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SUPERCYCLICITY OF ℓp-SPHERICAL AND TORAL ISOMETRIES ON BANACH SPACES

  • 투고 : 2016.09.10
  • 심사 : 2016.12.22
  • 발행 : 2017.07.31

초록

Let $p{\geq}1$ be a real number. A tuple $T=(T_1,{\ldots},T_n)$ of commuting bounded linear operators on a Banach space X is called an ${\ell}^p$-spherical isometry if ${\sum_{i=1}^{n}}{\parallel}T_ix{\parallel}^p={\parallel}x{\parallel}^p$ for all $x{\in}X$. The tuple T is called a toral isometry if each Ti is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n{\geq}1$, there is a supercyclic ${\ell}^2$-spherical isometric n-tuple on ${\mathbb{C}}^n$ but there is no supercyclic ${\ell}^2$-spherical isometry on an infinite-dimensional Hilbert space. In this article, we investigate the supercyclicity of ${\ell}^p$-spherical isometries and toral isometries on Banach spaces. Also, we introduce the notion of semicommutative tuples and we show that the Banach spaces ${\ell}^p$ ($1{\leq}p$ < ${\infty}$) support supercyclic ${\ell}^p$-spherical isometric semi-commutative tuples. As a result, all separable infinite-dimensional complex Hilbert spaces support supercyclic spherical isometric semi-commutative tuples.

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참고문헌

  1. S. I. Ansari and P. S. Bourdon, Some properties of cyclic operators, Acta Sci. Math. (Szeged) 63 (1997), no. 1-2, 195-207.
  2. M. Ansari, K. Hedayatian, B. Khani-Robati, and A.Moradi, Supercyclicity of joint isometries, Bull. Korean Math. Soc. 52 (2015), no. 5, 1481-1487. https://doi.org/10.4134/BKMS.2015.52.5.1481
  3. A. Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (1990), no. 2, 339-350.
  4. N. S. Feldman, Hypercyclic tuples of operators and somewhere dense orbits, J. Math. Anal. Appl. 346 (2008), no. 1, 82-98. https://doi.org/10.1016/j.jmaa.2008.04.027
  5. H. M. Hilden and L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1973/74), 557-565. https://doi.org/10.1512/iumj.1974.23.23046
  6. 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. https://doi.org/10.1142/S1793557115500229
  7. L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997), no. 3-4, 571-605.
  8. L. Kerchy, Hyperinvariant subspaces of operators with non-vanishing orbits, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1363-1370. https://doi.org/10.1090/S0002-9939-99-04842-X
  9. R. Soltani, K. Hedayatian, and B. Khani-Robati, On supercyclicity of tuples of operators, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 4, 1507-1516. https://doi.org/10.1007/s40840-014-0083-z