Powers of m-isometries (original) (raw)

Abstract

A bounded linear operator T on a Banach space X is called an (m, p)-isometry for a positive integer m and a real number p ≥ 1 if, for any

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What explains the stability of m-isometries under power operators?add

The research demonstrates that any power T^r of an (m, p)-isometry remains an (m, p)-isometry, providing a significant structural property within the class of m-isometries.

How do m-isometries relate to traditional isometries?add

The findings confirm that all m-isometries are nested within the framework of isometries, affirming that 1-isometries coincide with traditional isometries.

What key results arise from combining different powers of m-isometries?add

The study reveals that if T^r is an (m, p)-isometry and T^s is an (l, p)-isometry, then T^g is an (h, p)-isometry where g is the gcd of r and s, and h is the minimum of m and l.

What methodology aids in analyzing power properties of m-isometries?add

The paper employs recursive equations and characteristic polynomials to derive fundamental properties of m-isometries, enhancing the understanding of their stability and interrelations.

When can powers of m-isometries guarantee original operator properties?add

Theorems established indicate that if powers T^r and T^(r+1) are (m, p)-isometries, then the operator T itself must also qualify as an (m, p)-isometry.

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References (20)

  1. J. Agler. A disconjugacy theorem for Toeplitz operators. Amer. J. Math. 112 (1990), no. 1, 1-14.
  2. J. Agler, M. Stankus. m-isometric transformations of Hilbert space. I. Integral Equations Operator Theory 21 (1995), no. 4, 383-429.
  3. J. Agler, M. Stankus. m-isometric transformations of Hilbert space. II. Integral Equations Operator Theory 23 (1995), no. 1, 1-48.
  4. J. Agler, M. Stankus. m-isometric transformations of Hilbert space. III. Integral Equations Operator Theory 24 (1996), no. 4, 379-421.
  5. R. P. Agarwal. Difference equations and inequalities. Theory, methods, and applications, Monographs and Textbooks in Pure and Applied Mathematics, 155. Marcel Dekker, Inc., New York, 1992.
  6. A. Athavale. Some operator theoretic calculus for positive definite kernels. Proc. Amer. Math. Soc. 112(3) (1999), 701-708.
  7. F. Bayart. m-isometries on Banach spaces. Math. Nachr. to appear.
  8. T. Bermúdez, I. Marrero, A. Martinón. On the orbit of an m-isometry. Integral Equations and Operator Theory 64 (2009), 487-494.
  9. T. Bermúdez, A. Martinón, E. Negrín. Weighted shift operators which are m-isometries, Integral Equations and Operator Theory 68 (2010), 301-312.
  10. F. Botelho, J. Jamison. Isometric properties of elementary operators, Linear Algebra and its Applications 432 (2010), 357-365.
  11. W. G. Kelley, A. C. Peterson. Difference Equations. An Introduction with Applications. Academic Press, 1991.
  12. M. Faghih Ahmadi, K. Hedayatian. Hypercyclicity and supercyclicity of m-isometric opera- tors. Preprint.
  13. M. Faghih Ahmadi, K. Hedayatian. m-isometric weighted shifts and reflexivity of some op- erators. Preprint.
  14. J. Gleason, S. Richter. m-Isometric Commuting Tuples of Operators on a Hilbert space. Integral Equations and Operator Theory 56 (2006), 181-196.
  15. C. Helleings. Two-Isometries on Pontryagin Spaces. Integral Equations and Operator Theory 61 (2008), 211-239.
  16. S. Panayappan, S. K. Latha, Some Isometric Composition Operators. Int. J. Contemp. Math. Sciences 5 (2010), 615-621.
  17. S. M. Patel. 2-isometry operators, Glasnik Mat. 37 (57) (2002), 143-147.
  18. L. J. Patton, M. E. Robbins. Composition operators that are m-isometries. Houston J. Math. 31 (2005), 255-266.
  19. S. Richter. A representation theorem for cyclic analytic two-isometries. Tran. Amer. Math. Soc. 328 (1991), 325-349.
  20. O. A. M. Sid Ahmed. m-isometric operators on Banach spaces. Asian-European J. Math. 3 (2010), 1-19.