Estimating matching distance between spectra
by
Abdelkrim Nokrane
D\´epartement de Math\´ematiques, Universit\´e Cadi-Ayyad, B.P: 2390, Marrakech, Maroc

We show that if a, b are finite spectrum elements of an unital Banach algebra such that almost all convex combinations of a and b have a finite spectrum of cardinality is n, then the optimal matching distance between their spectra satisfies

D(s(a), s(b)) ≤ cn ( ∥a∥+∥b∥ )1-1/n ∥a-b∥1/n,

where c_n ≤ 16(1+1/n)n^1/n.

References

[1] B. Aupetit, A primer on Spectral Theory, Springer, 1991.

[2] R. Bhatia, L. Elsner and G. Krause, Bounds for the Variation of the Roots of a Polynomial and the Eigenvalues of a Matrix, Linear Algebra Appl. 142 (1990) 195-209.

[3] Y. Chen, A. Nokrane and T. Ransford, Estimates for the spectrum near algebraic elements, Linear Algebra Appl. 308 (2000) 153-161.

[4] G. Krause, Bounds for the variation of matrix eigenvalues and polynomial roots, Linear Algebra Appl. 208/209 (1994) 73-82.

[5] A. Nokrane and T. Ransford, Schwarz's Lemma for Algebroid Multifunctions, Complex Variables Theory Appl. 45 (2001) 183-196.

Date received: May 22, 2008