Factorizations and singular value estimates of operators with Gelfand-Shilov and Pilipovi\'c kernels

We prove that any linear operator with kernel in a Pilipovi\'c or Gelfand-Shilov space can be factorized by two operators in the same class. We also give links on numerical approximations for such compositions. We apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.


Introduction
The singular values and their decays are strongly related to possibilities of obtaining suitable finite rank approximations of the operators. For a linear and compact operator which acts between Hilbert spaces, the singular values are the eigenvalues in decreasing order of the modulus of the operator. If more generally, the linear operator T is continuous from the quasi-Banach space B 1 to (another) quasi-Banach space B 2 , then the singular value of order j ≥ 1 is given by where the infimum is taken over all linear operators T 0 from B 1 to B 2 of rank at most j − 1. (See Section 1 for notations.) It follows that T is compact, if and only if σ j (T ) decreases to zero as j tends to infinity, or equivalently, T can be approximated by finite rank operators with arbitrarily small errors.
In this paper we deduce estimates of σ j (T ) when B 1 and B 2 stays between small test function spaces, denoted by H s (R d ) and H 0,s (R d ), and their (large) duals. The spaces H s (R d ) and H 0,s (R d ) depend on the parameter s ≥ 0 and are obtained by imposing certain exponential type estimates on the Hermite coefficients of the Hermite series expansions of the involved functions. More precisely, the set H s (R d ) (H 0,s (R d )) consists of all f = α such that |c α | e −c|α| 1 2s for some (for every) c > 0. It follows that H s (R d ) and H 0,s (R d ) increase with s, and are continuously embedded and dense in S (R d ).
In [20] the spaces H s (R d ) and H 0,s (R d ) and their duals were characterized in different ways. For example, the images under the Bargmann transform were given, and it was proved that f ∈ H s (R d ) (f ∈ H s (R d )), if and only if f satisfies for some h > 0 (for every h > 0), where H = H d is the harmonic oscillator |x| 2 − ∆ on R d . In this context we recall that Pilipović introduced in [14] function spaces whose elements obey estimates of the form (0.2) for certain choices of s. For this reason, we call H s (R d ) and H 0,s (R d ) the Pilipović spaces of Roumieu and Beurling type, respectively, of degree s ≥ 0 (cf. in [20]). In [14], it is also proved that H s 1 (R d ) and H 0,s 2 (R d ) agree with the Gelfand-Shilov spaces S s 1 (R d ) and Σ s 2 (R d ), respectively, when s 1 ≥ 1 2 and s 2 > 1 2 , while H 0, 1 The family of Pilipović spaces therefore contains all Gelfand-Shilov spaces which are invariant under Fourier transformations.
In Section 4 we consider linear operators whose kernels belong to H s (R 2d ). We show that the singular values of such operator satisfies the estimate σ k (T, B 1 , B 2 ) e −ck 1 2ds (0. 3) for some c > 0, when B j stays between H s (R d ) and its dual. If the H s -spaces and their duals are replaced by H 0,s -spaces and their duals, then we also prove that (0.3) is true for every c > 0. Furthermore, if H s -spaces and their duals are replaced by Schwartz spaces and their duals, then we prove σ k (T, B 1 , B 2 ) k −N (0. 4) for every N ≥ 0, which should be available in the literature. These singular-value estimates are based on the fact that the operator classes here above possess convenient factorization properties, which are deduced in Section 3. More precisely, an operator class M is called a factorization algebra, if for every T ∈ M, there exist T 1 , T 2 ∈ M such that T = T 1 • T 2 . (In [21] the term decomposition algebra is used instead of factorization algebra.) Evidently, L (B), the set of continuous linear operators on the quasi-Banach space B is a factorization algebra, since we may choose T 1 as the identity operator and T 2 = T . A more challenging situation appears when M does not contain the identity operator, and in this situation it is easy to find operator classes which are not factorization algebras. For example, any Schatten-von Neumann class of finite order is not a factorization algebra.
