theory, Amer. Math. Soc., Providence, 2000.
  10. K.R. Davidson, The distance to the analytic Toeplitz operators, Illinois J. Math. 31 (1987), 265-273.
  11. J.A. Deddens, P.A. Fillmore, Reflexive linear transformations, Linear Algebra Appl. 10 (1975), 89-93.
  12. D. Hadwin, A general view of reflexivity, Trans. Amer. Math. Soc. 344 (1994), 325-360.
  13. D. Hadwin, E.A. Nordgren, Subalgebras of reflexive algebras, J. Operator Theory 7 (1982), 3-23.
  14. P.R. Halmos, L.J. Wallen, Powers of Partial Isometries, J. Math. Mech. 19 (1970), 657-663.
  15. K. Kliś, M. Ptak, Quasinormal operators are hyperreflexive, Banach Center Publ. 67 (2005), 241-244.
  16. K. Kliś, M. Ptak, \(k\)-hyperreflexive subspaces, Houston J. Math. 32 (2006), 299-313.
  17. J. Kraus, D.R. Larson, Reflexivity and distance formulae, Proc. Lond. Math. Soc. 53 (1986), 340-356.
  18. A.I. Loginov, V.I. Shulman, Hereditary and intermediate reflexivity of \(W^*\) algebras, Math. USSR-Izv. 9 (1975), 1189-1201.
  19. W.E. Longstaff, On the operation Alg Lat in finite dimensions, Linear Algebra Appl. 27 (1979), 27-29.
  20. V. Müller, M. Ptak, Hyperreflexivity of finite-dimensional subspace, J. Funct. Anal. 218 (2005), 395-408.
  21. K. Piwowarczyk, M. Ptak, On the hyperreflexivity of power partial isometries, Linear Algebra Appl. 437 (2012), 623-629.
  22. H. Radjavi, P. Rosenthal, Invariant Subspaces, Springer-Verlag, Berlin-Heidberg-New York, 1973.
  23. S. Rosenoer, Distance estimates for von Neumann algebras, Proc. Amer. Math. Soc. 86 (1982) 2, 248-252.
  24. D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511-517.
  • Kamila Piwowarczyk
  • Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland
  • Marek Ptak
  • Department of Applied Mathematics, University of Agriculture, ul. Balicka 253c, 30-198 Kraków, Poland
  • Communicated by P.A. Cojuhari.
  • Received: 2015-12-11.
  • Accepted: 2016-05-04.
  • Published online: 2016-10-29.
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Cite this article as:
Kamila Piwowarczyk, Marek Ptak, 2-hyperreflexivity and hyporeflexivity of power partial isometries, Opuscula Math. 36, no. 6 (2016), 799-806, http://dx.doi.org/10.7494/OpMath.2016.36.6.799

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