Binet’s formula for operator-valued recursive sequences and the operator moment problem
DOI:
https://doi.org/10.17398/2605-5686.40.2.181Keywords:
Binet formula, recursive operator-valued sequences, operator moment problem, representing measures, flat extensionAbstract
We derive a Binet-type formula for operator-valued sequences satisfying linear recurrence relations, extending the classical scalar case to the setting of bounded operators on Hilbert spaces. In this framework, we analyze the operator moment problem as an application, establishing new connections between recursive operator sequences and moment sequences.
Downloads
References
[1] S.K. Berberian, “Notes on spectral theory”, Van Nostrand Mathematical Studies, No. 5, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966.
[2] T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, “Introduction to algorithms”, Third edition, MIT Press, Cambridge, MA, 2009.
[3] R.E. Curto, A. Ech-charyfy, H. El Azhar, E.H. Zerouali, The local operator moment problem on R, Complex Anal. Oper. Theory 19 (2) (2025), Paper No. 25, 31 pp.
[4] E.B. Davies, “Linear operators and their spectra”,Cambridge Stud. Adv. Math., 106, Cambridge University Press, Cambridge, 2007.
[5] F. Dubeau, W. Motta, M. Rachidi, O. Saeki, On weighted r-generalized Fibonacci sequences, Fibonacci Quart. 35 (2) (1997), 102 – 110.
[6] K.-J. Engel, R. Nagel, “One-parameter semigroups for linear evolution equations”, Grad. Texts in Math., 194, Springer-Verlag, New York, 2000.
[7] K. Idrissi, E.H. Zerouali, Charges solve the truncated complex moment problem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 21 (4) (2018), no. 4, 1850027, 16 pp.
[8] K. Idrissi, E.H. Zerouali, Recursiveness approach to multi-dimensional moment problems, Ann. Funct. Anal. 13 (1) (2022), Paper No. 2, 19 pp.
[9] J.A. Jeske, Linear recurrence relations, Part I, Fibonacci Quart. 1 (2) (1963), 69 – 74.
[10] J.K. Hale, “Ordinary differential equations”, Dover Publications, 2009.
[11] W.G. Kelley, A.C. Peterson, “Difference equations: An introduction with applications”, Academic Press, Inc., Boston, MA, 1991.
[12] C. Levesque, On mth order linear recurrences, Fibonacci Quart. 23 (4) (1985) 290 – 293.
[13] M. Mlak, “Dilations of Hilbert space operators (general theory)”, Dissertationes Math. (Rozprawy Mat.) 153 (1978), 61 pp.
[14] M. Mouline, M. Rachidi, Application of Markov chains properties to r-generalized Fibonacci sequences, Fibonacci Quart. 37 (1) (1999), 34 – 38.
[15] G.N. Philippou, On the kth order linear recurrence and some probability applications, in “Applications of Fibonacci numbers” (San Jose, CA, 1986), Kluwer Academic Publishers, Dordrecht, 1988, 89 – 96..
[16] P. Pietrzycki J. Stochel, Subnormal nth roots of quasinormal operators are quasinormal, J. Funct. Anal. 280 (12) (2021), Paper No. 109001, 14 pp.
[17] P. Pietrzycki J. Stochel, Two-moment characterization of spectral measures on the real line, Canad. J. Math. 75 (4) (2023), 1369 – 1392.
[18] K. Schmüdgen, “Unbounded Self-adjoint operators on Hilbert space”, Grad. Texts in Math., 265, Springer, Dordrecht, 2012.
[19] R. Ben Taher, M. Rachidi, E.H. Zerouali, Recursive subnormal completion and the truncated moment problem, Bull. London Math. Soc. 33 (4) (2001), 425 – 432.
[20] W.J. Terrell, “Stability and stabilization. An introduction”, Princeton University Press, Princeton, NJ, 2009.
[21] Y. Saad, “Iterative methods for sparse linear systems”, Second edition, SIAM, Philadelphia, PA, 2003.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 The authors
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