If B above is a Hilbert space and M is the set of compact operators on B, then it follows by an application of the spectral theorem that M is a factorization algebra. It is also well-known that the set of linear operators with kernels in the Schwartz space is a factorization algebra (see e. g. [1,9,16,21,22]). Furthemore, similar facts hold true for the set of operators with kernels in a fixed Gelfand-Shilov space (cf. [21]).
In Section 3 we extend the latter property such that all Pilipović spaces are included. That is, we prove that the set of operators with kernels in a fixed (but arbitrarily chosen) Pilipović space is a factorization algebra.
Since the singular values of the operators under considerations either satisfy conditions of the form (0.3) or (0.4) for every N ≥ 0, it follows that the sequence {σ j (T )} ∞ j=1 belongs to ℓ p for every p > 0. This implies that any such operator is a Schatten-von Neumann operator of degree p for every p > 0.
Here we remark that the latter conclusions in the Gelfand-Shilov situation, were deduced in [21] in slight different ways, which enables to replace the quasi-Banach spaces B 1 and B 2 by convenient Hilbert spaces. The main property behind the latter reduction concerns [19,Proposition 3.8 The Schatten-von Neumann properties are then obtained in straightforward ways by the factorization properties in combination with the exact formulas, for Hilbert-Schmidt norms of operators acting between Hilbert spaces.
Our investigations also include analysis of operators with kernels in H ♭σ , H 0,♭σ , σ > 0, or their duals. These spaces were carefully investigated in [20] and are defined by imposing conditions of the form h |α| (α!) 1 2σ on the Hermite coefficients of the involved functions. In [20], these spaces are characterized in different ways. For example, it is here proved that the Bargmann transform is bijective from H ♭σ (R d ) to the set of all entire functions F on C d such that for some constant C > 0.
In Section 2 we deduce kernel theorems for operators with kernels in these spaces, or related distribution spaces. In Section 3 we show certain factorization properties of operators with kernels in H ♭σ or in H 0,♭σ . These factorization results are slightly weaker compared to what is deduced for operators with kernels in H s and H 0,s when s ≥ 0 is real.
Finally we apply these factorization properties in Section 4, to obtain singular value decompositions for operators with kernels in H ♭σ or in H 0,♭σ . In particular we show that if T is an operator on L 2 (R d ) with kernel in H ♭σ (R 2d ), then the singular values of T satisfy

Preliminaries
In this section we recall some basic facts. We start by discussing Pilipović spaces and their properties. Thereafter we consider suitable spaces of formal Hermite series expansions, and discuss their links with Pilipović spaces.
1.1. The Pilipović spaces. We start to consider spaces which are obtained by suitable estimates of Gelfand-Shilov or Gevrey type when using powers of the harmonic oscillator H We let and equip these spaces by projective and inductive limit topologies, respectively, of S h,s (R d ), h > 0. (Cf. [6,13,14,20].) In [13,14], Pilipović proved that if s 1 ≥ 1 2 and s 2 > 1 2 , then S s 1 (R d ) and Σ s 2 (R d ) agree with the Gelfand-Shilov spaces S s 1 (R d ) and Σ s 2 (R d ), respectively. (See e. g. [20] for notations.) On the other hand, S s 1 (R d ) and Σ s 2 (R d ) are trivially equal to {0} when s 1 < 1 2 and s 2 ≤ 1 2 , while any Hermite function h α fulfills (1.1) for every h > 0, when 0 < s ≤ 1 2 . Hence, when s ≥ 0, with inductive respective projective limit topologies of S ′ h,s (R d ), h > 0 (cf. [20]). 1.2. Spaces of Hermite series expansions. Next we recall the definitions of topological vector spaces of Hermite series expansions, given in [20]. As in [20], it is convenient to use the sets R ♭ and R ♭ when indexing our spaces.
