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The main result in this paper is the determination of the Fréchet derivative of an analytic function of a bounded operator, tangentially to the space of all bounded operators. Some applied problems from statistics and numerical analysis are included as a motivation for this study. The perturbation operator (increment) is not of any special form and is not supposed to commute with the operator at which the derivative is evaluated. This generality is important for the applications. In the Hermitian case, moreover, some results on perturbation of an isolated eigenvalue, its eigenprojection, and its eigenvector if the eigenvalue is simple, are also included. Although these results are known in principle, they are not in general formulated in terms of arbitrary perturbations as required for the applications. Moreover, these results are presented as corollaries to the main theorem, so that this paper also provides a short, essentially self-contained review of these aspects of perturbation theory.

Motivated by certain applications in numerical analysis
and, in particular, statistics, this paper deals with the Fréchet derivative of
an analytic function

Clearly

The perturbations

The
perturbation

A central theme
in perturbation theory concerns the perturbation of an isolated eigenvalue and
corresponding eigenprojection (see, e.g, the references mentioned
before). Some of the results are included, because they can be easily derived
from the main result on the Fréchet derivative by choosing a special function

As has already been mentioned in the beginning,

We will exclusively deal with infinite dimensional
Hilbert spaces and will not attempt to include the simpler finite dimensional
case in our formulation. The Fréchet derivative for arbitrary perturbations is
well known in the finite dimensional matrix case. This result and further
references can be found in the recent monograph by Bhatia [

Let us fix an arbitrary

The resolvent

The operators

Let

For

Applying a Neumann series expansion [

The upper bounds in (

It will be seen in Section

In this situation of commuting operators, Dunford and Schwartz [

Keeping the perturbation as before, we now restrict

Let the conditions of Theorem

Let us substitute the expansion (

The function

Next let us, for

For

Throughout this section, both

The region

For the Fréchet derivative of

The Fréchet derivative of

This follows by substitution of (

By Cauchy's integral formula

Some results
about the perturbation of

Under the assumptions (

In view of (

Next, let

Under the assumptions of Corollary

Let us first observe that

Regarding

For

Under the assumptions of Corollary

With the help of (

Let

All nonzero eignvalues of

The assumption that

In this section, we will sketch three applications: two in statistics and one in numerical analysis.

Let

It is the purpose to recover

Recently, there is an interest in certain econometric
models where the operator

An upper bound for this increase of the MISE can be
easily found from the results in Section

To find an upper bound for its MISE, let us first
observe that (

Hence, under suitable assumptions, estimation of the
kernel yields an extra term in the MISE of the input estimator which is of
order

Let

Next suppose that we are given a random sample

Because

For a precise definition, let

For an alternative description of these canonical
correlations, let us introduce the
operator

It is well known that the asymptotic distribution of
the eigenvalues and eigenfunctions of a random operator can be derived from the
asymptotic distribution of this random operator itself (see [

Result (

In Bakushinsky and Kokurin [

In their proof of this result, the authors need a
crucial upper bound. Under some additional assumptions, we want to derive this
upper bound as an immediate consequence of Theorem

Narrowing down the generality in Bakushinsky and
Kokurin [

The authors are grateful to the referee for some useful comments. For this research, D. S. Gilliam was supported by AFOSR Grant no. FA9550-04-1027 and F. H. Ruymgaart by NSF Grant no. DMS-0605167.