Rocky Mountain Journal of Mathematics

Norm estimates for functions of two non-commuting operators

Michael Gil'

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Analytic functions of two non-commuting bounded operators in a Banach space are considered. Sharp norm estimates are established. Applications to operator equations and differential equations in a Banach space are discussed.

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Rocky Mountain J. Math., Volume 45, Number 3 (2015), 927-940.

First available in Project Euclid: 21 August 2015

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Primary: 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 47A60: Functional calculus 47A62: Equations involving linear operators, with operator unknowns 47G10: Integral operators [See also 45P05]

Functions of non-commuting operators norm estimate operator equation


Gil', Michael. Norm estimates for functions of two non-commuting operators. Rocky Mountain J. Math. 45 (2015), no. 3, 927--940. doi:10.1216/RMJ-2015-45-3-927.

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  • R. Arens and A.P. Calderon, Analytic functions of several Banach algebra elements, Ann. Math. 62 (1955), 204–216.
  • D.S. Cvetkovic-Ilic, The solutions of some operator equations, J. Korean Math. Soc. 45 (2008), 1417–-1425.
  • Yu.L. Daleckii and M.G. Krein, Stability of solutions of differential equations in Banach space, American Mathematical Society, Providence, RI, 1974.
  • 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.
  • D.S. Djordjevic, Explicit solution of the operator equation $A^*X +X^*A = B$, J. Comp. Appl. Math. 200 (2007), 701–-704.
  • B.P. Duggal, Operator equations $ABA = A^2$ and $BAB = B^2$, Funct. Anal. Approx. Comp. 3 (2011), 9–-18.
  • 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.
  • F.R. Gantmacher, The matrix theory, Nauka, Moscow, 1967 (in Russian).
  • I.M. Gel'fand and G.E. Shilov, Some questions of theory of differential equations, Nauka, Moscow, 1958 (in Russian).
  • M.I. Gil', Estimates for norm of matrix-valued functions, Linear and Multilinear Alg. 35 (1993), 65–73.
  • ––––, Operator functions and localization of spectra, Lect. Notes Math. 1830, Springer-Verlag, Berlin, 2003.
  • ––––, Norms of functions of commuting matrices, Electr. J. Linear Alg. 13 (2005), 122–130.
  • ––––, Difference equations in normed spaces. Stability and sscillations, Math. Stud. 206, Elsevier, Amsterdam, 2007.
  • ––––, Estimates for entries of matrix valued functions of infinite matrices, Math. Phys. Anal. Geom. 11 (2008), 175–186.
  • ––––, Estimates for functions of two commuting infinite matrices and applications, Ann. Univ. Ferr. Sez. VII Sci. Mat. 56 (2010), 211–218.
  • ––––, Norms estimates for functions of two non-commuting matrices, Electr. J. Linear Alg. 22 (2011), 504–512.
  • ––––, Perturbations of operator functions in a Hilbert space, Comm. Math. Anal. 13 (2012), 108-–115.
  • R.A. Horn and C.R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.
  • M. Konstantinov, Da-Wei Gu, V. Mehrmann and P. Petkov, Perturbation theory for matrix equations, Stud. Comp. Math. 9, North Holland, 2003.
  • A.G. Mazko, Matrix equations, Spectral problems and stability of dynamic systems, Stability, Oscillations and Optimization of Systems, Scientific Publishers, Cambridge, 2008.
  • V. Müller, Spectral theory of linear operators, Birkhäusr Verlag, Basel, 2003.
  • A. Pietsch, Eigenvalues and $s$-numbers. Cambridge Univesity Press, Cambridge, 1987.
  • 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.
  • J.L. Taylor, Analytic functional calculus for several commuting operators, Acta Math. 125 (1970), 1–38.
  • R. Werpachowski, On the approximation of real powers of sparse, infinite, bounded and Hermitian matrices, Linear Alg. Appl. 428 (2008), 316–323.
  • 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.
  • 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.