Definition 1.1. The sets R ♭ and R ♭ are given by Moreover, beside the usual ordering in R, the elements ♭ σ in R ♭ and R ♭ are ordered by the relations x 1 < ♭ σ 1 < ♭ σ 2 < x 2 , when σ 1 < σ 2 , x 1 < 1 2 and x 2 ≥ 1 2 are real. Definition 1.2. Let p ∈ (0, ∞], s ∈ R ♭ , r ∈ R, ϑ be a weight on N d , and let Then, C such that c α = 0 for at most finite numbers of α; Let p ∈ (0, ∞], and let Ω N be the set of all α ∈ N d such that |α| ≤ N. Then the topology of ℓ 0 (N d ) is defined by the inductive limit topology of the sets ; c α = 0 when α = Ω N with respect to N ≥ 0, and whose topology is given through the quasinorms 2) Since any two quasi-norms on a finite-dimensional vector space are equivalent, it follows that these topologies are independent of p. Furthermore, the topology of ℓ ′ 0 (N d ) is defined by the quasi-semi-norms (1.2). It follows that ℓ ′ 0 (N d ) is a Fréchet space, and that the topology is independent of p.
Next we introduce spaces of formal Hermite series expansions For that reason we consider the mappings between sequences and formal Hermite series expansions. Definition 1.3. Let p ∈ (0, ∞], ϑ be a weight on N d , and let s ∈ R ♭ .
• the spaces and H ′ 0,s (R d ) (1.6) are the images of T in (1.5) under corresponding spaces in (1.4). Furthermore, the topologies of the spaces in (1.6) are inherited from corresponding spaces in (1.4).
, when f is given by (1.3).

By the definitions it follows that the inclusions
, when s, t ∈ R ♭ , s < t (1.7) hold true. The next result shows that the spaces in Definition 1.6 essentially agrees with the Pilipović spaces. We refer to [20] for the proof.
Remark 1.6. Let T be given by (1.5). Then are isometric bijections between Gelfand triples. (Cf. e. g. Section 2 in [20].) Finally, in Section 5 we apply the results from the first sections to obtain certain characterizations of operators with kernels in H s and H 0,s .

Kernel theorems
In this section we deduce suitable kernel theorems for operators between Pilipović spaces and their duals. Since the spaces under considerations can in convenient ways be formulated in terms of Hermite series expansions, we may easily reduce ourselves to kernel results for matrix operators, in similar ways as in e. g. [15]. We therefore begin with the latter case. Proposition 2.1. Let ϑ k be weight functions on N d k , k = 1, 2, ϑ(α, β) = ϑ 1 (β) −1 ϑ 2 (α), and let T be a linear and continuous map from . Then the following is true: extends uniquely to a linear and continuous map from Proof. The assertion (1) follows by straight-forward estimates and is left for the reader.

By the links between
, respectively, the previous proposition immediately gives the following. (Cf. Remark 1.6.) , and let T be a linear and continuous map from . Then the following is true: extends uniquely to a linear and con- We have now the following kernel results.
Theorem 2.3. Let s ∈ R ♭ , and let T be the linear and continuous map from , given by (2.2). Then the following is true: 3) holds true. The same holds true if the H s and H ′ s spaces are replaced by H 0,s and H ′ 0,s spaces, respectively, or by S and S ′ spaces, respectively.
, s ∈ R ♭ and let T be the linear and continuous map from , given by (2.2). Then the following is true: , then T extends uniquely to linear and continuous mappings from , then T extends uniquely to linear and continuous mappings from and σ r (α) = α r . The results follow from Proposition 2.2, and the facts that with suitable inductive limit topologies, and with suitable inductive and projective limit topologies.
Evidently, the assertions on S and S ′ in Theorems 2.3 and 2.4 are well-known. For the other cases, the results are straight-forward consequences of the nuclearity of H 1 [ϑ] (R d 2 × R d 1 ) (cf. e. g. [5] or [?]). For completeness we also write down some of the corresponding results in the matrix case. The proofs follow by similar arguments as for the proofs of Theorems 2.4 and 2.3, and are left for the reader.
