×

Estimates for functions of two commuting infinite matrices and applications. (English) Zbl 1205.15028

Summary: We consider a matrix valued function of two infinite commuting matrices generating bounded operators in a Banach space. Sharp estimates for the norm and entries are derived. Applications of the obtained estimates to differential equations are also discussed.

MSC:

15A27 Commutativity of matrices
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A60 Functional calculus for linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arens R., Calderon A.P.: Analytic functions of several Banach algebra elements. Ann. Math. 62, 204-216 (1955) · Zbl 0065.34802 · doi:10.2307/1969675
[2] Daleckii, Y.L., Krein, M.G.: Stability of solutions of differential equations in Banach space. Am. Math. Soc., Providence, RI (1974)
[3] Fritzsche B., Kirstein B., Lasarow A.: On Schwarz-Pick-Potapov block matrices of matrix-valued functions which belong to the extended Potapov class. Anal. München 22(3), 243-263 (2002) · Zbl 1033.47014
[4] Fritzsche B., Kirstein B., Lasarow A.: Orthogonal rational matrix-valued functions on the unit circle: Recurrence relations and a Favard-type theorem. Math. Nachr. 279(5-6), 513-542 (2006) · Zbl 1093.30032 · doi:10.1002/mana.200310376
[5] Gel’fand, I.M., Shilov, G.E.: Some Questions of Theory of Differential Equations. Nauka, Moscow (1958) (in Russian) · Zbl 0091.11104
[6] Gil’ M.I.: Estimates for norm of matrix-valued functions. Linear Multilinear Algebra 35, 65-73 (1993) · Zbl 0778.15015 · doi:10.1080/03081089308818242
[7] Gil’ M.I.: Stability of Finite and Infinite Dimensional Systems. Kluwer, Boston (1998) · Zbl 0916.93002
[8] Gil’ M.I.: Operator Functions and Localization of Spectra, Lectures Notes In Mathematics, vol. 1830. Springer, Berlin (2003) · Zbl 1032.47001
[9] Gil’ M.I.: Norm estimates for functions of two operators on tensor products of Hilbert spaces. Math. Nachr 281(8), 1-13 (2008)
[10] Gil’ M.I.: Estimates for entries of matrix valued functions of infinite matrices. Math. Phys., Anal. Geom. 11(2), 175-186 (2008) · Zbl 1194.47019 · doi:10.1007/s11040-008-9046-7
[11] Pao C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992) · Zbl 0777.35001
[12] Ryan R.A.: Introduction to Tensor Products of Banach Spaces. Springer, London (2002) · Zbl 1090.46001
[13] Taylor J.L.: Analytic functional calculus for several commuting operators. Acta Math. 125, 1-38 (1970) · Zbl 0233.47025 · doi:10.1007/BF02392329
[14] Verde-Star L.: Functions of matrices. Linear Algebra Appl. 406, 285-300 (2005) · Zbl 1083.15035 · doi:10.1016/j.laa.2005.04.016
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.