Eigenfunction expansions of ultradifferentiable functions and ultradistributions. II. Tensor representations

In this paper we analyse the structure of the spaces of coefficients of eigenfunction expansions of functions in Komatsu classes on compact manifolds, continuing the research in our previous paper. We prove that such spaces of Fourier coefficients are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on spaces of Fourier coefficients and characterise their adjoint mappings. In particular, the considered classes include spaces of analytic and Gevrey functions, as well as spaces of ultradistributions, yielding tensor representations for linear mappings between these spaces on compact manifolds.


Introduction
The present paper is a continuation of our paper [4] where we have characterised Komatsu spaces of ultradifferentiable functions and ultradistributions on compact manifolds in terms of the eigenfunction expansions related to positive elliptic operators.In particular, these classes include the spaces of analytic, Gevrey and smooth functions as well as the corresponding dual spaces of distributions and ultradistributions, in both Roumieu and Beurling settings.
In particular, this extended the earlier characterisations of analytic functions on compact manifolds in terms of the eigenfunction expansions by Seeley [23] (see also [22]), and characterisations of Gevrey spaces and ultradistributions on tori [24] and on compact Lie groups and homogeneous spaces [3].
For example, if E is a positive elliptic pseudo-differential operator on a compact manifold X without boundary and λ j denotes its eigenvalues in the ascending order, then smooth functions on X can be characterised in terms of their Fourier coefficients: where f (j, k) = f, e k j L 2 with e k j being the k th eigenfunction corresponding to the eigenvalue λ j (of multiplicity d j ), see (2.1).If X and E are analytic, the result of Seeley [23] can be reformulated as where ν is the order of the pseudo-differential operator E. In [4] we extended such characterisations to Gevrey classes and, more generally, to Komatsu classes of ultradifferentiable functions and the corresponding classes of ultradistributions.
In this paper we continue this analysis showing that the appearing spaces of coefficients with respect to expansions in eigenfunctions of positive elliptic operators are perfect spaces in the sense of the theory of sequence spaces (see e.g.Köthe [10]).Consequently, we obtain tensor representations for linear mappings between spaces of ultradifferentiable functions and the corresponding spaces of ultradistributions.Such discrete representations in a given basis are useful in different areas of timefrequency analysis, in partial differential equations, and in numerical investigations.Due to possible multiplicities of eigenvalues the mappings beget the tensor structure rather than the matrix one as it would be in the case of simple eigenvalues, and our results are new for both situations.
Our analysis is based on the global Fourier analysis on compact manifolds which was consistently developed in [5], with a number of subsequent applications, for example to the spectral properties of operators [7], or to the wave equations for the Landau Hamiltonian [20].The corresponding version of the Fourier analysis based on expansions with respect to biorthogonal systems of eigenfunctions of non-self-adjoint operators has been developed in [19], with a subsequent extension in [21].
The obtained characterisations of Komatsu classes found their applications, for example for the well-posedness problems for weakly hyperbolic partial differential equations [8].The spaces of coefficients of eigenfunction expansions in R n with respect to the eigenfunctions of the harmonic oscillator have been analysed in [9] , and the corresponding Komatsu classes have been investigated in [26].The original Komatsu spaces of ultradifferentiable functions and ultradistributions have appeared in the works [11,12,13] by Komatsu (see also Rudin [18]), extending the original works by Roumieu [17].The universality of the spaces of Gevrey functions on the torus has been established in [25].
The regularity properties of spaces of distributions and ultradistributions have been analysed in [15], and their convolution properties appeared in [14].
The characterisations in terms of the eigenfunction expansions provide for descriptions alternative to those using the classical Fourier analysis, with applications in the theory of partial differential equations, see e.g.[16].For some other applications of this type of analysis one can see e.g.[1,2].