Theorem 2.5. Let s ≥ 0 be real and let T be the linear and continuous map from . Then the following is true: , then T extends uniquely to linear and continuous mappings from , then T extends uniquely to linear and continuous mappings from . The same holds true if ℓ s , ℓ s and their duals are replaced by ℓ 0,s , ℓ 0,s and their duals, respectively, or by ℓ S and ℓ S and their duals, respectively. 9 Theorem 2.6. Let s ≥ 0 be real and let T be the linear and continuous map from . Then the following is true: (1) if T is a linear and continuous map from . The same holds true if ℓ s , ℓ s and their duals are replaced by ℓ 0,s , ℓ 0,s and their duals, respectively, or by ℓ S and ℓ S and their duals, respectively.

Factorizations of Pilipović and Gelfand-Shilov kernels, and pseudo-differential operators
In this section we deduce convenient factorization properties for operators with kernels in Pilipović spaces.
In what follows we use the convention that if T 0 is a linear and continuous operator from . Then T is said to be a Hermite diagonal operator if T = T 0 ⊗ g, where the Hermite functions are eigenfunctions to T 0 , and either d 2 = d 1 and g = 1, or d 2 > d 1 and g is a Hermite function.
Moreover, if T = T 0 ⊗ g is on Hermite diagonal form and T 0 is positive semi-definite, then T is said to be a positive semi-definite Hermite diagonal operator.
The first part of the following result can be found in [1,22] (see also [9,16] and the references therein for an elementary proof).
Theorem 3.2. Let s ∈ R, T be a linear and continuous operator from . Then the following is true: (1) If s ≥ 0 and K ∈ H s (R d 2 +d 1 ), then there are operators T 1 and T 2 with kernels K 1 ∈ H s (R d 0 +d 1 ) and is fixed and d 0 ≥ d j , then T j can be chosen as a positive semidefinte Hermite diagonal operator. (2) If s > 0 and K ∈ H 0,s (R d 2 +d 1 ), then there are operators T 1 and T 2 with kernels K 1 ∈ H 0,s (R d 0 +d 1 ) and K 2 ∈ H 0,s (R d 2 +d 0 ) respectively such that T = T 2 • T 1 . Furthermore, if j ∈ {1, 2} is fixed and d 0 ≥ d j , then T j can be chosen as a positive semidefinte Hermite diagonal operator.
The corresponding result for s = ♭ σ reads: Let σ > 0, T be a linear and continuous operator from with the kernel K. Then the following is true.
, respectively, and T = T 2 • T 0 • T 1 . Furthermore, T 1 and T 2 can be chosen as positive semi-definite Hermite diagonal operators; Analogously, an operator with kernel in H 0,s (R 2d ) (S (R 2d )) is sometimes called a regularizing operator with respect to H 0,s (R d ) (S (R d )).
Proof of Theorem 3.2. First we assume that d 0 = d 1 , and start to prove (1). Let h d,α (x) be the Hermite function on R d of order α ∈ N d . Then K posses the expansion where the coefficients a α,β satisfies sup α,β |a α,β e r(|α| for some r > 0. Now let z ∈ R d 1 , and Here δ α,β is the Kronecker delta. Then it follows that Hence, if T j is the operator with kernel K 0,j , j = 1, 2, then T = T 2 • T 1 . Furthermore, This implies that K 0,1 ∈ H s (R d 1 +d 1 ) and K 0,2 ∈ H s (R d 2 +d 1 ), and (1) follows with K 1 = K 0,1 and K 2 = K 0,2 , in the case d 0 = d 1 .
In order to prove (2), we assume that K ∈ H 0,s (R d 2 +d 1 ), and we let a α,β be the same as the above. Then (3.2) holds for any r > 0, which implies that if n ≥ 0 is an integer, then is finite.
