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\topmatter
\title\nofrills
Approximation of nonessential spectrum of transfer operators
\endtitle
\author
Viviane Baladi and Matthias Holschneider
\endauthor
\date
June 1998
\enddate
\address
V. Baladi: Section de Math\'ematiques, CH1211 Geneva 24,
SWITZERLAND
\endaddress
\email
Viviane.Baladi\@math.unige.ch
\endemail
\address
M. Holschneider:
Centre de Physique Th\'eorique (UPR 7061 CNRS) Luminy, Case 907
\newline
\phantom{vb M. Holschneider:} F13288 Marseille Cedex 9, FRANCE\newline
\phantom{vb} {\it and} IPGP, Laboratoire de g\'eomagn\'etisme,
F75252 Paris Cedex 5, FRANCE
\endaddress
\email
Matthias.Holschneider\@cpt.univmrs.fr
\endemail
\abstract
We give sufficient conditions
to approximate the ``nonessential'' spectrum
of a bounded operator $\LL$ acting on a Banach
space $\BB$ by part of the spectra of a sequence of compact
(or finite rank) operators $\LL_j=(\Id\Pi_j)\LL(\Id\Pi_j)$,
where $\Id\Pi_j$ is a suitable family of uniformly bounded
operators which approach the identity.
(By nonessential spectrum we mean here all the spectrum outside
of the disc of radius equal to the essential spectral
radius.) For this, we
combine the formulas
$$
\rho_{\text{ess}}({\LL}) =
\lim_{m \to\infty}(\limsup_{j\to\infty}({\inf})\\Pi_j{ \LL}^m\)^{1/m}=
\lim_{m \to\infty}(\limsup_{j\to\infty}({\inf})\ \LL^m\Pi_j\)^{1/m}\,
,
$$
for the essential spectral radius with nonstandard
perturbative results on the stability of
the nonessential spectrum of quasicompact operators.
We present concrete applications to transfer operators of smooth
expanding maps using multiresolution analysis
(large scale approximation projections).
\endabstract
\subjclass 41A30 42C15 47A10 47B38 58F19
\endsubjclass
\thanks
\phantom{vb} V.B. is partially supported by the Fonds National
Suisse de la Recherche Scientifique.
\endthanks
\endtopmatter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\head 1. Introduction
\endhead
Matrices, and more generally linear operators on infinitedimensional
vector spaces, are ubiquitous tools which permeate
pure and applied mathematics. A natural
problems, which has kept mathematicians
busy for centuries, is to determine, or at
least approximate, their spectrum (in infinitedimensional
situations, sometimes only a discrete part of it).
In this work, we are concerned with the infinitedimensional
(Banach space) situation, and we deal with bounded linear operators
which are not necessarily compact. Our main result (Proposition~3
in Section~2) is a list of conditions guaranteeing
that a subset of the eigenvalues of a sequence of
compact or finiterank operators $(\Id\Pi_j)\LL(\Id\Pi_j)$
(together with the corresponding
eigenspaces) converge to those eigenvalues of the original
operator $\LL$ which are outside of a disc containing the
essential spectrum. The simple proof combines a convenient
exact formula for the essential spectral radius (Theorem~1
from [H1]) with a nonstandard  and somewhat
unexpected ~ ~perturbative result (Theorem~2 from [BY]),
which had originally been used
to control the spectrum of randomly perturbed dynamical
systems.
Sequences of compact or finite rank operators $\Id\Pi_j$
for which our results
hold can sometimes be explicited in some cases
(sometimes via multiresolution analysis,
using wavelets).
In Section 3, we explain a specific dynamical systems setting where
our scheme works.
The linear operator there is the Ruelle transfer
operator associated to a differentiable uniformly
expanding dynamical system on a torus. (Transfer operators,
sometimes also called PerronFrobenius operators,
are very powerful tools to study the ergodic properties
of dynamical systems. We refer e.g.
to [R] and the references therein for the framework
of Section~3.) The Banach spaces
are Sobolev or H\"older spaces, and
the finiterank operators $\Id\Pi_j$ are constructed
using the Meyer [Me] orthonormal wavelet basis (it would be interesting
to actually run the algorithm on a computer).
An important and famous finitedimensional matrix
scheme used in ergodic theory of dynamical systems
(to approximate the physical, or ``SRB'' measure, together with its
rate of mixing) is the Ulam method. Recent numerical
and theoretical work has shown
that not only the maximal eigenvalue, but also further
spectral values of Ulam matrices approximate well
part the spectrum associated to various types of chaotic
dynamics (see in particular the sequence of papers
and effective algorithms of
Dellnitz and collaborators [DJ], and
the rigorous results of Hunt [Hu], and Froyland [Fr];
we also mention the recent paper
[MKY]~ see also [K1]  for similar approximation
results, together with quantified estimates on
the speed of convergence, finally [BIS] contains results obtained
using Theorem~2 below from [BY]).
It seems to us, however, that since Ulam matrices
are obtained by locally constant approximations, they
cannot describe the action of the dynamics on
observables smoother than H\"older or Lipschitz.
Our scheme, on the other hand, is applicable to a wider range of
smoothness classes.
\smallskip
We end this introduction by mentioning, in order of
expected difficulty, three directions for future research.
As soon as one proves that a mathematical
object can be approximated by a sequence, one obvious
question is the speed of convergence. For the case considered in
Section~3,
we believe that exponential speeds of convergence hold
(by analogy to the results in [KMY], e.g.).
A second natural problem consists in extending our
dynamical results from Section~3 to compact boundaryless
manifolds more general than the $n$torus $\Bbb T^n$.
This should be possible by developing and/or applying
the necessary multiresolution analysis.
Last, but definitely not least, we have limited ourselves
to uniformly expanding dynamical systems for which the
transfer operator has nice spectral properties when acting
on smooth function. When the dynamics is uniformly hyperbolic,
the inverse maps improve smoothness along unstable manifolds
but make functions less smooth along stable manifolds.
Although recent progress has been made in our understanding
of analytic
[Rg], but also differentiable [Li, Ki], hyperbolic settings, one
still does not have a good Banach
space framework for the transfer operator. Perhaps our approximation
scheme can be extended to the hyperbolic setting
via the use of ``directional'' Banach spaces.
(Further extensions to nonuniformly
hyperbolic dynamics would also be desirable.)
\smallskip
{\bf Acknowledgements:}
V.B. would like to thank Gilles Courtois and Gerhard
Keller for useful comments. She gratefully acknowledges the
hospitality of IHES where part of this work was done.
M.H. is thankful to the University of Geneva for its
kind hospitality.
%\bigskip
\head 2. Approximation of the discrete spectrum: two abstract results
\endhead
\smallskip
We first recall a few basic definitions and facts
(see [K] and [DS] for more information).
Let $(\BB, \\cdot \)$ be a Banach space, that will always
be assumed infinitedimensional. (Since we mostly have function
spaces in mind, we denote vectors in $\BB$ by $\varphi$, $\psi$ etc.).
Denote by $B(\BB)$ the set of bounded linear operators acting in
$\BB$ (noting $\\LL\$ for the operator norm of
$\LL \in B(\BB)$), by $K(\BB)\subset B(\BB)$ the ideal of compact
operators,
and by $F(\BB)\subset K(\BB)$ the ideal of finite rank operators.
