NORM EQUALITIES AND INEQUALITIES FOR TRIDIAGONAL PERTURBED TOEPLITZ OPERATOR MATRICES

Jiajie Wang; Yanpeng Zheng; Zhaolin Jiang

doi:10.11948/20210489

2023 Volume 13 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(2): 671-683

Next Article Previous Article

Jiajie Wang, Yanpeng Zheng, Zhaolin Jiang. NORM EQUALITIES AND INEQUALITIES FOR TRIDIAGONAL PERTURBED TOEPLITZ OPERATOR MATRICES[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 671-683. doi: 10.11948/20210489

Citation:

Jiajie Wang, Yanpeng Zheng, Zhaolin Jiang. NORM EQUALITIES AND INEQUALITIES FOR TRIDIAGONAL PERTURBED TOEPLITZ OPERATOR MATRICES[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 671-683. doi: 10.11948/20210489

NORM EQUALITIES AND INEQUALITIES FOR TRIDIAGONAL PERTURBED TOEPLITZ OPERATOR MATRICES

1.
School of Mathematics and Statistics, Linyi University, Linyi, 276000, China
2.
School of Automation and Electrical Engineering, Linyi University, Linyi, 276000, China

Corresponding authors: Email: 937262674@qq.com(J. Wang); Email: zhengyanpeng0702@sina.com(Y. Zheng); Email: jzh1208@sina.com(Z. Jiang)
Fund Project: The research was supported by the Natural Science Foundation of Shandong Province(No.ZR2020QA035), the National Natural Science Foundation of China(No.12001257) and the PhD Research Foundation of Linyi University(No. LYDX2018BS067)

Abstract

Abstract

Tridiagonal perturbed Toeplitz operator matrices is a class of important structured matrices. In this paper, we present several norm equalities and inequalities for this class of matrices. The special norms we consider include the usual operator norm and the Schatten $ p $-norms. Moreover, pinching type inequalities are also discussed for general weakly unitarily invariant norms. The proofs feature the special structure of tridiagonal perturbed Toeplitz operator matrices.
- Tridiagonal perturbed Toeplitz operator matrix /
- unitarily invariant norm /
- equality /
- inequality
MSC: 15A09, 15A15, 65F50

FullText(HTML)