We let and define inductively Then I j ∩ I k = ∅ when j = k, and and by the definitions it follows that I j is a finite set for every j.
Next assume that d 0 > d 1 , and let d = d 0 − d 1 ≥ 1. Then we set K 1 (z 0 , y) = K 0,1 (z 1 , y)h d,0 (z) and K 2 (x, z 0 ) = K 0,2 (x, z 1 )h d,0 (z), where K 0,j are the same as in the first part of the proofs, z 1 ∈ R d 1 and z ∈ R d , giving that z 0 = (z 1 , z) ∈ R d 0 . We get The assertion (1) now follows in the case d 0 > d 1 from the equivalences and Since the same equivalences hold after the H s spaces have been replaced by H 0,s spaces, the assertion (2) also follows in the case d 0 > d 1 , and the theorem follows in the case d 0 ≥ d 1 .
It remains to prove the result in the case d 0 ≥ d 2 . By taking the adjoint, the rules of j = 1 and j = 2 are interchanged, and the result follows when d 0 ≥ d 2 as well. The proof is complete.

We let
and define inductively Then I j,m ∩ I j,n = ∅ when m = n, and m≥1 I j,m = N d j .
and by the definitions it follows that I j,m is a finite set for every m.
In the same way, (3.13) follows in the case k = 2.
Remark 3.5. Let σ > 0 and T ≥ 0 be a Hermite diagonal operator on L 2 (R d ) with kernel K in H ♭σ . By the proof of Theorem 3.3, Then there are Hermite diagonal operators T 1 ≥ 0 and T 2 ≥ 0 on L 2 (R d ) with kernels K 1 and K 2 such that In fact, if K is given by (3.7) with d 1 = d 2 = d, it suffices to let K 1 and K 2 be given by (3.9) and (3.10), with and a 2,α,β = (a α,β ) 1/2 (α!β!) 1 4σ .
Remark 3.6. From the construction of K 1 and K 2 in the proofs of Theorems 3.2 and 3.3, it follows that it is not so complicated for using numerical methods when obtaining approximations of candidates to K 1 and K 2 . In fact, K 1 and K 2 are formed explicitly by the elements of the matrix for T , when the Hermite functions are used as basis for S , S s and Σ s .
The following result is an immediate consequence of Theorem 3.2 and the fact that the map a → K a,t is continuous and bijective on S s 1 (R 2d ), and on Σ s 2 (R 2d ), for every s 1 ≥ 1/2, s 2 > 1/2 and t ∈ R.
Remark 3.8. Extensions of Theorem 3.7 to the case s 1 , s 2 ≥ 0 is not so smooth, because the Pilipović spaces which are not Gelfand-Shilov spaces are not invariant under dilations. However, if A is a real d × d matrix and a ∈ S (R 2d ) is such that the kernel belongs to H s (R 2d ), then we may apply Theorem 3.2 in this situation as well. Therefore, let G A,s (R 2d ) (G 0,A,s (R 2d )) be the set of all a ∈ S (R 2d ) such that the map in (3.14) belongs to H s (R 2d ) (H 0,s (R 2d )). If a ∈ G A,s (R 2d ) (a ∈ G 0,A,s (R 2d )), then there are a 1 , a 2 ∈ G A,s (R 2d ) (a 1 , a 2 ∈ G 0,A,s (R 2d )) such that a = a 1 # A a 2 .

Singular value estimates and Schatten-von Neumann properties for operators with Gelfand-Shilov kernels
In this section we use Theorem 3.2 to obtain estimates of the form (0.3) for operators T with kernels in Pilipović spaces of order s, provided B 1 and B 2 stay between the given Pilipović space and its dual. In particular it follows that any such operator belongs to any Schattenvon Neumann class.
We start by recalling the definition of quasi-Banach spaces. Let B be a vector space. A quasi-norm · B on B is a non-negative and real-valued function on B which is non-degenerate in the sense and fulfills for some constant D ≥ 1 which is independent of f, g ∈ B. Then B is a topological vector space when the topology for B is defined by · B , and B is called a quasi-Banach space if B is complete under this topology.