For $\LL \in B( \BB)$ the {\it resolvent set} of $\LL$
is the set of complex numbers $z$ so that
$\LLz\, \Id:\BB \to \BB$ is an invertible operator with
a bounded inverse $(\LLz \, \text{Id})^{1} \in B(\BB)$.
The {\it spectrum} $\sigma( \LL)$
of $\LL$ is the set of $z\in \complex$
which are not in the resolvent set of $\LL$.
The {\it spectral radius} $\rho(\LL)$ of $\LL$ is
$$
\rho(\LL) =
\sup \{ z \text{ s.t. } z \in \sigma( \LL ) \} \, .
$$
As is well known, the spectral radius of $\LL$ can be obtained as the
following limit:
$$
\rho(\LL) = \lim_{m \to\infty} \ {\LL}^m \^{1/m} \, .
$$
An element $z\in \sigma( \LL)$ is an {\it eigenvalue} of $\LL$
if $\LLz\Id$ is not invertible. The {\it geometric multiplicity} of
an eigenvalue $z$ is
the dimension $1 \le m_1(z) \le \infty$ of its {\it eigenspace}
$\{ \varphi \in \BB \text{ s.t. } (\LL z) \varphi = 0\}$, and its {\it
(algebraic) multiplicity} is the dimension
$m_1(z) \le m_2(z) \le \infty$
of the generalised
eigenspace $\{ \varphi \in \BB \text{ s.t. } \exists m \ge 1 \, ,
(\LL z)^m \varphi = 0\}$. The supremum $1\le i(z) \le \infty$
of those $m$ which occur in the definition of the
generalised eigenspace of an eigenvalue $z$
is called the {\it index} of $z$.
The {\it essential spectral radius} $\rho_{\text{ess}}(\LL)$ of
$\LL$ is the largest number $\kappa\ge 0$ such that
any $\lambda\in \sigma( \LL)$ with modulus $\lambda > \kappa$
is an isolated eigenvalue of finite (algebraic) multiplicity of $\LL$.
We sometimes use the informal terminology ``nonessential spectrum''
or ``discrete spectrum'' to denote the spectrum of
$\LL$ outside of the disc of radius $\rho_{\text{ess}}(\LL)$.
There exist several
definitions for the {\it essential spectrum} of a linear bounded
operator.
{\it Browder's} [Br, Section 6] {\it essential spectrum} is the set
of those $z\in \complex$ such that at least one of
the three following possibilities holds: $z$ is a limit point
of $\sigma(\LL)$, or $(\LLz \, \Id) \BB$ is not closed,
or the generalised eigenspace $\{ \varphi \in \BB \text{ s.t. }
\exists m \ge 1\, ,
(\LL z)^m \varphi = 0\}$ has infinite dimension.
{\it Wolf's} [W] {\it essential spectrum} is the set
of those $z\in \complex$ such that
$\LL  z \, \Id$ is not Fredholm. In general the Wolf and
Browder essential spectrum do not coincide
(as noted in [N], the complement of the Browder essential
spectrum is the set of those components of the complement
of the Wolf essential spectrum which meet the resolvent set).
Our definition for the essential spectral radius
is consistent both with Browder's and Wolf's definition
of essential spectrum as we explain now.
Firstly, $\rho_{\text{ess}}(\LL)$ is the radius of the
smallest disc containing the Browder essential
spectrum (because of [Br, Lemma 17, p. 110]).
Secondly, $\rho_{\text {ess}} (\LL)$ is also
the radius of the smallest disc containing the Wolf essential
spectrum. This second property can be deduced from
two facts: on the one hand
$z$ is in the Wolf essential spectrum if
and only if $\LL z\, \Id$ is invertible modulo $K(\BB)$
if and only if $\LLz \Id$ is invertible modulo $F(\BB)$,
see [L, Chapter IX, Theorem 6]. On the other hand Nussbaum's
formula [N] states that the radius
$\rho_{\text{ess}} (\LL)$ of the smallest
disc containing the Browder essential spectrum coincides with
$$
\rho_{\text{ess}} (\LL)=\lim_{m \to\infty}(\inf\{\ \LL^m\KK\
\, \text{ s.t. }\, \KK \in K(\BB)\})^{1/m} \, .\tag{2.1}
$$
By the above equivalent formulations of the Wolf
spectrum, Nussbaum's formula can be modified to
$$
\rho_{\text{ess}}(\LL) = \lim_{m \to\infty}(\inf\{\\LL^m\FF\\,
\text{ s.t. } \, \FF \in F(\BB)\})^{1/m} \, .\tag{2.2}
$$
This is a nontrivial result since the set $F(\BB)$
of finite rank operators is not necessarily dense in the ideal
$K(\BB)$ of compact operators.
(The paper [F] of Fried was extremely useful in clarifying the
above points.)
We denote by $\BB^*$ the dual space of the Banach
space $\BB$, i.e., the space
of bounded linear functionals $\nu : \BB \to \complex$.
(Having in mind mainly complex measures and distributions we write
$\nu$, $\mu$ for elements of $\BB^*$.) For $\nu \in \BB^*$,
we use the notation $\nu(\varphi)=(\nu  \varphi)$.
For $\LL\in B(\BB)$, we write $\LL^*\in B(\BB^*)$, for
the dual operator defined by
$(\LL^* \nu  \varphi )= (\nu  \LL (\varphi))$.
Recall that operators of finite rank on $\BB$ can be written as
$$
\psi \mapsto \sum_{\alpha\in A} \nu_\alpha(\psi)\varphi_\alpha \, ,
$$
where $A$ is a finite index set, $\nu_\alpha\in \BB^*$, and
$\varphi_\alpha\in \BB$.
Our first abstract result is
a list of exact formulas (probably wellknown ``in spirit'')
for the essential spectral radius
(see [H1] for proofs).
They hold for Banach spaces for which
suitable families of bounded operators $\Pi_j$ converging to zero
and for which $(\text{Id} \Pi_j) \LL$ is compact exist.
Such families of operators can be constructed
via multiresolution analysis for many classical function spaces
(most notably Sobolev, but more generally Triebel, and partly
BesovH\"older classes), of
periodic functions on the torus $\Bbb T^n$, say
(see e.g. [H1, H2],
see also Section 3 below for applications to transfer
operators, where the $\Id\Pi_j$ are in fact projections).
\definition{Definition (Compact approximation of the identity for
$(\LL, \BB)$)}
A sequence of operators $\Id\Pi_j \in B(\BB)$, for $j \in \integer^+$,
is a compact uniformly bounded
approximation of the identity for $(\LL,\BB)$ if:
$$
\eqalign{
&(i) \
\exists \quad \const>0 \text{ s.t. } \\Pi_j\\leq \const\, ,
\forall j \in \integer^+\, ;
\cr\,
&(ii) \
(\Id  \Pi_j)\LL \text{ is compact,} \,
\forall j \in \integer^+\,;
}
$$
and
$$
(iii)\ \BB_0=\BB_0(\{\Pi_j\}) =
\{\varphi \in \BB\text{ s.t. } \lim_{j\to\infty}\Pi_j (\varphi )= 0\}
\text{ is dense in } \BB \, .
$$
\enddefinition
We shall also need a dual notion:
\definition{Definition ($*$compact approximation of the identity
for $(\LL,\BB)$)}
A sequence of operators $\Id\Pi_j \in B(\BB)$, for
$j \in \integer^+$,
is a $*$compact uniformly bounded
approximation of the identity for $(\LL,\BB)$ if it
satisfies $(i)$ together with:
$$
\eqalign{
&(ii^*)\
\LL (\Id  \Pi_j) \text{ is compact, } \, \forall j \in \integer^+\, ;
}
$$
and
$$
(iii^*)\
\BB_0^* =\BB_0^*(\{\Pi_j\})= \{ \nu\in \BB^*\text{ s.t. }
\lim_{j \to\infty}\Pi_j^* (\nu) = 0\} \text{ is dense in } \BB^* \, .
$$
\enddefinition
(Note that a sequence of operators $\Id\Pi_j$ is a $*$compact
uniformly
bounded approximation of the identity for $(\LL,\BB)$ if and only
if the $\Id\Pi_j^*$ form a compact uniformly bounded approximation
of the identity for $(\LL^*, \BB^*)$.)
