Commutable Matrix-Valued Functions and Operator-Valued Functions

Year 2020, Volume: 3 Issue: 4, 225 - 235, 22.12.2020

Abdelaziz Maouche

https://doi.org/10.33434/cams.759336

Abstract

A simple expression is established for an analytic commutable matrix-valued function. Then a characterization of two by two functional commutative matrices is proven. Finally, a family of analytic normal compact operators on a Hilbert space, which commute with their derivatives, is shown to be functionally commutative.

Keywords

Riesz projection, Analytic matrix-valued function, Commutable matrices, Eigenvalue, Holomorphic operator-valued function, Resolvent, Spectrum

Supporting Institution

Sultan Qaboos university

References

• [1] B. Aupetit, A Primer On Spectral Theory, Universitext, Springer-Verlag, 1991.
• [2] H. Baumgartel, Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications, 15, Birkhauser, 1985.
• [3] K. Clancy, I. Gohberg, Factorization of matrix functions and singular integral operators, Oper. Theory Adv. Appl., 3, Birkhauser Verlag (Basel) 1981.
• [4] J. Dieudonne, Sur un theoreme de Schwertfeger, Ann. Polon. Math. 24(1974), 87 - 88.
• [5] J.C. Evard, Conditions for a vector subspace E(t) and for a projector P(t) not to depend on t: , Lin. Alg. Appl. 91(1987), 121-131.
• [6] J. C. Evard, On matrix functions which commute with their derivative, Lin. Alg. Appl. 68(1985), 145 - 178.
• [7] S. Goff, Hermitian function matrices which commute with their derivative, Lin. Alg. Appl. 36(1981), 33 - 40.
• [8] I. Gohberg, J. Leiter, Holomorphic Operator Functions of one Variable and Applications, Oper. Theory Adv. Appl., 192, 2009.
• [9] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of linear operators, 1, Birkhauser 1990.
• [10] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980.
• [11] C.S. Kubrusly, Spectral Theory of Bounded Linear Operators, Birkhauser 2020.
• [12] A. Maouche, Functional commutativity of analytic families of self adjoint compact operators on a Hilbert space, Commun. Adv. Math. Sci., 3(1) (2020), 9 - 12.
• [13] M. Reed, B. Simon, Modern Methods of Mathematical Physics, Academic Press, 1975.
• [14] F. Rellich, Perturbation Theory of Eigenvalue Problems, Institute of Mathematical Sciences, New York, 1950.
• [15] H. Schwertfeger, Sur les matrices permutables avec leur deriv´ee, Rend. Sem. Mat. Univ. Politec. Torino. 11(1952), 329 - 333.