Abstract
This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface
𝔏
:
[
a
,
b
]
×
[
c
,
d
]
→
Φ
0
(
U
,
V
)
{\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)}
,
(
λ
,
μ
)
↦
𝔏
(
λ
,
μ
)
{(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)}
, depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where
Φ
0
(
U
,
V
)
{\Phi_{0}(U,V)}
stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see
[T. Kato,
Perturbation Theory for Linear Operators, 2nd ed.,
Class. Math.,
Springer, Berlin, 1995, Chapter 2, Section 5]).
A Trace Formula for the Index of B-Fredholm Operators