Stronger results will hold for sequences of
operators satisfying certain hierarchical
constraints (which hold in particular in the setting of
Section 3):
\definition{Definition (Hierarchical compact approximation of the
identity
for $(\LL,\BB)$)}
A sequence of operators $\Id\Pi_j\in B(\BB)$
is called a {\it hierarchical} compact
(respectively $*$compact) approximation of
the identity for $(\LL,\BB)$ if it satisfies $(i)$,
$(ii)$ and $(iii)$ (respectively $(ii^*), (iii^*)$) together with
$$
(iv)\ \Pi_{j}\Pi_{j+1} = \Pi_{j+1}\Pi_j = \Pi_{j+1} \, , \quad
\forall \, j \in \integer^+ \, .\tag{2.3}
$$
The operator $\LL$ is said to
act {\it in scales}, respectively
$*${\it scales}, for $k$ with respect to the hierarchical compact
approximation of the identity, if there is $k \in \integer^+$ such that
$$
\eqalign{
&(v) \ \lim_{j \to\infty} \\Pi_{j+k}\LL(\Id\Pi_j)\
= 0
\, ;
\cr\,
\text{ respectively }
\cr
&(v^*) \
\lim_{j \to\infty}\(\Id\Pi_j)\LL\Pi_{j+k}\
= 0 \, .
} \tag{2.4}
$$
If $(v)$ (respectively $(v^*)$) in
\thetag{2.4} holds for $k=0$ then $\LL$ is said to
act {\it exactly} (respectively {\it *exactly})
{\it in scales} on $\{\Pi_j\}$.
\enddefinition
\proclaim{Theorem 1 (Holschneider [H1], 1996)}
Let $\LL\in B(\BB)$. Suppose there is a
compact uniformly bounded approximation of the identity
$\{\Id\Pi_j \}$ for $(\LL, \BB)$.
(i.e., satisfying $(i, ii, iii)$).
Then $\BB_0(\{\Pi_j\})=\BB$, and
$$
\rho_{\text{ess}}(\LL) =
\lim_{m \to\infty}(\lim\sup_{j \to\infty}\\Pi_j \LL^m\)^{1/m}
=
\lim_{m \to\infty}(\liminf_{j\to\infty}\\Pi_j \LL^m\)^{1/m} \,
.\tag{2.5}
$$
If there is $\{\Id\Pi_j \}$ a
$*$compact uniformly bounded
approximation of the identity in $(\LL,\BB)$
(i.e., satisfying $(i,ii^*,iii^*)$),
then $\BB_0^*(\{\Pi_j\})=\BB^*$ and
$$
\rho_{\text{ess}}({\LL}) =
\lim_{m \to\infty}(\limsup_{j\to\infty}\\LL^m\Pi_j\)^{1/m}
= \lim_{m \to\infty}(\liminf_{j\to\infty}\\LL^m\Pi_j\)^{1/m} \,
.\tag{2.6}
$$
If there is a hierarchical compact
(or $*$compact) approximation of
the identity $\{\Id\Pi_j\}$,
and if there is $k \in \integer^+$
such that $\LL$ acts in scales for $k$ on $\{\Pi_j\}$
(i.e., $(i, iv)$
and either $(ii,iii,v)$ or $(ii^*iii^*,v^*)$ hold), then any of the
four limits in \thetag{2.52.6} coincide with
$\rho_{\text{ess}}(\LL)$.
\endproclaim
\remark{Remark 1}
In the last assertion of the theorem,
if $\Pi_j$ satisfies $\\Pi_j\ \leq 1$,
then the interior limit actually exists. Indeed, for any bounded
$\AA$
we have by $(iv)$
$$
\\AA \Pi_j\ = \\AA\Pi_{j1}\Pi_j\ \leq \\AA\Pi_{j1}\ \, .
$$
Therefore the sequence $\\AA \Pi_j\$ is monotonically decreasing. The
same
argument applies to $\\Pi_j \AA\$.
\endremark
\remark{Remark 2}
If the sequence $\Pi_j$ satisfies only $(i)$ and $(ii)$
(respectively $(ii^*)$) then the Nussbaum
formula \thetag{2.1} clearly gives the upper bounds
in \thetag{2.5}, respectively \thetag{2.6}
(see also the proof in the Appendix).
In some applications neither $(iii)$ nor $(iii^*)$
can be assumed to hold, but an assumption similar
to the one appearing in [K2], and that we state
now, may be used. (Note that a key idea to obtain
lower bounds for the essential spectral radius by
constructing almost eigenvalues was contained
in the beautiful short paper [Ma] of Mather.)
Suppose that $\Pi_j$ and
$\LL$ are such that $(i, ii)$ hold and
that there is a double sequence of infinite
dimensional closed spaces $V^m_j \subset \Pi_j \BB$,
for $j, m \in \integer^+$ and a constant $C > 0$ so that
for each fixed $m$ and all sequences $\varphi_j^m \in V_j^m$ with
$\\varphi_j \=1$ we have
$$
\limsup_{j \to \infty}
\\LL^m \varphi_j \\ge C \limsup_{j \to \infty} \\LL^m \Pi_j\\,
.\tag{2.7}
$$
Then
$$
\rho_{\text{ess}} (\LL) = \lim_{m \to \infty} (\limsup_{j\to \infty}
\\LL^m \Pi_j\)^{1/m} \, .
$$
(See [H2, Theorem 3.3, and also
Section~9] for an analysis of transfer operators
acting on homogeneous
$(s+\alpha)$H\"older spaces, with
$0 < \alpha < 1$ and $s \in \integer^+$,
using condition \thetag{2.7}.)
\endremark
\medskip
The idea to use families of projection operators to exploit the
essential
spectrum is not new. For example, Persson [Pe] gives a formula for
the lower bound of the essential spectrum of a selfadjoint operator
in $L^2(\real^n)$ by using restrictions of the operator to complements
of compact sets of $\real^n$. More recently, Keller [K3] developed
a theory of quasinuclear operators and applied it to
construct dynamical Fredholm determinants associated
to transfer operators in an analytic setting (his Proposition~2.2
is related to our Theorem~1, but it requires additional assumptions
on the approximating projections, in particular the nontrivial
hypothesis [K3,(2.12)]).
