Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXAMPLES OF m-ISOMETRIC TUPLES OF OPERATORS ON A HILBERT SPACE

  • Gu, Caixing (Department of Mathematics California Polytechnic State University)
  • Received : 2017.03.11
  • Accepted : 2017.08.10
  • Published : 2018.01.01
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Abstract

The m-isometry of a single operator in Agler and Stankus [3] was naturally generalized to the m-isometric tuple of several commuting operators by Gleason and Richter [22]. Some examples of m-isometric tuples including the recently much studied Arveson-Drury d-shift were given in [22]. We provide more examples of m-isometric tuples of operators by using sums of operators or products of operators or functions of operators. A class of m-isometric tuples of unilateral weighted shifts parametrized by polynomials are also constructed. The examples in Gleason and Richter [22] are then obtained by choosing some specific polynomials. This work extends partially results obtained in several recent papers on the m-isometry of a single operator.

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References

  1. J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 1, 608-631. https://doi.org/10.1007/BF01694057
  2. J. Agler, W. Helton, and M. Stankus, Classification of hereditary matrices, Linear Algebra Appl. 274 (1998), 125-160. https://doi.org/10.1016/S0024-3795(97)00307-8
  3. J. Agler and M. Stankus, m-Isometric transformations of Hilbert space. I, Integral Equations Operator Theory 21 (1995), no. 4, 383-429. https://doi.org/10.1007/BF01222016
  4. J. Agler and M. Stankus, m-isometric transformations of Hilbert space. II, Integral Equations Operator Theory 23 (1995), no. 1, 1-48. https://doi.org/10.1007/BF01261201
  5. J. Agler and M. Stankus, m-isometric transformations of Hilbert space. III, Integral Equations Operator Theory 24 (1996), no. 4, 379-421. https://doi.org/10.1007/BF01191619
  6. A. Athavale, Some operator theoretic calculus for positive definite kernels, Proc. Amer. Math. Soc. 112 (1991), no. 3, 701-708. https://doi.org/10.1090/S0002-9939-1991-1068114-8
  7. W. Arveson, Subalgebra of C*-algbebra. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159-228. https://doi.org/10.1007/BF02392585
  8. C. A. Berger and B. L. Shaw, Self-commutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc. 79 (1973), 1193-1199. https://doi.org/10.1090/S0002-9904-1973-13375-0
  9. F. Bayart, m-isometries on Banach spaces, Math. Nachr. 284 (2011), no. 17-18, 2141-2147. https://doi.org/10.1002/mana.200910029
  10. T. Bermudez, A. Martinon, and V. Muller, (m, q)-Isometries on metric spaces, J. Operator Theory 72 (2014), no. 2, 313-329. https://doi.org/10.7900/jot.2013jan29.1996
  11. T. Bermudez, A. Martinon, and J. A. Noda, An isometry plus a nilpotent operator is an m-isometry, J. Math. Anal. Appl. 407 (2013), no. 2, 505-512. https://doi.org/10.1016/j.jmaa.2013.05.043
  12. T. Bermudez, A. Martinon, and J. A. Noda, Products of m-isometries, Linear Algebra Appl. 438 (2013), no. 1, 80-86. https://doi.org/10.1016/j.laa.2012.07.011
  13. T. Bermudez, A. Martinon, and E. Negrin, Weighted shift operators which are m-isometries, Integral Equations Operator Theory 68 (2010), 301-312. https://doi.org/10.1007/s00020-010-1801-z
  14. T. Bermudez, A. Martinon, V. Muller, and J. Noda, Perturbation of m-isometries by nilpotent operators, Abstr. Appl. Anal. 2014 (2014), Art. ID 745479, 6 pp.
  15. F. Botelho and J. Jamison, Isometric properties of elementary operators, Linear Algebra Appl. 432 (2010), no. 1, 357-365. https://doi.org/10.1016/j.laa.2009.08.013
