The algebra of harmonic functions for a matrixvalued transfer operator
Abstract.
We analyze matrixvalued transfer operators. We prove that the fixed points of transfer operators form a finite dimensional algebra. For matrix weights satisfying a lowpass condition we identify the minimal projections in this algebra as correlations of scaling functions, i.e., limits of cascade algortihms.
Key words and phrases:
transfer operator, spectral radius, algebra, completely positive2000 Mathematics Subject Classification:
37C40, 37A55, 42C40Research supported in part by a grant from the National Science Foundation DMS 0457491 and the Research Council of Norway, project number NFR 154077/420
Contents
1. Introduction
In this paper we study a class of harmonic functions which come from the theory of dynamical systems. While there is an analogy to the classical theory of harmonic functions for Laplace operators, this analogy has two steps: First the continuous variable Laplace, or elliptic operators in PDE admit a variety of discrete approximations leading to random walk models, with variable coefficients PDEs corresponding to variable weight functions in the discrete version.
As a result of work by numerous authors, and motivation from applications, there is now a rich harmonic analysis which is based on a certain transfer operator. It is based on path models, paths originating in a compact space, and with finite branching. The simplest instances come from endomorphisms in compact spaces , onto, but with finitely branched inverses. A weight function on is prescribed, and the corresponding transfer operator (alias Ruelle operator) is denoted .
Our motivation comes mainly from problems in wavelet analysis, but transfer operators are ubiquitous in applied mathematics: As it turns out, a study of transfer operators has now emerged as a subject of independent interest, with numerous applications. Examples: operator theory (locating the essential spectrum) [CI91, Hel96]; algebras [MS06]; groupoids [MSS06, KR06]; infinite determinants [BR96, Rue02]; anisotropic Sobolev spaces [Bal05]; tiling spaces [MM05]; number theory (zeta functions) [Hen02, HM02, HM04, HMM05]; ergodic theory [CHR97, DPU96], dynamics [CR98, CR03, Pol01]; optimization [Hag05, KL99]; and quantum statistical mechanics [Ara80, Rue92].
Our present paper uses tools from at least three areas, algebras, dynamics, and wavelets:
1. We will be using both algebras, and their representations, especially an important family of matrix algebras.
2. Dynamical systems. The connection between the algebras are relevant to the kind of dynamical systems which are built on branchinglaws. The reason for this is that the spectral properties of the transfer operator and of the associated eigenspace algebras are connected to the ergodic properties of the dynamical system.
3. Wavelet analysis. The connection to basis constructions using wavelets is this: The context for wavelets is a Hilbert space , where may be where is a dimension, for the line (signals), for the plane (images), etc. The more successful bases in Hilbert space are the orthonormal bases ONBs, but until the mid 1980s, there were no ONBs in which were entirely algorithmic and effective for computations.
Originating with [Law90], a popular tool for deciding whether or not a candidate for a wavelet basis is in fact an ONB uses a wavelet version of the transfer operator, still based on fold branching laws, but now with the branching corresponding to frequency bands. The wavelet Ruelle operator weights input over branching possibilities, and the weighting is assigned by a prescribed scalar function , the modulus squared of a lowpass filter function, often called . We are interested in the top part of the spectrum of , a distinguished eigenspace for ; an infinitedimensional version of the so called PerronFrobenius problem from finitedimensional matrix theory.
This is especially useful for wavelets that are initialized by a single function, called the scaling function. These are called the multiresolution analysis (MRA) wavelets, or for short the MRAwavelets. But there are multiwavelets (i.e., more than one scaling function) for example for localization in frequency domain, where the MRAwavelets do not suffice, where it will by necessity include more than one scaling function. And then asking for an ONB is not feasible, but instead frame wavelets are natural.
We attack this problem by introducing a matrix version of the weight function ; so our is no longer scalar valued, but rather matrixvalued; the size of the matrices depending on an optimal number of scaling functions. We show that the space of harmonic functions for the Ruelle operator with matrix valued weights acquires the structure of a algebra. It serves to decide ONB vs frame properties, and the stability properties needed in applications, for example in the analogue to digital signal problem.
A transfer operator, also called Ruelle operator, is associated to a finitetoone endomorphism and a weight function , and it is defined by
for functions on . Here is a compact Hausdorff space, and is a nonnegative continuous function on . The function is said to be normalized if
Transfer operators have been extensively used in the analysis of discrete dynamical systems [Bal00] and in wavelet theory [BJ02].
In multivariate wavelet theory (see for example [JS99] for details) one has an expansive integer matrix , i.e., all eigenvalues have , and a multiresolution structure on , i.e., a sequence of closed subspaces of such that

, for all ;

is dense in ;

;

if and only if ;