\bigskip
Theorem~1 will be used in combination with the following result
on spectral approximations (which was originally proved in
view of understanding small stochastic perturbations):
\proclaim{Theorem 2 (BaladiYoung [BY], 1993)}
For $\LL \in B(\BB)$, let $\LL_j \in B(\BB)$
be a sequence of operators with
$$
\\LL_j \\le \const \, , \forall \ j \in \integer^+\, \text{ and }
\lim_{j \to \infty} \LL_j (\varphi) = \LL (\varphi)\, ,
\forall \, \varphi \in \BB \, .
$$
Assume that for any $\kappa > \rho_{\text{ess}}(\LL)$,
there is $m_0 \in \integer^+$ such that for
each $m \ge m_0$,
there is $j_0(m)$ such that for all $j \ge j_0(m)$
$$
\ \LL^m  \LL^m_j \ \le \kappa^m \, .\tag{2.8}
$$
Then for any $\kappa > \rho_{\text{ess}}$ there is $j_0(\kappa)$ such
that
the essential spectral radius $\rho_{\text{ess}} (\LL_j) < \kappa $
for each $j \ge j_0$.
Furthermore, writing $X$, respectively
$X_j$ (with $j\ge j_0$), for the (finitedimensional)
direct sum of the generalised
eigenspaces associated to the eigenvalues of $\LL$,
respectively $\LL_j$, of modulus
larger than $\kappa$, and denoting the corresponding
spectral projections by $\Bbb P$, $\Bbb P_j$, we have
\roster
\item
The norms $\\Bbb P\Bbb P_j\$ tend to zero as $j \to \infty$.
More precisely, there is $\delta > 0$
such that
$$
\\Bbb P\Bbb P_j\\le\exp^{\delta m(j)} \ , \, \forall \, j\ge j_0 \,
,
\tag{2.9}
$$
where
$$
m(j)=\max \{ m \in \integer^+ {\text{ s.t. }} j \ge j_0(m)\} \, .
$$
\item
The Hausdorff distance between the spectrum of $\LL_X$ and
that of $\LL_j_{X_j}$ tends to zero as $j \to \infty$.
More precisely, for all
$j \ge j_0$
$$
\text{HD} (\sigma(\LL_X), \sigma(\LL_j_{X_j}))
\le \const \, (C^{(m(j))}_X(j) + C^{(1)}_X (j)) ^{1/d} \, ,\tag{2.10}
$$
where $d\ge 1$ is the maximum of the indices of eigenvalues of
$\LL$ of modulus larger than $\kappa$ and
$$
C^{(m)}_X(j) =\max \Sb \varphi \in X \\ \\varphi \=1 \endSb
\ (\LL^m_j  \LL^m) \varphi\\, .\tag{2.11}
$$
(Note that $C^{(m(j))}_X (j) \le \const \kappa^{m(j)}$ for
$j \ge j_0$.)
\item
If $\varphi \in X$ is an eigenvector for $\LL$ and an eigenvalue
$\lambda$ of algebraic multiplicity
$d_a$ and index $d_i\ge 1$ , then for each $j \ge j_0$ the operator
$\LL_j$ has
$1\le \ell\le d_a$ eigenvectors $\varphi_{i,j}$ for eigenvalues
$\lambda_{i,j}$
($i=1, \ldots, \ell$),
with sum over $i=1,\ldots, \ell$ of the algebraic
multiplicities of $\lambda_{i,j}$ equal to $d_a$, and
$$
\max_{1\le i \le \ell}
\max (\lambda\lambda_{i,j} , \\varphi\varphi_{i,j}\)
\le \const (C^{(m(j))}_X(j)+ C^{(1)}_X (j)) ^{1/d_i} \, .\tag{2.12}
$$
\endroster
\endproclaim
Theorem 2 is obtained by combining Lemmas~1, 2 and 3 from Section~2
in [BY], noting that assumptions (A.1) and (A.3) there follow from
the definition of the essential spectral radius, while assumption
(A.2) in [BY] is just our hypothesis \thetag{2.8}.
\bigskip
The main new result of this paper is obtained by putting together
Theorem 1 and Theorem 2:
\proclaim{Proposition 3}
For $\LL\in B(\BB)$, let $\{\Id\Pi_j\}$
be either both a compact {\it and} a $*$compact uniformly bounded
approximation of the identity,
or a compact {\it or} $*$compact uniformly bounded
hierarchical approximation of the identity on which
$\LL$ acts in scales for some $k$,
i.e., either (a) [$(i,ii, ii)$ {\it and} $(i,ii^*, iii^*)$],
{\it or} (b) $(i,ii, iii, iv, v)$, {\it or}
(c) $(i,ii^*, iii^*, iv, v^*)$ hold.
Assume further that for some large enough $M \in \integer^+$,
$\LL^M$ acts exactly in scales
on the sequence $\Pi_j$
(More precisely: in case (a) we assume [$(v)$ {\it and} $(v^*)$] for
$k=0$,
in case (b) we assume $(v)$ for $k=0$, and in case
(c) the convergence $(v^*)$ for $k=0$.)
Then for any fixed $\kappa > \rho_{\text{ess}}(\LL)$,
the spectrum and generalised
eigenspaces of the compact operators
$$
\LL_j = (1\Pi_j) \LL (1\Pi_j)
$$
outside of the disc of radius $\kappa$
converge to those of $\LL$
in the sense of \therosteritem{1}\therosteritem{3}
of Theorem~2 (including bounds \thetag{2.92.12}).
\endproclaim
Obviously, Theorems 1 and 2 as well as Proposition~3
are of interest only
when $\rho_{\text{ess}} (\LL) < \rho(\LL)$ (especially
when there are ``many'' eigenvalues between
$\rho_{\text{ess}} (\LL)$ and $\rho(\LL)$).
Proposition 3 is especially interesting if
$\LL(1\Pi_j)$ or $(1\Pi_j)\LL$ are finite rank operators.
Section ~3 contains specific
examples where both conditions in this paragraph are satisfied.
\smallskip
\demo{Proof of Proposition 3}
We start by replacing $\LL$ by $\MM=\LL^M$ and $\LL_j$
by $\MM_j=(1\Pi_j)\LL^M(1\Pi_j)$, at the end of the proof we shall
see how to recover results about $\LL$ itself.
(Note that if $(ii)$ respectively $(ii^*)$
is satisfied for $\LL$ then it also holds for $\LL^M$.
Iterating assumption $(iv)$ does not lead into difficulties.)
The only thing which requires checking is assumption \thetag{2.8}
of Theorem~2. For this, we observe that the
conditions for the strongest statement
of Theorem~1 are satisfied. Thus
$$
\rho_{\text{ess}} (\LL^M)
= \lim_{m \to \infty} (\limsup_{j \to \infty} \ \Pi_j \MM^m\)^{1/m}=
\lim_{m \to \infty} (\limsup_{j \to \infty} \ \MM^m\Pi_j \)^{1/m}\, .
$$
In other words, for any $\tilde \kappa$
with $\tilde \kappa > \rho_{\text{ess}}(\LL^M)$,
there is $m_0\ge 1$ such that for
all $m\ge m_0$ and each $\epsilon > 0$ there is
$j_1(m)$ such that for all $j \ge j_1(m)$
$$
\\Pi_j \MM^m \ \le \tilde \kappa ^{m} +\epsilon \text{ and }
\\MM^m \Pi_j \ \le \tilde\kappa ^{m} +\epsilon \, .\tag{2.13}
$$
(We may, and will, take $\epsilon=\tilde\kappa^{m}$.)
