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http://dx.doi.org/10.4134/JKMS.j170183

EXAMPLES OF m-ISOMETRIC TUPLES OF OPERATORS ON A HILBERT SPACE  

Gu, Caixing (Department of Mathematics California Polytechnic State University)
Publication Information
Journal of the Korean Mathematical Society / v.55, no.1, 2018 , pp. 225-251 More about this Journal
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.
Keywords
isometry; m-isometry; multivariable weighted shift; Drury-Arveson space;
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1 F. Bayart, m-isometries on Banach spaces, Math. Nachr. 284 (2011), no. 17-18, 2141-2147.   DOI
2 T. Bermudez, A. Martinon, and V. Muller, (m, q)-Isometries on metric spaces, J. Operator Theory 72 (2014), no. 2, 313-329.   DOI
3 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.   DOI
4 T. Bermudez, A. Martinon, and J. A. Noda, Products of m-isometries, Linear Algebra Appl. 438 (2013), no. 1, 80-86.   DOI
5 T. Bermudez, A. Martinon, and E. Negrin, Weighted shift operators which are m-isometries, Integral Equations Operator Theory 68 (2010), 301-312.   DOI
6 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.
7 F. Botelho and J. Jamison, Isometric properties of elementary operators, Linear Algebra Appl. 432 (2010), no. 1, 357-365.   DOI
8 F. Botelho, J. Jamison, and B. Zheng, Strict isometries of arbitrary orders, Linear Algebra Appl. 436 (2012), no. 9, 3303-3314.   DOI
9 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.   DOI
10 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.
11 S. W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300-304.   DOI
12 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.   DOI
13 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.   DOI
14 C. Gu, Elementary operators which are m-isometries, Linear Algebra Appl. 451 (2014), 49-64.   DOI
15 C. Gu, The (m, q)-isometric weighted shifts on $l_p$ spaces, Integral Equations Operator Theory 82 (2015), no. 2, 157-187.   DOI
16 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.   DOI
17 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.   DOI
18 C. Gu, Products of (m, p)-isometric tuples of operators on a Banach space, in preparation.
19 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.   DOI
20 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.
21 B. P. Duggal, Tensor product of n-isometries, Linear Algebra Appl. 437 (2012), no. 1, 307-318.   DOI
22 A. Olofsson, A von Neumann-Wold decomposition of two-isometries, Acta Sci. Math. (Szeged) 70 (2004), no. 3-4, 715-726.
23 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.   DOI
24 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.
25 T. Le, Algebraic properties of operator roots of polynomials, J. Math. Anal. Appl. 421 (2015), no. 2, 1238-1246.   DOI
26 S. McCullough, Sub-Brownian operators, J. Operator Theory 22 (1989), no. 2, 291-305.
27 S. McCullough and B. Russo, The 3-isometric lifting theorem, Integral Equations Operator Theory 84 (2016), no. 1, 69-87.   DOI
28 S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205-220.
29 S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991), no. 1, 325-349.   DOI
30 B. Russo, Lifting commuting 3-isometry tuples, Oper. Matrices 11 (2017), no. 2, 397-433.
31 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.
32 S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147-189.
33 J. Agler and M. Stankus, m-isometric transformations of Hilbert space. II, Integral Equations Operator Theory 23 (1995), no. 1, 1-48.   DOI
34 J. Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 1, 608-631.   DOI
35 J. Agler, W. Helton, and M. Stankus, Classification of hereditary matrices, Linear Algebra Appl. 274 (1998), 125-160.   DOI
36 J. Agler and M. Stankus, m-Isometric transformations of Hilbert space. I, Integral Equations Operator Theory 21 (1995), no. 4, 383-429.   DOI
37 J. Agler and M. Stankus, m-isometric transformations of Hilbert space. III, Integral Equations Operator Theory 24 (1996), no. 4, 379-421.   DOI
38 A. Athavale, Some operator theoretic calculus for positive definite kernels, Proc. Amer. Math. Soc. 112 (1991), no. 3, 701-708.   DOI
39 W. Arveson, Subalgebra of C*-algbebra. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159-228.   DOI
40 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.   DOI