Abstract
In this paper we collect some results about arithmetic progressions of higher order, also called polynomial sequences. Those results are applied to (m, q)-isometric maps.
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Agarwal R.P.: Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics, vol. 155. Marcel Dekker, Inc., New York (1992)
Agler J.: A disconjugacy theorem for Toeplitz operators. Am. J. Math. 112(1), 1–14 (1990)
Agler J., Stankus M.: m-isometric transformations of Hilbert space. I. Integr. Equ. Oper. Theory 21(4), 383–429 (1995)
Agler J., Stankus M.: m-isometric transformations of Hilbert space. II. Integr. Equ. Oper. Theory 23(1), 1–48 (1995)
Agler J., Stankus M.: m-isometric transformations of Hilbert space. III. Integr. Equ. Oper. Theory 24(4), 379–421 (1995)
Alonso J.: Arithmetic sequences of higher order. Fibinacci Q. 14(2), 147–152 (1976)
Bayart F.: m-isometries on Banach spaces. Math. Nachr. 284, 2141–2147 (2011)
Bermúdez T., Díaz-Mendoza C., Martinón A.: Powers of m-isometries. Studia Math. 208, 249–255 (2012)
Bermúdez T., Marrero I., Martinón A.: On the orbit of an m-isometry. Integr. Equ. Oper. Theory 64, 487–494 (2009)
Bermúdez, T., Martinón, A., Müller, V.: (m, q)-isometries on metric spaces. J. Operator Theory 72, 313–329 (2014)
Bermúdez, T., Martinón, A., Müller, V., Noda, J.A.: Perturbation of an m-isometry by a nilpotent. Abstr. Appl. Anal. Article ID 745479 (2014)
Bermúdez T., Martinón A., Negrín E.: Weighted shift operators which are m-isometries. Integr. Equ. Oper. Theory 68, 301–312 (2010)
Bermúdez T., Martinón A., Noda J.: Products of m-isometries. Linear Algebra Appl. 438, 80–86 (2013)
Bermúdez T., Martinón A., Noda J.: An isometry plus an nilpotent operator is an m-isometry. J. Math. Anal. Appl. 407, 505–512 (2013)
Bermúdez, T., Martinón, A., Noda, J.A.: On \({\infty}\)-inverses (in preparation)
Botelho F.: On the existence of n-isometries on \({\ell_p}\) spaces. Acta Sci. Math. (Szeged) 76, 183–192 (2010)
Dlab V.: Arithmetic progressions of higher order. Teach. Math. Comput. Sci. 9(2), 225–239 (2011)
Duggal B.P., Müller V.: Tensor product of left n-invertible operators. Studia Math. 215(2), 113–125 (2013)
Gu C., Stankus M.: Some results on higher order isometries and symmetries: products and sums with a nilpotent operator. Linear Algebra Appl. 469, 500–509 (2015)
Hoffmann P., Mackey M., Searcóid M.O.: On the second parameter of an (m, p)-isometry. Integr. Equ. Oper. Theory 71, 389–405 (2011)
Le T.: Algebraic properties of operator roots of polynomials. J. Math. Anal. Appl. 421(2), 1238–1246 (2015)
Kelley W.G., Peterson A.C.: Difference Equations: an Introduction with Applications. Academic Press, London (1991)
Sid Ahmed, O.A.M.: m-isometric operators on Banach spaces. Asian Eur. J. Math. 3, 1–19 (2010)
Sid Ahmed O.A.M.: Some properties of m-isometries and m-invertible operators on Banach spaces. Acta Math. Sci. Ser. B Engl. Ed. 32(2), 520–530 (2012)
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Bermúdez, T., Martinón, A. & Noda, J.A. Arithmetic Progressions and Its Applications to (m, q)-Isometries: A Survey. Results. Math. 69, 177–199 (2016). https://doi.org/10.1007/s00025-015-0470-2
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DOI: https://doi.org/10.1007/s00025-015-0470-2