For a constant $\widehat C> 0$ (independent of
$j$ and $m$) to be defined
below, and for each fixed $m \ge m_0(\tilde\kappa)$, we set ,
$$
\delta =\delta(m)= {1\over (m1) (\widehat C \\MM\)^{m1}} \, ,
$$
and take $j_2\ge \max(j_0(m), j_1(m))$ large enough so
that
$$
\ \Pi_j \MM (1\Pi_j)\ \le \delta \, \tilde\kappa^m \tag{2.14}
$$
for all
$j \ge j_2$ (recall that $\MM$ acts exactly in scales).
We then have the bounds
$$
\eqalign
{
\\MM^m\MM^m_j\&= \\MM^m  [(1\Pi_j)\MM(1\Pi_j)]^m\\cr
&=
\\MM^m  (1\Pi_j) \MM^m (1\Pi_j) \cr
&\qquad\qquad

\sum_{k=1}^{m1} (1\Pi_j) \MM^k (2\Pi_j) \Pi_j \MM
[(1\Pi_j)^2\MM]^{mk1}
(1\Pi_j)\\cr
&\le
\ \Pi_j \MM^m \Pi_j +
(1\Pi_j) \MM^m \Pi_j + \Pi_j \MM^m (1\Pi_j) \\cr
&\qquad\qquad\qquad+ (m1) \delta \tilde\kappa^m (\widehat C
\\MM\)^{m1}
\cr
&\le
(\sup_j \\Pi_j\)^2 2 \tilde\kappa^m +
2 (\sup_j \\Pi_j\) (1+\sup_j \\Pi_j\) 2\tilde\kappa^m \cr
&\qquad\qquad\qquad
+\tilde\kappa^m \, ,
}
$$
(we used $\hat C = (1+\sup_j \\Pi_j\) (2+\sup_j \\Pi_j\)^2$)
obtaining an estimate
which is clearly equivalent to \thetag{2.8} for $\MM=\LL^M$
since all constant factors are uniform in $m$.
\smallskip
We now explain how to prove our statements for the
original operator $\LL$.
First note that, since $\rho_{\text{ess}} (\LL^M) =
(\rho_{\text{ess}} (\LL))^M$, if $\tilde \kappa > \rho_{\text{ess}}
(\LL)$
then $\kappa=
\tilde \kappa^M > \rho_{\text{ess}} (\LL^M)$.
Also, since the spectral projection associated to
$\LL$ and an eigenvalue $\lambda$ coincides with
that of $\LL^M$ and $\lambda^M$,
the assertion regarding $\\Bbb P \Bbb P_j\$ in \therosteritem{1}
of Theorem 2 is clearly valid.
Since $X$ is finitedimensional, we may conclude
by applying perturbation theory of finite dimensional
matrices
as in the proof of Lemma 3 in [BY, pp.~361362].
(Note that algebraic multiplicity is preserved
when taking powers of an operator, whereas the index, which is the
size of the largest Jordan block, may only
decrease.)
\qed
\enddemo
\smallskip
{\bf Nonconvergence of the determinants}
\smallskip
In Section~3 we shall discuss situations where
Proposition~3 furnishes us with a sequence of finite rank operators
$\LL_j$ whose eigenvalues of large enough modulus (together with the
corresponding
generalised eigenspaces) converge to the corresponding data for
a given noncompact bounded operator $\LL$. It is tempting
to consider the associated sequence of ``Fredholm determinants''
$d_j(z)=\det (\Id  z \LL_j)$. The functions
$d_j(z)$ are polynomials, of degree increasing
with $j$, and whose zeroes of small enough modulus converge
as $j\to \infty$ to the inverse eigenvalues of $\LL$.
In many situations, however, the functions $d_j(z)$ do {\it not
converge} as holomorphic functions in any disc. (The ``small (essential)
spectrum'' of $\LL$ seems to be the reason for this lack of
convergence.)
Such counterexamples can be obtained in the framework of
smooth expanding maps of the circle, by considering a sequence
$\Pi_j$ associated to finite smoothness.
\smallskip
\bigskip
%\newpage
\head 3. Dynamical transfer operators and wavelet approximations: an
application
\endhead
A typical situation where Theorem 1 applies is when $\LL$ is the
transfer
operator associated to a smooth expanding map of the $n$dimensional
torus
$\Bbb T^n$, the Banach space is
a space of smooth distributions,
a Sobolev space, or more generally
a Triebel space (see [H2]),
and the $\Pi_j$ are obtained by orthogonal
projections on a multiresolution analysis
(see [Me]).
We now give precise definitions and statements, without striving
for the fullest generality.
\medskip
{\bf Definition of the transfer operator}
\smallskip
Let $f : \Bbb T^n \to \Bbb T^n$ be a $C^\infty$ uniformly expanding
map (i.e., there exists $\gamma > 1$ with
$\Df v \\ge \gamma \v\$ for each $x \in \Bbb T^n$
and each $v \in T_x \Bbb T^n$). Let $g : \Bbb T^n \to \complex$
be a $C^\infty$ function. The Ruelle transfer operator $\LL$ associated
to the pair $(f, g)$ is defined (e.g. on $L^2(\Bbb T^n)=L^2(\Bbb T^n,
d\mu)$,
with $\mu$ Lebesgue measure on the torus) by
$$
\LL \varphi (x) =\sum_{f(y)=x}
{g(y) \varphi(y) \over \det Df(y)} \, .\tag{3.1}
$$
(The choice $g\equiv1$ leads to the usual PerronFrobeniustype transfer
operator, for which there exists a maximal eigenfunction
which is the density of the unique absolutely continuous invariant
probability measure for $f$.)
\medskip
{\bf Definition of the hierarchical approximation
of the identity}
\smallskip
We now give an explicit example of operators $\Pi_j$ that will satisfy
the assumptions of Proposition~3.
We start with an orthonormal wavelet basis for $L^2(\real^n)$,
e.g., the Meyer basis obtained from multiresolution
analysis of $L^2(\real^n)$ (see [Me]).
This is a set of $2^n1$ functions
$\psi_k$ ($k=1, \ldots, 2^n1$), in the Schwartz space
$S(\real^n)$ of rapidly decreasing functions
$$
S(\real^n)= \{ \varphi \in L^2(\real^n) \text{ s.t. }
\sup_{\real^n } x^m \partial^p \varphi(x)  < \infty \, ,
\forall p, m\in (\integer^+)^n \}
$$
such that all moments of each $\psi_k$ vanish (i.e.,
$\int_{\real^n} x^m \psi_k(x) \, d\mu =0$), and
such that
$$
\{ \psi_k^{j, \ell} = 2^{jn/2} \psi_k (2^j x\ell) \text{ s.t. }
k=1, \ldots, 2^n1 \, , j \in \integer, \ell \in \integer^n \}
$$
is an orthonormal basis of $L^2(\real^n)$.
Moreover, the $\psi_k$ have compactly supported
Fourier transforms.