The paper in organised as follows.In Section 2 we briefly recall the constructions leading to the global Fourier analysis on compact manifolds.In Section 3 we very briefly recall the relevant definitions from the theory of sequence spaces.In Section 4 we present the main results of this paper and their proofs.In Section 5 we first recall the definitions for Beurling version of the spaces and then give the statement of the corresponding adjointness Theorem 4.7 in this case.
In this paper we adopt the notation N 0 = N ∪ {0}.

Fourier analysis on compact manifolds
Let X be a closed C ∞ -manifold of dimension n endowed with a fixed measure dx.We first recall an abstract statement from [5,Theorem 2.1] giving rise to the Fourier analysis on L 2 (X).
Theorem 2.1.Let H be a complex Hilbert space and let H ∞ ⊂ H be a dense linear subspace of H. Let {d j } j∈N 0 ⊂ N and let {e k j } j∈N 0 ,1≤k≤d j be an orthonormal basis of H such that e k j ∈ H ∞ for all j and k.Let H j := span{e k j } d j k=1 , and let P j : H → H j be the orthogonal projection.For f ∈ H, we denote f (j, k) := (f, e k j ) H and let f (j) ∈ C d j denote the column of f (j, k), 1 ≤ k ≤ d j .Let T : H ∞ → H be a linear operator.Then the following conditions (i)-(iii) are equivalent.
(i) For each j ∈ N 0 , we have T (H j ) ⊂ H j .
(ii) For each l ∈ N 0 there exists a matrix σ(l) ∈ C d l ×d l such that for all e k j , T e k j (l, m) = σ(l) mk δ jl .(iii) If in addition all e k j are in the domain of T * , then for each l ∈ N 0 there exists a matrix σ(l) ∈ C d l ×d l such that for all f ∈ H ∞ we have The matrices in (ii) and (iii) coincide.
The equivalent properties (i)-(iii) follow from the condition: (iv) For each j ∈ N 0 , we have T P j = P j T on H ∞.
If, in addition, T extends to a bounded operator T ∈ L(H) then (iv) is equivalent to (i)-(iii).
Under the assumptions of Theorem 2.1 we have the direct sum decomposition k=1 , and we have d j = dim H j .Here we will consider H = L 2 (X) for a compact manifold X with H j being the eigenspaces of an elliptic positive pseudo-differential operator E.
The eigenvalues of E (counted without multiplicities) form a sequence λ j , j ∈ N, which we order so that 0 =: λ 0 < λ 1 < λ 2 < ... For each eigenvalue λ j , there is the corresponding finite dimensional eigenspace H j of functions on X, which are smooth due to the ellipticity of E. We set Since the operator E is elliptic, it is Fredholm, hence also d 0 < ∞.
We denote by Ψ ν +e (X) the space of positive elliptic pseudo-differential operators on order ν > 0 on M.Here we recall a useful relation between the sequences λ j and d j of eigenvalues of E ∈ Ψ ν +e (X) and their multiplicities from [5].Proposition 2.2.Let X be a closed manifold of dimension n, and let E ∈ Ψ ν +e (X), with ν > 0. Then there exists a constant C > 0 such that we have for all j ≥ 1.Moreover, we also have For f ∈ L 2 (X), by definition we have the Fourier series decomposition The Fourier coefficients of f ∈ L 2 (X) with respect to the orthonormal basis {e k j } are denoted by We denote the space of Fourier coefficients by is a complete orthonormal system of L 2 (X) we have the Plancherel formula where we interpret f as an element of the space We endow l 2 (N 0 , Σ) with the norm We can think of F = F X as of the Fourier transform which is an isometry form L 2 (X) onto l 2 (N 0 , Σ).The inverse of this Fourier transform can be then expressed by thus thinking of the Fourier transforn always as a column vector.In particular, we think of as of a column, and we notice that e k j (l, m) = δ jl δ km .

Sequence spaces and sequential linear mappings
We briefly recall that a sequence space E is a linear subspace of The dual E (α-dual in the terminology of G. Kothe [10]) is a sequence space defined by which converges absolutely by the definition of E.