There exist such that forms an orthonormal basis for .
The functions are called scaling functions, and their Fourier transforms satisfy the following scaling equation:
where are some periodic functions on .
The orthogonality of the translates of imply the following QMF equation:
where is the matrix . When the translates of the scaling functions are not necessarily orthogonal one still obtains the following relation: if we let
then the matrix satisfies the following property:
(1.1) 
i.e., is a fixed point for the matrixvalued transfer operator . The fixed points of a transfer operator are also called harmonic functions for this operator. Thus the orthogonality properties of the scaling functions are directly related to the spectral properties of the transfer operator .
This motivates our study of the harmonic functions for a matrixvalued transfer operator. The onedimensional case (numbers instead of matrices) was studied in [BJ02, Dut04b, Dut04a]. These results were then extended in [DJ06a, DJ06b], by replacing the map on the torus , by some expansive endomorphism on a metric space.
Here we are interested in the case when the weights defining the transfer operator are matrices, just as in equation (1.1). We keep a higher level of generality because of possible applications outside wavelet theory, in areas such as dynamical systems or fractals (see [DJ06a, DJ06b]). However, for clarity, the reader should always have the main example in mind, where on the torus. The quotient map defines a simply connected covering space, and lifts to the dilation on .
In Section 2 we introduce the main notions. Since we are especially interested in continuous harmonic maps we used the language of vector bundles (see also [PR04]). Packer and Rieffel introduced projective multiresolution analyses (PMRA’s) in [PR04]. In their formalism, the scaling spaces correspond to sections in vector bundles over . Motivated by their work, we introduce transfer operators that act on bundlemaps on vectorbundles. This gives a way to construct new PMRA’s.
In Section 3 we perform a spectral analysis of the transfer operator, prove that it is quasicompact and give an estimate of the essential spectral radius. Section 4 contains one of the main results of the paper: the continuous harmonic functions form a algebra, with the usual addition, multiplication by scalars and adjoint, and with the multiplication defined by a certain spectral projection of .
In Section 5 we define the refinement operator. This operates at the level of the covering space , and extends the usual refinement operator from wavelet theory (see [BJ02]). The correlations of the scaling functions correspond to fixed points of the refinement operator. We give the intertwining relation between the transfer operator and the refinement operator in Theorem 5.5. With this relation we will see that the fixed points of the transfer operator correspond to the fixed points of the refinement operator, i.e., to scaling functions.
In Section 6 we consider the case of lowpass filters. In the onedimensional case the lowpass condition amounts to , i.e., is maximal. This is needed in order to obtain solutions of the scaling equation in . In the matrix case, the appropriate lowpass condition was introduced in [JS99] under the name of the condition (see Definition 6.1). If this condition is satisfied we show that the associated scaling functions exist in our case as pointwise limits of iterates of the refinement operator (Theorem 6.2). Their correlation functions give minimal projections in the algebra of continuous harmonic functions (Theorem 6.5).
In Theorem 6.7 we show that, if the peripheral spectrum of the transfer operator is “simple”, and we make an appropriate choice of the starting point, then the iterates of the refinement operator converge strongly.
2. Transfer operators
We describe now our setting, starting with the main example, the one used in multivariate wavelet theory.
Example 2.1.
Let and let be a discrete subgroup such that is compact, i.e. is a fullrank lattice, and and let such that and such that if is an eigenvalue of then . Then there exists a norm on and a number such that for all .
Indeed, let and define as
By the spectral radius formula and the root test, this converges and defines a norm on such that for all .
Let denote the quotient map and define a map as . The map is a regular covering map with deck transformations . We have that and denote this number by . If is the normalized Haar measure on , then
(2.1) 
i.e., is strongly invariant. Finally, there exists a normalization of the Haar measure on , say such that for every .
We are mainly concerned with the above situation, but our results apply to a more general setting:
2.1. The setting
Let be a locally compact metric space with an isometric and properly discontinuous action of a discrete group such that is compact.
Let be the quotient covering map.
Suppose is a strictly expansive homeomorphism on , i.e., for some
Let be the fixed point of and .
Assume that there exists such that is a normal subgroup, and for every , .
Let , , for all .
Moreover, let and be regular measures on and such that is strongly invariant as in (2.1), and
(2.2) 
Let be a Lipschitz continuous dimensional complex vectorbundle over with a hermitian metric, i.e. a map that restricts to positive definite sesquilinear forms on each fiber. We say that is Lipschitz continuous if there exists a trivializing system of bundlemaps such that every is Lipschitz continuous on .
Let denote the continuous sections in . By the SerreSwan theorem [Ati89], this is a projective module and the endomorphisms on are exactly the bundlemaps on acting as . In fact, equipped with the norm and the pointwise involution with respect to the form is a algebra.
Let be the pullback of the vector bundle by the map , i.e.,
Assumption: We assume that is a trivial bundle.
This is always the case if is contractible. We claim that we can take to be the trivial bundle with the canonical Hermitian inner product.
Given linearly independent sections in such that is Lipschitz continuous for every we get a Lipschitz continuous bundle isomorphism such that , where , are the canonical sections in the trivial bundle . We equip the product bundle with the standard inner product on and with pullback of the inner product on . Let denote the pointwise polar decomposition.
is positive and invertible, i.e. the holomorphic functional calculus on Banach algebras tells us that we can apply the square root and still get a bounded operator. This means that is a bounded operator on the Lipschitz continuous . Now defines a Lipschitz continuous bundle isomorphism that is isometric over each . This means that we can identify with the trivial product bundle equipped with the ordinary inner product.
Let denote the set of such that is Lipschitz continuous. is a projective module of finite rank and a Banach space with the norm
Let denote the endomorphisms on . is dense in and a Banach algebra with the usual operator norm
Let denote the pullback of along (see [Ati89, 1.1]) and let
be a bundle map. The pullback of the inner product on gives an inner product on and we get a unique such that for every pair .
Let be a complete system of representatives for the right cosets of in , with , and define as
(2.3) 
we see that , and for all .
Definition 2.2.
We define the transfer operator associated to as the operator acting on such that
An element is called harmonic for the transfer operator , if . Define
Moreover, let , and let denote the ’th coordinate of .
A computation yields the following identity:
We will use repeatedly the following CauchySchwarz type inequality:
Lemma 2.3.
If then
Proof.
The space with the sesquilinear map
form an Hilbert module.
If then by the CauchySchwarz inequality for Hilbert Modules
The claim follows with , and . ∎
If there exists that is positive, invertible and harmonic with respect to , then there exists a such that ; this implies that for all , , and with Lemma 2.3, for all :
so the existence of such an element implies that .
We assume from now on that satifies the following conditions.