Let $\PP : S(\real^n )\to L^2(\Bbb T^n)$ be the periodisation
operator
$$
\PP \varphi (x) = \sum_{\ell \in \integer^n}\varphi(x+\ell) \, .
$$
Then we get an orthonormal basis of $L^2(\Bbb T^n)$ by considering
$$
\{
P \psi^{j,\ell}_k \, , k =1, \ldots, 2^n1\, , j \in \integer^\, ,
\ell \in \{ 0, \ldots 2^{j} 1\}^n
\} \cup \{ \psi_0 \equiv 1\} \, .
$$
Finally, we define the sequence $\Pi_j : L^2(\Bbb T^n) \to L^2(\Bbb
T^n)$,
for $j \in \integer^+$,
by setting $\Id\Pi_j$ to be the (finiterank)
orthogonal projection on the finite dimensional space
generated by the $\PP \psi_k^{j', \ell}$
for $k =1, \ldots, 2^n1$, $j \le j' \le 0$, and
$\ell \in \{ 0, \ldots , 2^{j'}1\}^n$:
$$
\eqalign
{
(\Id\Pi_j) (\varphi)
&= \int_{\Bbb T^n} \varphi \, \psi_0 \, d\mu\cr
&\qquad\qquad +
\sum_{k =1}^{ 2^n1} \sum_{j'=j}^ 0 \sum_{\ell \in \{ 0, \ldots ,
2^{j'}1\}^n}
\PP \psi_k^{j', \ell} \cdot
\int_{\Bbb T^n} \varphi \, \overline{\PP \psi_k^{j', \ell}}\, d\mu
}\tag{3.2}
$$
\smallskip
{\bf Two (scales of) Banach spaces}
\smallskip
We shall consider the transfer operator $\LL$
acting on two scales of Banach spaces (the sequence
of projections just introduced will work for both).
The first is the wellknown scale of Sobolev spaces
$H^s(\Bbb T^n)$, for fixed $s \in \real^+$, i.e.,
$$
\eqalign{
H^s(\Bbb T^n) = \{ \varphi \in L^2(\Bbb T^n) \text{ s.t. }
\\varphi\_{H^s}:=
\sqrt{
\hat \varphi(0)^2+
\sum_{m \in \integer^n} m^{2s} \hat \varphi(m)^2 } <\infty \}
\, . }
\tag{3.3}
$$
where we used the notation $\hat \varphi(m)$,
for the Fourier coefficients of $\varphi$, and we write
$m=\sqrt{\sum_i m_i^2}$.
The second is derived from
the H\"older  Zygmund scale of spaces $\Lambda^\alpha(\Bbb T^n)$.
This is the space of periodic functions $f\in C^{[s]}(\Bbb T^n)$,
for $s=[s]+\{s\}$,
$[s]\in\Bbb N$, $\{s\}\in[0,1)$, for which for all multiindices
$\beta$,
$\beta= [s]$ we have
$$
\Vert \varphi\Vert_\infty + \sup_{\beta \, , x \, , y}
{\partial^\beta \varphi(x)\partial^\beta
\varphi(y)\overxy^{\{s\}}} < \infty\tag{3.4}
$$
The lefthand side defines the norm of $\Lambda^s(\Bbb T^n)$
Here we suppose $s\notin\Bbb N$, otherwise we have to use
the Zygmund spaces (see, e.g., [Tr] for a definition).
In Lemma~4 below we will use
the closure of $C^\infty(\Bbb T^n)$ in $\Lambda^s(\Bbb T^n)$,
denoted by $\lambda^s(\Bbb T^n)$.
The point is that both Sobolev and
H\"older scales of spaces (together with many others) are well
characterised through wavelet coefficients. More precisely, a
distribution
$\varphi$ is in $H^s(\Bbb T^n)$ if and only if its {\it wavelet
coefficients}
$$
\alpha^{j,\ell}_k:=({\Cal P}\psi^{j,\ell}_k  \varphi)
$$
satisfy
$$
\sqrt{\sum_{j,k,\ell}
2^{2 j s}  \alpha^{j,\ell}_k ^2 }< \infty \, .
$$
The lefthand side defines a norm equivalent to the standard Sobolev
norm.
The scale $\Lambda^s(\Bbb T^n)$ is characterised in wavelet space via
$$
\sup_{j,k,\ell} 2^{s j}  \alpha^{j,\ell}_k  < \infty
$$
Again we have equivalence of norms.
The closed subspace $\lambda^s(\Bbb T^n)$ consists of precisely those
functions
for which, in addition,
$$
\lim_{j\to\infty} \max_{\ell,k} 2^{s j}  \alpha^{j,\ell}_k = 0 \, .
$$
The projections $\Id\Pi_j$ from \thetag{3.2} are finiterank
on both the Sobolev and H\"older spaces.
Since our transfer operator $\LL$ is associated with a
smooth map $f$ and weight $g$, it is also bounded on
these spaces (see [H1]). As the following key
technical lemma shows, we may apply Proposition~3:
\proclaim{Lemma 4} {\bf (Exact scaling for suitable Banach spaces)}
Let $\LL$ be a transfer operator
associated to a smooth expanding map $f$ and a smooth weight
$g$ as in \thetag{3.1}, acting on
a Banach space $\BB=H^s(\Bbb T^n)$ or
$\BB = \lambda^s(\Bbb T^n)$. Let $\Id \Pi_j$ be the sequence
of finiterank projections defined by \thetag{3.2}.
Then $\Pi_j$ and $\LL$ satisfy properties $(i, ii, iii, iv, v)$.
Furthermore, there is $M \in \integer^+$ such that
$\LL^M$ acts exactly in scales on the hierarchical sequence
$\Pi_j$.
\endproclaim
\smallskip
\demo{Proof of Lemma 4}
\smallskip
Assertions $(ii)$ and $(iv)$ are obvious consequences of the
definitions. Properties $(i)$ and $(iii)$ follow from the above
wavelet
characterisation of the scales $H^s(\Bbb T^n)$ and $\lambda^s(\Bbb T^n)$
in wavelet space.
(Note that property $(iii)$ does not hold for $\Lambda^s(\Bbb T^n)$.)
To prove the exact scaling property we recall a result proved in
[H1, Sections 78].
There it was shown that the transfer operator may be written as
$$
{\Cal L} = {\Cal L}^\partial + {\Cal R} \, ,
$$
where $\Cal R$ is smoothing: it is
continuous from $H^r(\Bbb T^n)$ to
$H^{r+1}(\Bbb T^n)$, and from $\lambda^r(\Bbb T^n)$ to
$\lambda^{r+1} (\Bbb T^n)$ for all $r>1$.
${\Cal L}^\partial$
is a linearised version of $\Cal L$, its precise definition can
be found in [H1, Lemma 7.3].
(Note that the operators analogous to $\Cal L^\partial$
and $\RR$ in Sections 78 of [H1] were defined in
the wavelet coordinates, while here we work with their
versions in the
original space $\Bbb T^n$; also, [H1] considered the
non periodic case of $\real^n$, this does
not lead to difficulties.) It suffices to recall that $\Cal L^\partial$
is
completely determined by the derivatives of the
dynamics $f$ at all points of $\Bbb T^n$.
In particular, since, by hypothesis $\Cal L$ is obtained
by composing with uniform contractions,
${\Cal L}^\partial$ maps functions supported by a disk in Fourier space
to functions supported by a $\gamma$times smaller disk,
where $\gamma >1 $ is our bound for the expansion rate of the dynamics.