References

[1]	W. Bani-Domi, F. Kittaneh and M. Shatnawi, New norm equalities and inequalities for certain operator matrices, Math. Inequalities Appl., 2020, 23, 1041–1050. Google Scholar
[2]	R. Bhatia and F. Kittaneh, Clarkson inequalities with several operators, Bull. London Math. Soc., 2004, 36, 820–832. doi: 10.1112/S0024609304003467 CrossRef Google Scholar
[3]	R. Bhatia and F. Kittaneh, Norm inequalities for partitioned operators and an application, Math. Ann., 1990, 287, 719–726. doi: 10.1007/BF01446925 CrossRef Google Scholar
[4]	R. Bhatia, W. Kahan and R. Li, Pinchings and norms of scaled triangular matrices, Linear Multilinear Algebra, 2002, 50, 15–21. doi: 10.1080/03081080290011674 CrossRef Google Scholar
[5]	R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. Google Scholar
[6]	S. Compernolle, L. Chibotaru and A. Coulemans, Eigenstates and transmission coefficients of finite-sized nanotubes, J. Chem. Phys., 2003, 119, 2854–2873. doi: 10.1063/1.1587691 CrossRef Google Scholar
[7]	W. B. Domi and F. Kittaneh, Norm equalities and inequalities for operator matrices, Linear Algebra Appl., 2008, 429, 57–67. doi: 10.1016/j.laa.2008.02.004 CrossRef Google Scholar
[8]	Y. Fu, X. Jiang, Z. Jiang and S. Jhang, Properties of a class of perturbed Toeplitz periodic tridiagonal matrices, Comp. Appl. Math., 2020, 39, 1–19. doi: 10.1007/s40314-019-0964-8 CrossRef Google Scholar
[9]	Y. Fu, X. Jiang, Z. Jiang and S. Jhang, Inverses and eigenpairs of tridiagonal Toeplitz matrix with opposite-bordered rows, J. of Appl. Anal. Comput., 2020, 10(4), 1599–1613. Google Scholar
[10]	Y. Fu, X. Jiang, Z. Jiang and S. Jhang, Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns, Spec. Matrices, 2020, 8, 131–143. doi: 10.1515/spma-2020-0012 CrossRef Google Scholar
[11]	M. Gu, J. Li, H. Sun, et al., Spectral signatures of the surface anomalous Hall effect in magnetic axion insulators, Nat. Commun., 2021, 12, 3524. doi: 10.1038/s41467-021-23844-z CrossRef Google Scholar
[12]	I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Monographs, vol. 18, American Mathematical Society, Providence, RI, 1969. Google Scholar
[13]	O. Hirzallah and F. Kittaneh, Non-commutative Clarkson inequalities for n-tuples of operators, Integr. Equat. Oper. Th., 2008, 60, 369–379. doi: 10.1007/s00020-008-1565-x CrossRef Google Scholar
[14]	Z. Jiang, Y. Qiao and S. Wang, Norm equalities and inequalities for three circulant operator matrices, Acta Mathematica Sinica, English Series, 2017, 33(4), 571–590. doi: 10.1007/s10114-016-5607-z CrossRef Google Scholar
[15]	Z. Jiang and T. Xu, Norm estimates of ω-circulant operator matrices and isomorphic operators for ω-circulant algebra, Sci. China Math., 2016, 59(2), 351–366. doi: 10.1007/s11425-015-5051-z CrossRef Google Scholar
[16]	P. D. Kvam1, J. R. Busemeyer and T. J. Pleskac, Temporal oscillations in preference strength provide evidence for an open system model of constructed preference, Sci. Rep., 2021, 11, 8169. doi: 10.1038/s41598-021-87659-0 CrossRef Google Scholar
[17]	E. Kissin, On Clarkson-McCarthy inequalities for n-tuples of operators, Proc. Amer. Math. Soc., 2007, 135, 2483–2495. doi: 10.1090/S0002-9939-07-08710-2 CrossRef Google Scholar
[18]	T. Kostyrko, M. Bartkowiak and G. D. Mahan, Reflection by defects in a tight-binding model of nanotubes, Phys. Rev. B, 1999, 59, 3241–3249. Google Scholar
[19]	L. Lakatos, L. Szeidl and M. Telek, Introduction to Queueing Systems with Telecommunication Applications, 2 Eds., Springer Publishing Company, Incorporated, 2019. Google Scholar
[20]	L. Molinari, Transfer matrices and tridiagonal-block Hamiltonians with periodic and scattering boundary conditions, J. Phys. A, 1997, 30, 983–997. doi: 10.1088/0305-4470/30/3/021 CrossRef Google Scholar
[21]	G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 1992, 13, 707–728. doi: 10.1137/0613045 CrossRef Google Scholar
[22]	L. G. Molinari, Determinants of block tridiagonal matrices, Linear Algebra Appl., 2008, 429, 2221–2226. doi: 10.1016/j.laa.2008.06.015 CrossRef Google Scholar
[23]	E. D. Nabben, Decay rates of the inverse of nonsymmetric tridiagonal and band matrices, SIAM J. Matrix Anal. Appl., 1999, 20, 820–837. doi: 10.1137/S0895479897317259 CrossRef Google Scholar
[24]	N. P. D. Sawaya, et al., Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians, NPJ Quantum. Inform., 2020, 6(1), 1–13. doi: 10.1038/s41534-019-0235-y CrossRef Google Scholar
[25]	N. S. Savage, Describing the movement of molecules in reduced-dimension models, Commun. Biol., 2021, 4, 689. doi: 10.1038/s42003-021-02200-3 CrossRef Google Scholar
[26]	N. M. Schnerb and D. R. Nelson, Winding numbers, complex currents and non-Hermitian localization, Phys. Rev. Lett., 1998, 80, 5172–5175. doi: 10.1103/PhysRevLett.80.5172 CrossRef Google Scholar
[27]	T. Sogabe, On a two-term recurrence for the determinant of a general matrix, Appl. Math. Comput., 2007, 187, 785–788. Google Scholar
[28]	Y. Wei, Y. Zheng, Z. Jiang and S. Shon, The inverses and eigenpairs of tridiagonal Toeplitz matrices with perturbed rows, J. App. Math. Comput., 2022, 68(1), 623–636. doi: 10.1007/s12190-021-01532-x CrossRef Google Scholar
[29]	Y. Wei, X. Jiang, Z. Jiang, et al., On inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners, Journal of Applied Analysis and Computation, 2020, 10(1), 178–191. Google Scholar
[30]	Y. Wei, Y. Zheng, Z. Jiang, et al., A study of determinants and inverses for periodic tridiagonal Toeplitz matrices with perturbed corners involving Mersenne numbers, Mathematics, 2019, 7(10), 893. doi: 10.3390/math7100893 CrossRef Google Scholar
[31]	Y. Wei, X. Jiang, Z. Jiang, et al., Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices, Advances in Difference Equations, 2019, 2019(1), 410. doi: 10.1186/s13662-019-2335-6 CrossRef Google Scholar
[32]	C. Yue, Y. Xu, Z. Song, et al., Symmetry-enforced chiral hinge states and surface quantum anomalous Hall effect in the magnetic axion insulator $Bi_{2-x}Sm_x Se_3$, Nat. Phys., 2019, 15, 577–581. doi: 10.1038/s41567-019-0457-0 CrossRef $Bi_{2-x}Sm_x Se_3$" target="_blank">Google Scholar
[33]	H. Yamada, Electronic localization properties of a double strand of DNA: a simple model with long range correlated hopping disorder, Internat. J. Modern Phys. B, 2004, 18, 1697–1716. doi: 10.1142/S0217979204024884 CrossRef Google Scholar