  16. F. Botelho, J. Jamison, and B. Zheng, Strict isometries of arbitrary orders, Linear Algebra Appl. 436 (2012), no. 9, 3303-3314. https://doi.org/10.1016/j.laa.2011.11.022
  17. M. Cho, S. Ota, and K. Tanahashi, Invertible weighted shift operators which are m-isometries, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4241-4247. https://doi.org/10.1090/S0002-9939-2013-11701-6
  18. R. E. Curto and F. H. Vasilescu, Automorphism invariance of the operator-valued Possion transform, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 65-78.
  19. S. W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300-304. https://doi.org/10.1090/S0002-9939-1978-0480362-8
  20. B. P. Duggal, Tensor product of n-isometries, Linear Algebra Appl. 437 (2012), no. 1, 307-318. https://doi.org/10.1016/j.laa.2012.02.017
  21. M. Faghih-Ahmadi and K. Hedayatian, m-isometric weighted shifts and reflexivity of some operators, Rocky Mountain J. Math. 43 (2013), no. 1, 123-133. https://doi.org/10.1216/RMJ-2013-43-1-123
  22. J. Gleason and S. Richter, m-isometric commuting tuples of operators on a Hilbert spaces, Integral Equations Operator Theory 56 (2006), no. 2, 181-196. https://doi.org/10.1007/s00020-006-1424-6
  23. C. Gu, Elementary operators which are m-isometries, Linear Algebra Appl. 451 (2014), 49-64. https://doi.org/10.1016/j.laa.2014.03.020
  24. C. Gu, The (m, q)-isometric weighted shifts on $l_p$ spaces, Integral Equations Operator Theory 82 (2015), no. 2, 157-187. https://doi.org/10.1007/s00020-015-2234-5
  25. C. Gu, On (m, p)-expansive and (m, p)-contractive operators on Hilbert and Banach spaces, J. Math. Anal. Appl. 426 (2015), no. 2, 893-916. https://doi.org/10.1016/j.jmaa.2015.01.067
  26. C. Gu, Functional calculus for m-isometries and related operators on Hilbert spaces and Banach spaces, Acta Sci. Math. (Szged) 81 (2015), no. 3-4, 605-641. https://doi.org/10.14232/actasm-014-550-3
  27. C. Gu, Products of (m, p)-isometric tuples of operators on a Banach space, in preparation.
  28. C. Gu and M. Stankus, Some results on higher order isometries and symmetries: products and sums with a nilpotent, Linear Algebra Appl. 469 (2015), 500-509. https://doi.org/10.1016/j.laa.2014.09.044
  29. P. Hoffmann and M. Mackey, (M, p)-isometric and (m, ${\infty}$)-isometric operator tuples on normed spaces, Asian-Eur. J. Math. 8 (2015), no. 2, 1550022, 32 pp.
  30. P. Hoffmann, M. Mackey, and M. Searcoid, On the second parameter of an (m, p)-isometry, Integral Equations Operator Theory 71 (2011), no. 3, 389-405. https://doi.org/10.1007/s00020-011-1905-0
  31. N. P. Jewell and A. R. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory 1 (1979), no. 2, 207-223.
  32. T. Le, Algebraic properties of operator roots of polynomials, J. Math. Anal. Appl. 421 (2015), no. 2, 1238-1246. https://doi.org/10.1016/j.jmaa.2014.07.074
  33. S. McCullough, Sub-Brownian operators, J. Operator Theory 22 (1989), no. 2, 291-305.
  34. S. McCullough and B. Russo, The 3-isometric lifting theorem, Integral Equations Operator Theory 84 (2016), no. 1, 69-87. https://doi.org/10.1007/s00020-015-2240-7
  35. A. Olofsson, A von Neumann-Wold decomposition of two-isometries, Acta Sci. Math. (Szeged) 70 (2004), no. 3-4, 715-726.
  36. S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205-220.
  37. S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991), no. 1, 325-349. https://doi.org/10.1090/S0002-9947-1991-1013337-1
  38. B. Russo, Lifting commuting 3-isometry tuples, Oper. Matrices 11 (2017), no. 2, 397-433.
  39. A. L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, pp. 49-128. Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974.
  40. S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147-189.