On the other hand, the image
of $(\text{Id}  \Pi_j)$ consists of functions supported by a disk
of radius $\leq c\,2^j$ in Fourier space, whereas the image of
$\Pi_j$ is supported outside a disk of radius $C\, 2^j$ with some
$C>c>0$. Upon replacing ${\LL}$ by
${\LL}^m$ ($m$ depends on $C/c$) we see that
$$
\Pi_j ({\LL}^m)^\partial (\hbox{Id}  \Pi_j)= 0 \, .
$$
It remains to analyse the effect of
$\RR$. Since ${\Cal R}$ is acting in scales, it is
continuous from $H^{s1}(\Bbb T^n)$ to
$H^{s}(\Bbb T^n)$ and from $\lambda^{s1}(\Bbb T^n)$ to
$\lambda^s(\Bbb T^n)$.
It can thus be written as ${\RR}= \Gamma {\RR}^\prime$,
where $\Gamma$ is
defined via $\Gamma : {\Cal P}\psi^{j,\ell}_k \to 2^{j}{\Cal
P}\psi^{j,\ell}_k$.
Thus ${\RR}^\prime$ is continuous from $H^{s}(\Bbb T^n)$ to
$H^{s}(\Bbb T^n)$. Now, by definition
$$
\Vert \Pi_j \Gamma\Vert_{H^s(\Bbb T^n)} =
\Vert \Pi_j \Gamma\Vert_{\lambda^s(\Bbb T^n)} \le
\text{const } \,
2^{j} \to 0 \quad (j\to\infty) \, ,
$$
and the exact scaling property follows.
\qed
\enddemo
\smallskip
\remark{Remark 3}
For $y\in \Bbb T^n$ we introduce the bounded linear operator
$\DD_{f,y}$
acting either on the Sobolev space $H^s( \real^n)$
or the derived from H\"older space $\lambda^s(\Bbb T^n)$ (defined
analogously
to \thetag{3.33.4}) by
$$
\DD_{f,y} \varphi = {\varphi \circ D (f^{1}_{f(y)})\over
\det Df(y)} \, , \tag{3.5}
$$
(where the inverse branch $f^{1}_{f(y)}$ is unambiguously defined
by $f^{1}(f(y))=y$). For
each $m\ge 1$ an operator $\DD_{f^m,y}$ can
be defined similarly as in \thetag{3.5}.
In fact, Holschneider [H1, Theorems 21.2.2]
applies Theorem~1 in wavelet coordinates to show a
more explicit formula for the essential spectral radius. The following
bounds are easy consequences of his formula:
$$
\rho_{\text{ess}} (\LL)
\le
\cases
\lim_{m \to \infty}
\biggl (
\sup_{x \in \Bbb T^n}
\sum_{f^m(y)=x} g^{(m)} (y) \, \\DD_{f^m, y} \_{\lambda^s(\real^n)}
\biggr )^{1/m} &\cr
\qquad\qquad\qquad \qquad\qquad \text{ for }
\BB=\lambda^s(\Bbb T^n)\, , & \cr
\lim_{m \to \infty}
\biggl (
\sup_{x \in \Bbb T^n}
\sum_{f^m(y)=x} g^{(m)} (y)^2 \, \\DD_{f^m, y} \^2_{H^s(\real^n)}
\biggr )^{1/(2m)} &\cr
\qquad\qquad\qquad\qquad\qquad \text{ for }
\BB=H^s(\Bbb T^n)\, , & \cr
\endcases
\tag{3.6}
$$
(for $m \ge 1$, we write
$g^{(m)}(x)=\prod_{i=0}^{m1} g(f^i (x))$).
Using that each inverse branch of $f^m$ is a contraction
by $\gamma^{m}$, it is not difficult to show that for any $y$
$$
\\DD_{f^m, y} \_{\lambda^s(\real^n)}\le \gamma^{ms}
\text{ and, for } s> {n \over 2} \, ,
\\DD_{f^m, y} \_{H^s(\real^n)}\le \gamma^{m(sn/2)}
\, .\tag{3.7}
$$
For $n=1$, the bound obtained from
\thetag{3.6} and the refined version
$\\DD_{f^m, y} \_{\lambda^s(\real^n)}\le (f^m)'(y)^{s}$ of
\thetag{3.7}
is similar to the onedimensional
exact formula for $\Lambda^s([0,1])$ with $0 < s < 1$ in [BJL].
(See also [CI] for $s\ge 1$, and, in a
onedimensional bounded variation setting, [K2], for earlier
and different expressions.) CampbellLatushkin [CL] and
GundlachLatushkin [GL] have recently obtained,
via a different (Oseledec theorem) approach,
other exact
expressions of the
essential spectral radius in higher
dimensional smooth expanding settings.
If $g$ is nonnegative, the spectral radius of $\LL$ acting on $C^s$
functions
(for any $s \ge 0$) is just (see [R])
$$
\rho (\LL)
=\lim_{m \to \infty}
\biggl (
\sup_{x \in \Bbb T^n}
\sum_{f^m(y)=x} {g^{(m)} (y) \over \det Df^m(y)}
\biggr )^{1/m} \, .
$$
Therefore, since there is an eigenvalue equal to $\rho(\LL)$
with a nonegative eigenfunction in $C^\infty(\Bbb T^n)\subset H^s(\Bbb
T^n)
\cap \lambda^s(\Bbb T^n)$ (use that $f,g$ are
$C^\infty$, see [R]),
the essential spectral radius of $\LL$ on $\lambda^s(\Bbb T^n)$
satisfies
$$
\rho_{\text{ess}} (\LL) \le \rho(\LL)/\gamma^s < \rho(\LL)
$$
(this double inequality was proved by Ruelle in [R] for $\LL$ acting
on $C^s(\Bbb T^n)$). Similarly, for $s > n/2$,
the essential spectral radius of $\LL$ on $H^s(\Bbb T^n)$
satisfies
$$
\rho_{\text{ess}} (\LL) \le \rho(\LL)/\gamma^{sn/2} < \rho(\LL) \, .
$$
\endremark
\remark{Remark 4}
Ordinary Fourier analysis could be used in
the Sobolev space framework, in particular, the analogue
of Lemma~4 would be true. However, Fourier
series would not be suitable for the H\"older space analysis,
or for other Banach spaces in which multiresolution
analysis and wavelets are applicable. Also, in situations where
Fourier and wavelet analysis are both applicable,
numerical algorithms based on wavelets usually converge
much faster. (See, e.g., [Me].)
\endremark
\remark{Remark 5}
Because our $C^\infty$ assumptions on $(f,g)$,
our transfer operator $\LL$ preserves any H\"older
or Sobolev space. We have chosen to work with
an orthonormal wavelet basis which is $r$regular
for each $r > 0$, and which can thus be applied to
approximate the ``nonessential'' spectrum of
$\LL$ on any $H^s(\Bbb T^n)$. Since the essential
spectral radius of $\LL$ on $H^s(\Bbb T^n)$
is a monotone function of $s$ which tends to zero
as $s \to \infty$, in fact {\it all} the eigenvalues of
$(\text{Id}\, \Pi_j) \LL (\text{Id}\, \Pi_j)$ will
converge to eigenvalues of $\LL$ for eigenfunctions
in $\cap_{s > 0} H^s(\Bbb T^n)$. (In other words, we
only see the ``embedded smooth spectrum'' of $\LL$.)
