Positivity 11 (2007), 41–55
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010041-15, published online October 13, 2006
Optimal Weyl-type Inequalities for Operators
in Banach Spaces
Bernd Carl and Aicke Hinrichs
Abstract. Let (s
)beans-number sequence. We show for each k =1, 2,...
and n ≥ k + 1 the inequality
≤ (k +1)
between the eigenvalues and s-numbers of a compact operator T in a Banach
space. Furthermore, the constant (k +1)
is optimal for n = k + 1 and
k =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 num-
bers form a minimal multiplicative s-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
Since A. Pietsch developed the axiomatic approach to s-numbers of operators in
Banach spaces in [P1], there emerged an extensive literature dealing with inequal-
ities between eigenvalues and s-numbers as well as between several s-numbers.
Following the basic results on eigenvalue distributions in [JKMR], [KRT], and,
especially, of H. K¨onig, Pietsch (cf. [P3]) was able to prove an important inequality
between eigenvalues and Weyl numbers of operators in Banach spaces, see Section
3. Good constants in this inequality were given by H. K¨onig (cf. [K1, K2, K3]).
Various problems on inequalities between eigenvalues and s-numbers have arisen
(cf. [K3, K4, P4]).
In the present paper we prove inequalities between the product of absolute
values of the ﬁrst n eigenvalues and the product of ks-numbers s
, where n−k+1 ≤
Research of the second author was supported by the DFG Emmy-Noether grant Hi 584/2-3.