Some results on higher orders quasi-isometries
The purpose of the present paper is to pursue further study of a class of linear bounded operators, known as $n$-quasi-$m$-isometric operators acting on an infinite complex separable Hilbert space ${\mathcal H}$. We give an equivalent condition for any $T$ to be $n$-quasi-$m$-isometric operator. Using this result we prove that any power of an $n$-quasi-$m$-isometric operator is also an $n$-quasi-$m$-isometric operator. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are $n$-quasi-$m$-isometries for a positive integer $r$, then T is an $n$-quasi-$m$-isometric operator. We study the sum of an $n$-quasi-$m$-isometric operator with a nilpotent operator. We also study the product and tensor product of two $n$-quasi-$m$-isometries. Further, we define $n$-quasi strict $m$-isometric operators and prove their basic properties.
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- [1] J. Agler and M. Stankus, m-isometric transformations of Hilbert space. I, Integral
Equations Operator Theory 21, 383–429, 1995.
- [2] T. Bermúdez, A. Martinón, V. Müller and J. A. Noda, Perturbation of m-Isometries
by Nilpotent Operators, Abstr. Appl. Anal. 2014, Article ID 745479, 6 pages, 2014.
- [3] T. Bermúdez, A. Martinón and J.A. Noda, Products of m-isometries, Linear Algebra
Appl. 438, 80–86, 2013.
- [4] T. Bermúdez, A. Martinón and J.A. Noda, An isometry plus a nilpotent operator is
an m-isometry. Applications, J. Math. Anal. Appl. 407 (2), 505-512, 2013.
- [5] T. Bermúdez, A. Martinón and J.A. Noda, Arithmetic Progressions and Its Applications
to (m, q)-Isometries: A Survey. Results Math. 69, 177-199, 2016.
- [6] T. Bermúdez, C.D. Mendoza and A. Martinón, Powers of m-isometries, Studia Math.
208 (3), 2012.
- [7] F. Botelho, J. Jamison and B. Zheng, Strict isometries of arbitrary orders, Linear
Algebra Appl. 436, 3303–3314, 2012.
- [8] M. Ch¯o, S. Óta and K. Tanahashi, Invertible weighted shift operators which are misometries,
Proc. Amer. Math. Soc. 141 (12), 4241-4247, 2013.
- [9] B.P. Duggal, Tensor product of n-isometries, Linear Algebra Appl. 437, 307-318,
2012.
- [10] C. Gu, Elementary operators which are m-isometries, Linear Algebra Appl. 451,
49-64, 2014.
- [11] C. Gu, Structures of left n-invertible operators and their applications, Studia Math.
226 (3), 189-211, 2015.
- [12] C. Gu, Functional calculus for m-isometries and related operators on Hilbert spaces
and Banach spaces, Acta Sci. Math. (Szeged) 81, 605–641, 2015.
- [13] C. Gu, Examples of m-isometric tuples of operators on a Hilbert space, J. Korean
Math. Soc. 55 (1), 225–251, 2018.
- [14] C. Gu and M. Stankus, Some results on higher order isometries and symmetries:
Products and sums with a nilpotent operator, Linear Algebra Appl. 469, 500-509,
2015.
- [15] J. Kyu Han, H. Youl Lee and W. Young Lee, Invertible completions of 2 × 2 upper
triangular operator matrices, Proc. Amer. Math. Soc. 128, 119–123, 1999.
- [16] O.A. Mahmoud Sid Ahmed, m-isometric operators on Banach spaces, Asian-
European J. Math. 3 (1), 19 pages, 2010.
- [17] O.A. Mahmoud Sid Ahmed, Generalization of m-partial isometries on a Hilbert
spaces, Int. J. Pure Appl. Math. 104 (4), 599–619, 2015.
- [18] S. Mecheri and S. M. Patel, On quasi-2-isometric operators, Linear Multlinear Algebra
66 (5), 1019–1025, 2018.
- [19] S. Mecheri and T. Prasad, On n-quasi-m-isometric operators, Asian-Eur. J. Math. 9
(3), 1650073, 8 pages, 2016.
- [20] S.M. Patel, A note on quasi-isometries, Glas. Mat. 35 (55), 307–312, 2002.
- [21] S.M. Patel, 2-isometric operators, Glas. Mat. 37 (57), 141–145, 2002.
- [22] S.M. Patel, A note on quasi-isometries II, Glas. Mat. 38 (58), 111-120, 2003.
- [23] M.A. Rosenblum, On the operator equation BX − XA = Q, Duke Math. J. 23,
263-269, 1956.
- [24] A. Saddi and O. A. Mahmoud Sid Ahmed, m-partial isometries on Hilbert, spaces
Intern. J. Funct. Anal. Operators Theory Appl. 2 (1), 67-83, 2010.
- [25] L. Suciu, Quasi-isometries in semi-Hilbertian spaces, Linear Algebra Appl. 430, 2474–
2487, 2009.