×
Gil’, Michael

Norm estimates for functions of two non-commuting operators. (English) Zbl 1339.47019

Rocky Mt. J. Math. 45, No. 3, 927-940 (2015).
The author considers analytic functions of two non-commuting Banach space operators and proves some norm estimates. Applications to some operator equations as well as some differential equations in a Banach space are given.
Reviewer: Cătălin Badea (Villeneuve d’Ascq)

Cited in 1 Document

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A60 Functional calculus for linear operators
47A62 Equations involving linear operators, with operator unknowns
47G10 Integral operators

Keywords:

functions of non-commuting operators; norm estimate; operator equation
PDF BibTeX XML Cite
\textit{M. Gil'}, Rocky Mt. J. Math. 45, No. 3, 927--940 (2015; Zbl 1339.47019)
Full Text: DOI Euclid
  • logo of hbz
  • logo of WorldCat

References:

[1] R. Arens and A.P. Calderon, Analytic functions of several Banach algebra elements , Ann. Math. 62 (1955), 204-216. · Zbl 0065.34802
[2] D.S. Cvetkovic-Ilic, The solutions of some operator equations , J. Korean Math. Soc. 45 (2008), 1417–1425. · Zbl 1165.47012
[3] Yu.L. Daleckii and M.G. Krein, Stability of solutions of differential equations in Banach space , American Mathematical Society, Providence, RI, 1974.
[4] M. Dehghan and M. Hajarian, The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations , Rocky Mountain J. Math. 40 (2010), 825-848. · Zbl 1198.15011
[5] D.S. Djordjevic, Explicit solution of the operator equation \(A^*X +X^*A = B\) , J. Comp. Appl. Math. 200 (2007), 701–704. · Zbl 1113.47011
[6] B.P. Duggal, Operator equations \(ABA = A^2\) and \(BAB = B^2\) , Funct. Anal. Approx. Comp. 3 (2011), 9–18. · Zbl 1258.47028
[7] B. Fritzsche, B. Kirstein and A. Lasarow, Orthogonal rational matrix-valued functions on the unit circle : Recurrence relations and a Favard-type theorem , Math. Nachr. 279 (2006), 513-542. · Zbl 1093.30032
[8] F.R. Gantmacher, The matrix theory , Nauka, Moscow, 1967 (in Russian).
[9] I.M. Gel’fand and G.E. Shilov, Some questions of theory of differential equations , Nauka, Moscow, 1958 (in Russian).
[10] M.I. Gil’, Estimates for norm of matrix-valued functions , Linear and Multilinear Alg. 35 (1993), 65-73. · Zbl 0778.15015
[11] —-, Operator functions and localization of spectra , Lect. Notes Math. 1830 , Springer-Verlag, Berlin, 2003. · Zbl 1032.47001
[12] —-, Norms of functions of commuting matrices , Electr. J. Linear Alg. 13 (2005), 122-130. · Zbl 1098.15021
[13] —-, Difference equations in normed spaces. Stability and sscillations , Math. Stud. 206 , Elsevier, Amsterdam, 2007.
[14] —-, Estimates for entries of matrix valued functions of infinite matrices , Math. Phys. Anal. Geom. 11 (2008), 175-186. · Zbl 1194.47019
[15] —-, Estimates for functions of two commuting infinite matrices and applications , Ann. Univ. Ferr. Sez. VII Sci. Mat. 56 (2010), 211-218. · Zbl 1205.15028
[16] —-, Norms estimates for functions of two non-commuting matrices , Electr. J. Linear Alg. 22 (2011), 504-512. · Zbl 1223.15030
[17] —-, Perturbations of operator functions in a Hilbert space , Comm. Math. Anal. 13 (2012), 108–115. · Zbl 1282.47015
[18] R.A. Horn and C.R. Johnson, Topics in matrix analysis , Cambridge University Press, Cambridge, 1991. · Zbl 0729.15001
[19] M. Konstantinov, Da-Wei Gu, V. Mehrmann and P. Petkov, Perturbation theory for matrix equations , Stud. Comp. Math. 9 , North Holland, 2003. · Zbl 1025.15017
[20] A.G. Mazko, Matrix equations, Spectral problems and stability of dynamic systems , Stability, Oscillations and Optimization of Systems, Scientific Publishers, Cambridge, 2008. · Zbl 1152.93300
[21] V. Müller, Spectral theory of linear operators , Birkhäusr Verlag, Basel, 2003.
[22] A. Pietsch, Eigenvalues and \(s\)-numbers . Cambridge Univesity Press, Cambridge, 1987. · Zbl 0615.47019
[23] Qing-Wen Wang and Chang-Zhou Dong, The general solution to a system of adjointable operator equations over Hilbert \(C^*\) -modules , Oper. Matr. 5 (2011), 333–350. · Zbl 1227.47007
[24] J.L. Taylor, Analytic functional calculus for several commuting operators , Acta Math. 125 (1970), 1-38. · Zbl 0233.47025
[25] R. Werpachowski, On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices , Linear Alg. Appl. 428 (2008), 316-323. · Zbl 1156.65040
[26] Xiqiang Zhao and Tianming Wang, The algebraic properties of a type of infinite lower triangular matrices related to derivatives , J. Math. Res. Expo. 22 (2002), 549-554. · Zbl 1021.15016
[27] Bin Zhou, James Lam and Guang-Ren Duan, On Smith-type iterative algorithms for the Stein matrix equation , Appl. Math. Lett. 22 (2009), 1038-1044. · Zbl 1179.15016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.