Abstract and Applied Analysis

Subnormal Weighted Shifts on Directed Trees and Composition Operators in $L^2$>-Spaces with Nondensely Defined Powers

Piotr Budzyński, Piotr Dymek, Zenon Jan Jabłoński, and Jan Stochel

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Abstract

It is shown that for every positive integer n there exists a subnormal weighted shift on a directed tree (with or without root) whose nth power is densely defined while its ( n + 1 )th power is not. As a consequence, for every positive integer n there exists a nonsymmetric subnormal composition operator C in an L 2-space over a σ-finite measure space such that Cn is densely defined and C n + 1 is not.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 791817, 6 pages.

Dates
First available in Project Euclid: 7 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412687021

Digital Object Identifier
doi:10.1155/2014/791817

Mathematical Reviews number (MathSciNet)
MR3173291

Zentralblatt MATH identifier
1259.35155

Citation

Budzyński, Piotr; Dymek, Piotr; Jabłoński, Zenon Jan; Stochel, Jan. Subnormal Weighted Shifts on Directed Trees and Composition Operators in $L^2$>-Spaces with Nondensely Defined Powers. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 791817, 6 pages. doi:10.1155/2014/791817. https://projecteuclid.org/euclid.aaa/1412687021


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References

  • M. Naimark, “On the square of a closed symmetric operator,” Doklady Akademii Nauk SSSR, vol. 26, pp. 866–870, 1940.
  • P. R. Chernoff, “A semibounded closed symmetric operator whose square has trivial domain,” Proceedings of the American Mathematical Society, vol. 89, no. 2, pp. 289–290, 1983.
  • K. Schmüdgen, “On domains of powers of closed symmetric operators,” Journal of Operator Theory, vol. 9, no. 1, pp. 53–75, 1983.
  • N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, vol. 2, Dover, New York, NY, USA, 1993.
  • A. Brown, “On a class of operators,” Proceedings of the American Mathematical Society, vol. 4, pp. 723–728, 1953.
  • J. Stochel and F. H. Szafraniec, “On normal extensions of unbounded operators. II,” Acta Universitatis Szegediensis, vol. 53, no. 1-2, pp. 153–177, 1989.
  • Z. J. Jabłoński, I. B. Jung, and J. Stochel, “Normal extensions escape from the class of weighted shifts on directed trees,” Complex Analysis and Operator Theory, vol. 7, no. 2, pp. 409–419, 2013.
  • E. A. Coddington, “Formally normal operators having no nor-mal extensions,” Canadian Journal of Mathematics, vol. 17, pp. 1030–1040, 1965.
  • P. Budzyński, Z. J. Jabłonski, I. B. Jung, and J. Stochel, “Unbounded subnormal composition operators in \emphL$^{2}$-spaces,” http://arxiv.org/abs/1303.6486.
  • P. Budzyński, Z. J. Jabłoński, I. B. Jung, and J. Stochel, “On unbounded composition operators in \emphL$^{2}$-spaces,” Annali di Matematica Pura ed Applicata, 2012.
  • K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht, The Netherlands, 2012.
  • Z. J. Jabłoński, I. B. Jung, and J. Stochel, “Weighted shifts on directed trees,” Memoirs of the American Mathematical Society, vol. 216, no. 1017, p. viii+107, 2012.
  • Z. J. Jabłoński, I. B. Jung, and J. Stochel, “A non-hyponormal operator generating Stieltjes moment sequences,” Journal of Functional Analysis, vol. 262, no. 9, pp. 3946–3980, 2012.
  • M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Reidel, Dordrecht, The Netherlands, 1987.
  • J. Weidmann, Linear Operators in Hilbert Spaces, vol. 68, Springer, New York, NY, USA, 1980.
  • J. B. Conway, The Theory of Subnormal Operators, vol. 36, American Mathematical Society, Providence, RI, USA, 1991.
  • J. Stochel and F. H. Szafraniec, “On normal extensions of unbounded operators. I,” Journal of Operator Theory, vol. 14, no. 1, pp. 31–55, 1985.
  • J. Stochel and F. H. Szafraniec, “On normal extensions of unbounded operators. III. Spectral properties,” Kyoto University. Research Institute for Mathematical Sciences. Publications, vol. 25, no. 1, pp. 105–139, 1989.
  • J. Stochel and F. H. Szafraniec, “The complex moment problem and subnormality: a polar decomposition approach,” Journal of Functional Analysis, vol. 159, no. 2, pp. 432–491, 1998.
  • R. B. Ash, Probability and Measure Theory, Harcourt/Academic Press, Burlington, Mass, USA, 2nd edition, 2000.
  • P. Budzyński, Z. J. Jabłonski, I. B. Jung, and J. Stochel, “Unbounded weighted composition operators in \emphL$^{2}$-spaces,” http://arxiv.org/abs/1310.3542.
  • P. Budzyński, Z. J. Jabłoński, I. B. Jung, and J. Stochel, “Unbounded subnormal weighted shifts on directed trees,” Journal of Mathematical Analysis and Applications, vol. 394, no. 2, pp. 819–834, 2012.
  • P. Budzyński, Z. J. Jabłoński, I. B. Jung, and J. Stochel, “Unbounded subnormal weighted shifts on directed trees. II,” Journal of Mathematical Analysis and Applications, vol. 398, no. 2, pp. 600–608, 2013. \endinput