If we had chosen a wavelet basis of a given regularity $r>0$,
and let $\LL$ act on $H^s(\Bbb T^n)$ for $r \ge s > 0$, then
the eigenvalues of modulus greater than the
essential spectral radius of $\LL$ acting
on $H^r(\Bbb T^n)$ (and only them) obtained by the approximation
scheme are guaranteed to exhaust the eigenvalues
of $\LL$ acting on $H^s(\Bbb T^n)$ in the corresponding
annulus (their
eigenfunctions will in fact lie in $H^r (\Bbb T^n)
\subset H^s(\Bbb T^n)$).
Finally, one could extend the results of [H1], and
therefore the results of the present paper, to
the case when the dynamics and weights involved in
the construction of the transfer operator have
a given finite regularity (this would restrict the
regularity of the Banach spaces which can be considered).
\endremark
\bigskip
\head Appendix: Proof of the abstract spectral radius theorem (Theorem
1)
\endhead
We reproduce for the reader's convenience
the proof of Theorem~1, adapted from [H1].
\demo {Proof of Theorem 1}
We first suppose that the compactness
condition $(ii^*)$ and the density property $(iii^*)$ hold
and show \thetag{2.6} (the proof of \thetag{2.5} assuming
$(ii)$ and $(iii)$ is completely analogous and is left to the reader).
We
consider only the ``$\limsup$'' and leave the ``$\liminf$''part to the
reader. We decompose the argument into five steps:
{\it Step 1}: We have for all $m$,
$$
\inf\{\\LL^m \KK\\text{ s.t. } \KK\ \text{ compact}\} \leq
\\LL^m\Pi_j\ \, .
$$
Indeed, we may write
$$
\LL^m  \LL^m\Pi_j = \LL^m(\Id\Pi_j) \, .
$$
The right hand side is a compact operator, since it contains a compact
factor
by hypothesis $(ii)$, and since $\LL$ is bounded.
{\it Step 2}: We have $\BB_0^*= \BB^*$. Indeed, suppose
$\BB_0^*\owns \psi_k\to \psi\in \BB^*$.
Then for every $\epsilon>0$, we find $K$ such that $k\geq K$ implies
$\{\psi_k\psi}\\leq \epsilon$. It follows that
$$
\\Pi_j^t \psi\\leq\\Pi_j^* \psi_k\ + \ \Pi_j^*(\psi\psi_k)\
\leq \ \Pi_j^* \psi_k\ + \const\epsilon \to \const\epsilon
\quad (j \to\infty) \, .
$$
Since $\epsilon$ was arbitrary, the statement follows.
{\it Step 3}: For all $\KK\in F(\BB)$, we have
$
\lim_{j \to\infty} \\KK \Pi_j\ = 0
$.
Indeed, for all $s$ with $\s\=1$ we find
$$
\\KK\Pi_j \psi \ = \{\sum_\alpha (\nu_\alpha  \Pi_j \psi )
\psi_\alpha\
\leq \sum_\alpha \\Pi_j^* \nu_\alpha}\_{\BB^*}\{\nu_\alpha}\_\BB \,
.
$$
This tends to $0$ as $j \to\infty$, since, by Step~2, $\BB_0^*=\BB^*$,
and since the sum contains only finitely many terms.
{\it Step 4}: We have for all ${\KK}\in F(\BB)$
$$
\\LL  \KK\
\geq {1\over\const}\,\lim\sup_{j \to\infty} \\LL\Pi_j\ \, .
$$
Here $\infty>\const\geq 1$ follows from condition $(i)$.
Indeed, since $\\Pi_j\\leq \const$, we find
$$
\(\LL \KK)\Pi_j\\leq \const \\LL  \KK\ \, ,
$$
for each $j$. Now
$$
\(\LL  \KK)\Pi_j\\geq \\LL\Pi_j\  \\KK\Pi_j\ \, .
$$
We may take the limit superior $j \to\infty$ and obtain the stated
estimate.
{\it Step 5}: We have for all $m$, upon replacing $\LL$ by $\LL^m$,
$$
\const^{1/m}\, (\lim\sup_{j \to\infty}\\LL^m\Pi_j\)^{1/m}\leq
(\inf\{\\LL^m{\KK}\\text{ s.t. } {\KK \text{ finite rank}}\})^{1/m}
\, ,
$$
and
$$
(\inf\{\\LL^m{\KK}\\text{ s.t. } {\KK\ \text{compact}}\})^{1/m}
\leq (\lim\sup_{j\to\infty}\\LL^m\Pi_j\)^{1/m} \, .
$$
We now may go to the limit $m \to\infty$, showing
\thetag{2.52.6}.
\smallskip
Assuming now that $\Pi_j$ is hierarchical
and that $(iii^*), (v^*)$ hold, we
prove the last assertion of Theorem~1 in three steps:
{\it Step 6}: Note that if $\AA$ satisfies $(v^*)$ for some $k$, it
also
satisfies it for $k+1$ (and hence for all $k^\prime>k$). Indeed, we
may write as before
$$
\(\Id \Pi_j)\AA\Pi_{j+k+1}\ =
\(\Id \Pi_j)\AA\Pi_{j+k}\Pi_{j+k+1}\
\leq \const\(\Id \Pi_j)\AA\Pi_{j+k}\ \, .
$$
The last expression tends to $0$ by hypothesis on $\AA$. An analogous
argument
applies for the second limit.
{\it Step 7}: The set of bounded operators satisfying $(v^*)$
for a given $k$ forms an algebra.
Indeed if both $\MM$ and $\LL$ satisfy condition $(v^*)$,
then their sum and scalar
multiples obviously satisfy it, and we are left to check the product. By
Step~$6$ we may suppose that $\LL$ and $\MM$ satisfy $(v^*)$ for the
same $k$.
Then we have, thanks to $(iv)$,
$$
\eqalign{
&\(\Id \Pi_j) \LL \MM \Pi_{j+2k+1}\\cr
&= \(\Id \Pi_j)\LL[\Pi_{j+k} + (\Id\Pi_{j+k})
(\Id\Pi_{j+k+1})]{\MM}\Pi_{j+2k+1}\ \, .}
$$
Using the triangular inequality and $(i)$, the last expression may be
bounded above by
$$
\leq \const\\MM\\(\Id \Pi_j)\LL\Pi_{j+k} \ +
(1+\const)^2\\LL\ \(\Id\Pi_{j+k+1}){\MM}\Pi_{j+2k+1}\ \, .
$$
By hypothesis, both expressions tend to $0$ as $j \to\infty$.
{\it Step 8}: Conclusion. For any $\MM \in B(\BB)$
satisfying $(v^*)$ and thus in particular for $\MM=\LL^m$, we have
$$
\\MM\Pi_{j+k}\\leq \\Pi_j\MM\Pi_{j+k}\
+ \(\Id\Pi_j)\MM\Pi_{j+k}\
$$
In the limit $j \to\infty$, we obtain
$$
\lim\sup_{j \to\infty} \\MM\Pi_{j+k}\ \leq
\lim\sup_{j \to\infty}\\Pi_j\MM\\ \Pi_{j+k}\\leq \const\,
\lim\sup_{j \to\infty} \\Pi_j\MM\ \, .
$$
In the same way way we obtain
$$
\lim\sup_{j \to\infty}\\Pi_{j+k}\MM\ \leq \const
\lim\sup_{j \to\infty} \\MM\Pi_j\ \, .
$$
Since the constant appearing in the righthand side
does not depend on $\MM$, the
stated equality of all limits follows.
\qed
\enddemo
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\enddocument