Isometric NN-Jordan weighted shift operators

Isometric NN-Jordan weighted shift operators

A bounded linear operator T on a Hilbert space is an isometric N -Jordan operator if it can be written as A + Q, where A is an isometry and Q is a nilpotent of order N such that AQ = QA. In this paper, we will show that the only isometric N -Jordan weighted shift operators are isometries. This answers a question recently raised.

___

  • [1] Agler J. A disconjugacy theorem for Toeplitz operators. Am J Math 1990; 112: 1-14.
  • [2] Agler J, Stankus M. m-Isometric transformations of Hilbert space. I. Integr Equ Oper Theory 1995; 21: 383-429.
  • [3] Agler J, Stankus M. m-Isometric transformations of Hilbert space. II. Integr Equ Oper Theory 1995; 23: 1-48.
  • [4] Agler J, Stankus M. m-Isometric transformations of Hilbert space. III. Integr Equ Oper Theory 1996; 24: 379-421.
  • [5] Athavale A. Some operator-theoretic for positive definite kernels. Proc Amer Math Soc 1991; 112: 701-708.
  • [6] Berm´udez T, Martin´on A, Negr´ın E. Weighted shift operators which are m-isometries. Integr Equ Oper Theory 2012; 68: 301-312.
  • [7] Berm´udez T, Martin´on A, Noda JA. An isometry plus a nilpotent is an m-isometry. Applications. J Math Anal Appl 2013; 407: 505-512.
  • [8] Ch¯o M, Ota S, Tanahashi K. Invertible weighted shift operators which are ˆ m-isometries. Proc Amer Math Soc 2013; 141: 4241-4247.
  • [9] Conway JB. The Theory of Subnormal Operators. Amer Math Soc Surveys and Monographs 36, Providence, 1991.
  • [10] Faghih-Ahmadi M, Hedayatian K. m-isometric weighted shifts and reflexivity of some operators. Rocky Mountain J Math 2013; 43: 123-133.
  • [11] Yarmahmoodi S, Hedayatian K, Yousefi B. Supercyclicity and hypercyclicity of an isometry plus a nilpotent. Abstract and Applied Analysis, 2011; Article ID 686832: 11 pages.