is Lipschitz continuous.

Remark 2.4.
Since is the trivial bundle, we have . This means that defines a bundle map . Now for some map . defines a map such that where is the unique element map such that for every and . Therefore we can identify with .
3. The peripheral spectrum
The essential spectral radius of a bounded linear operator on a Banach space is the infimum of positive numbers such that and implies is an isolated eigenvalue of finite multiplicity. If the essential spectral radius of is strictly less than the spectral radius then is said to be quasicompact.
Whenever is a complete metric space and we define the Ball measure of noncompactness of , say to be
If is a Banach space with unit ball and is a bounded linear operator on , define
The following theorem is due to Nussbaum [Nus70]
Theorem 3.1.
exists and equals the essential spectral radius of .
The corollary is due to Hennion [Hen93]
Corollary 3.2.
If is a Banach space with a second norm and an operator such that

is a compact operator.

For every there exist positive numbers and such that and
then the essential spectral radius of is less than .
Proof.
Let , and . Since is relatively compact with respect to there exists a sequence for every such that
If then
Thus can be covered by . Such a sequence can be found for arbitrary , i.e. and
∎
We intend to give an estimate of the essential spectral radius of . First we need some lemmas.
Lemma 3.3.
If is Lipschitz continuous, there exists a such that
For every and .
Proof.
Since is expansive, the maps are contractive, with contraction constant .
By the CauchySchwarz inequality , this is dominated by
Since for any positive matrix, we obtain
(3.1) 
This, with , implies that our expression is bounded by
∎
Lemma 3.4.
There exist such that
for every and
Lemma 3.5.
The map
defines an equivalent norm on .
Proof.
A straightforward computation gives the following estimate
On the other hand, we can find a finite cover of by open sets such that the restriction of to these sets is trivial. By the Lebsgue covering lemma there exists an such that whenever have , there exists a in this cover such that . We can find then a unit vector such that
Moreover, for every with , we can construct a section such that for every with , and, by constructing a Lipschitz partition of unity subordinated to our cover, we can assume that is uniformly bounded by some constant that does not depend on .
Now
If then
so we get a constant such that
∎
Theorem 3.6.
The essential spectral radius of is less than . Since , the spectral radius is at most .
Moreover, in the main example with where is a linear map on , the essential spectral radius is less than the spectral radius of .
Proof.
By the Lemmas 3.4 and 3.5 we get such that
in the general case, and
in the case with the linear map .
By the ArzelaAscoli theorem, bounded sets in are precompact in . This implies that bounded sets in are precompact in . The claim now follows from Corollary 3.2 and the spectral radius formula. ∎
Theorem 3.7.
The Cesaro means converges with respect to the norm for every . The map