Abstract.Let X be a Banach space, A a closed linear operator on X , and X\ , ... , A" isolated eigenvalues of A of finite multiplicity. If P is a projection on X such that X\,... ,Xn belong to the resolvent of the compression of A on the range of P it is easy to see that dimN(P) > max{dim N{X¡ -A): I dim NÍA) if Ap is injective.The collection of projections Poní with P(DiA)) ç DÍA) will be denoted by PaÍX) . The main result of this note is the following:Theorem. Let A be a closed linear operator in a Banach space X and let Xx, ... , Xn be isolated eigenvalues of A of finite (algebraic) multiplicity. Let Ci be a subset of C such that £2 ^ C ¿i«¿z £2 n (TÍA) = {Xx, ... , Xn\. If C £ C\£2 then there is a projection P £ PaÍX) such that £2 ç píAP), o(Ap) = tr(^)\£2u{C} and dim A(P) = max dim MX -A).
A€£2In [ 1, 2] Islamov proved a similar result for finite-dimensional perturbations of A. It is well known that many results on compact or finite-dimensional perturbations have analogues in terms of compressions to subspaces of finite codimension, see [5,6,7]. The theorem above is another example of this relationship.
Let
γ
(
S
)
\gamma (S)
denote the reduced minimum modulus of a linear operator
S
S
acting in a complex Banach space
X
X
, and let
I
I
denote the identity on
X
X
. In this paper it is shown that for a (not necessarily bounded) Fredholm operator
T
T
acting in
X
X
, the limit
lim
n
→
∞
γ
(
T
n
)
1
/
n
{\lim _{n \to \infty }}\gamma {({T^n})^{1/n}}
exists and is equal to the supremum of all positive numbers
δ
\delta
such that the dimension of the null space and the codimension of the range of
T
−
λ
I
T - \lambda I
are constant on
0
>
|
λ
|
>
δ
0 > |\lambda | > \delta
.
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