Optimal Weyl-type Inequalities for Operators in Banach Spaces.

Abstract??Let (s_n) be ans-number sequence. We show for eachk= 1, 2, . . . and n ?k+ 1 the inequalitybetween the eigenvalues ands-numbers of a compact operatorTin a Banach space. Furthermore, the constant (k+ 1)^1/2is optimal forn=k+ 1 andk= 1, 2, . . .. This inequality seems to be an appropriate tool for estimating the first single eigenvalues. On the other hand we prove that the Weyl numbers form a minimal multiplicatives-number sequence and by a well-known inequality between eigenvalues and Weyl numbers due to A. Pietsch they are very good quantities for investigating the optimal asymptotic behavior of eigenvalues. [ABSTRACT FROM AUTHOR]