We abuse the notation by also writing a : E → C for this mapping.
Definition 3.2.A mapping f : E → F between two sequence spaces is called a sequential linear mapping if (i) f is algebraically linear, (ii) for any v ∈ F , the composed mapping v • f ∈ E.

Tensor representations for Komatsu classes and their α-duals
Let M k be a sequence of positive numbers such that In a sequence of papers [11,12,13] Komatsu investigated classes of ultradifferentiable functions on R n associated to the sequence M k , namely, the spaces of functions Ψ ∈ C ∞ (R n ) such that for every compact K ⊂ R n there exist h > 0 and a constant C > 0 such that holds for all multi-indices α ≥ 0. Similar to the case of usual distributions given a space of ultradifferentiable functions satisfying (4.1) we can define a space of ultradistributions as its dual.
We now recall the analogous definition of the Komatsu ultradifferentiable functions Γ {M k } (X) and its α-dual Γ {M k } (X) ∧ .Here, as before, X is a compact manifold without boundary and E ∈ Ψ ν +e (X) with ν > 0. Definition 4.1.The class Γ {M k } (X) is the space of C ∞ functions φ on X such that there exist h > 0 and C > 0 such that we have where ν ∈ N is the order of the positive elliptic pseudo-differential operator E.
In [4] we have characterised the class Γ {M k } (X) in terms of the eigenvalues of the operator E. We assume that (M.3)For some l, C l > 0 we have k! ≤ C l l k M k , for all k ∈ N 0 .
In the sequel, for w l ∈ C d l we write where Example 4.2.As an example, for the (Gevrey-Roumieu) class of ultradifferentiable functions γ s (X) = Γ {(k!) s } (X), 1 < s < ∞, we have M(r) ≃ r 1/s .This is also true for s = 1, characterising the class of analytic functions if the manifold is analytic.The class γ s (X) coincides with the usual Gevrey class of functions on a manifold X defined in terms of their localisations.
Based on Theorem 4.1 we can then write For φ ∈ Γ {M k } (X) we will write φ ≈ φ(l) l∈N 0 so that Γ {M k } (X) can be thought of as a sequence space, but it will be convenient to view it as a subspace of Σ defined in (2.2), taking into account the dimensions of the eigenspaces of the operator E.
Next we recall the definition of the α-dual of the space Γ {M k } (X) (following [4]).
The α-dual of the space Γ {M k } (X) of ultradifferentiable functions, denoted by We also recall the following characterisations of the α-duals established in [4].
(iii) for every L > 0 there exists K L > 0 such that We will now show that the space Γ {M k } (X) is perfect.In the proof as well as in further proofs the following estimate will be useful: This estimate follows, for example, from the local Weyl law [6, Theorem 5.1], see also [5,Lemma 8.5].
Theorem 4.4.Let X be a compact manifold and assume conditions (M.0), (M.1), (M.2), (M.3).Then Γ {M k } (X) is a perfect space, that is, we have where To prove this we first establish the following lemma: Proof of Lemma 4.5.=⇒: For L > 0 we consider the echelon space where By the diagonal transform we have D L ∼ = l ∞ , and since l ∞ is a perfect space so we have D L ∼ = l 1 , and it is given by Let 1 ≤ p < q ≤ ∞ and let a ∈ C d×d .Then we have the estimates see e.g.[3, Lemma 3.2] for a simple proof.In particular, we have d −1 ||a|| l 1 ≤ ||a|| HS ≤ d||a|| l 1 for a ∈ C d×d .Here we also note the estimate: for every q, L > 0 and δ > 0 there exists C > 0 such that (4.4) λ q l e −δM Lλ exp(M(Lλ where L 2 = L √ AH, where A, H > 0 in (M.2).The above claim will be true if we can show that exp(2M(Lλ ).This follows from property (M.2). ⇐=: Converse follows similarly using estimates (4.3).
We can now prove Theorem 4.4.
