Analytic Families of Self-Adjoint Compact Operators Which Commute with Their Derivative

Year 2020, Volume: 3 Issue: 1, 9 - 12, 25.03.2020

Abdelaziz MAOUCHE

https://doi.org/10.33434/cams.627282

Cited By: 3

Abstract

Spectral properties of analytic families of compact operators on a Hilbert space are studied. The results obtained are then used to establish that an analytic family of self-adjoint compact operators on a Hilbert space $\mathcal{H},$ which commute with their derivative, must be functionally commutative.

Keywords

Compact operator, Spectral decomposition, Analytic projection, Functional commutativity

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, Vol. 15, Birkhauser, 1985.
• [3] R. Dautrey and Jacques-Louis Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 3, Spectral Theory and Applications. Springer 2000.
• [4] J.-C. Evard, “Conditions for a vector subspace E(t) and for a projector P(t) not to depend on t: ”, Lin. Alg. and Its Applications. 91 (1987), 121-131.
• [5] J.-C. Evard, “On matrix functions which commute with their derivative”, Lin. Alg. and Its Applications. 68 (1985), 145-178.
• [6] S. Goff, “Hermitian function matrices which commute with their derivative”, Lin. Alg. and Its Applications 36 (1981), 33-40.
• [7] T. Kato, Perturbation Theory for Linear Operators, Springer, 1980.
• [8] C.S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, Birkhauser, 2012.
• [9] F. Rellich, Perturbation Theory of Eigenvalue Problems, Institute of Mathematical Sciences, New York, 1950.