Proof of Theorem 4.4.We always have Since φ(l) = w l , it follows that there eists C > 0 such that . By Theorem 4.1, we have φ ∈ Γ {M k } (X).Hence we have shown that Next we proceed to prove the equivalence of two expressions for the duality.
Proof.=⇒: The proof is straightforward, following from the estimate ⇐=: We will be using the equality We consider the second term in the above inequality, that is, (4.8) for some C > 0 and L > 0. Then using (4.8) in (4.7) we get (4.9) Now for some C ′′ > 0 and L 2 > 0, we have i.e, u ∈ Γ {M k } (X).This is true since where L 2 = L √ AH , with A, H are constants in condition (M.2).Now since w ∈ Γ {M k } (X), so from (4.9) we have

Furthermore, the adjoint mapping
Let us summarise the ranges for indices in the used notation as well as give more explanation to (4.12).For f : Γ where we view f kj as a matrix, f kj ∈ C d k ×d j , and the product of the matrices has been explained in (4.13).
Remark 4.8.Let us now briefly describe how the tensor ∧ , and we can write Then for each 1 ≤ l ≤ d j we set (4.17) the l th component of the vector (v ki •f ) j ∈ C d j .The formula (4.17) will be shown in the proof of Theorem 4.7.In particular, since for φ ∈ Γ {M k } (X) we have f (φ) ∈ Γ {N k } (X), it will be a consequence of (4.28) and (4.29) later on that f kjli φ(j, l), so that the tensor (f kjli ) is describing the transformation of the Fourier coefficients of φ into those of f (φ).Another meaning of condition (4.11) is that if for each k ∈ N 0 and 1 ≤ i ≤ d k we define ∧ .Condition (4.12) is the continuity condition saying that for To prove Theorem 4.7 we first establish the following lemma.
Lemma 4.9.Let (f kjli ) k,j∈N 0 ,1≤l≤d j ,1≤i≤d k be an infinite tensor satisfying (4.11) and (4.12).Then for any u Proof of Lemma 4.9.Let u ∈ Γ {M k } (X) and u ≈ ( u(l)) l∈N 0 .Define u n := u (n) (l) l∈N 0 by setting Then for any w ∈ Γ {M k } (X), u − u n , w → 0 as n → ∞.This is true since ∧ and from (4.11) and (4.12) we have where , j ∈ N 0 , We can then conclude that v • f ∈ Γ {M k } (X) ∧ and we have in view of the Theorem 4.4.The same is true for the dual space Γ {N k } (X) ∧ .So then this argument gives The proof is complete.
Proof of Theorem 4.7.Let us assume first that the mapping f : Γ Let u 1 = ( u 1 (p)) p∈N 0 be such that for some j, l where j ∈ N 0 , 1 ≤ l ≤ d j we have u 1 (p, q) = 1, p = j, q = l, 0, otherwise. Then We now first show that The way in which f has been defined we have We can then write (f u) In particular using the definition of u 1 and (4.22) we get Then we consider the series So we proved that if (f kjli ) satistfies then for any u ∈ Γ {M k } (X) and v ∈ Γ {N k } (X) ∧ we have from (4.23) and (4.25) respectively, that is, For the dual space and for the α-dual, the following statements are equivalent: ∧ ; (iii) there exists L > 0 such that we have holds for all l ∈ N 0 .
Again we note that given this characterisation of α-duals, one can prove that they are perfect, in a way similar to the proof of Theorem 4.4, namely, that (5.2) .
Finally we can state the counterpart of the adjointness Theorem 4.7.
Furthermore, the adjoint mapping f : Γ (N k ) (X) → Γ (M k ) (X) defined by the formula f (v) = v • f is also sequential, and the transposed matrix (f kj ) t represents f , with f and f related by f (u), v = u, f(v) .
The proof of Theorem 5.2 is similar to the corresponding proof in Theorem 4.7 for the spaces Γ {M k } (X), so we omit the repetition.