Let B 1 and B 2 be (quasi-)Banach spaces, and let T be a linear map between B 1 and B 2 . Then recall that the singular values of order j ≥ 1 of T is given by (0.1), where the infimum is taken over all linear operators T 0 from B 1 to B 2 with rank at most j − 1. Therefore, σ 1 (T ) equals T B 1 →B 2 , and σ k (T ) are non-negative which decrease with k.
For any p ∈ (0, ∞], I p (B 1 , B 2 ), the set of Schatten-von Neumann operators of order p is the set of all linear and continuous operators T from B 1 to B 2 such that In the following result we show that the singular values for operators T K with kernels K in Pilipović spaces or Schwartz spaces, obey estimates of the form We observe that (5) should be available in the literature. Then the following is true: (1) if K ∈ H s (R d 2 +d 1 ), and B 1 and B 2 are quasi-Banach spaces such that B 1 ֒→ H ′ s (R d 1 ) and H s (R d 2 ) ֒→ B 2 , then (4.2) holds for some c > 0. In particular, T K ∈ I p (B 1 , B 2 ); (2) if K ∈ H 0,s (R d 2 +d 1 ), and B 1 and B 2 are quasi-Banach spaces such that B 1 ֒→ H ′ 0,s (R d 1 ) and H 0,s (R d 2 ) ֒→ B 2 , then (4.2) holds for every c > 0. In particular, T K ∈ I p (B 1 , B 2 ); (3) if K ∈ H ♭σ (R d 2 +d 1 ), and B 1 and B 2 are quasi-Banach spaces such that B 1 ֒→ H ′ 1/2 (R d 1 ) and H 1/2 (R d 2 ) ֒→ B 2 , then (4.3) holds for some c > 0. In particular, T K ∈ I p (B 1 , B 2 ); (4) if K ∈ H 0,♭σ (R d 2 +d 1 ), and B 1 and B 2 are quasi-Banach spaces such that B 1 ֒→ H ′ 0,1/2 (R d 1 ) and H 0,1/2 (R d 2 ) ֒→ B 2 , then (4.3) holds for every c > 0. In particular, T K ∈ I p (B 1 , B 2 ); (5) if K ∈ S (R d 2 +d 1 ), and B 1 and B 2 are quasi-Banach spaces such that B 1 ֒→ S ′ (R d 1 ) and S (R d 2 ) ֒→ B 2 , then (4.4) holds for every N > 0. In particular, T K ∈ I p (B 1 , B 2 ).
We need some preparations for the proof. First we recall that if B j , j = 0, 1, 2, are quasi-Banach spaces and T j are linear and continuous mappings from B j−1 to B j , j = 1, 2, then and In fact, if Ω j,l (k) is the set of all linear operators from B j to B l with rank at most k − 1, then which gives (4.6). In the same way (4.5) is obtained. (See also [17].) Proof of Theorem 4.1. We only prove (1), (3) and (5). The assertions (2) and (4) follow by similar arguments and are left for the reader.
(1) First we prove the result with d 0 = max(d 1 , d 2 ) in place of d. By Theorem 3.2 we get where the kernels K 1 , K 2 and K 3 of the operators T K 1 , T K 2 and T K 3 belong to H s (R d 0 +d 1 ), H s (R d 0 +d 0 ) and H s (R d 2 +d 0 ), respectively. Furthermore, we may assume that T K 2 is a positive semi-definite Hermite diagonal operator. It follows that T K 1 is continuous from B 1 to L 2 (R d 0 ), and T K 3 is continuous from L 2 (R d 0 ) to B 2 . Hence, by (4.5) and (4.6) it suffices to prove that if T = T K 2 , then By the constructions we have where 0 ≤ c α e −c|α|  It remains to prove that we may replace d 0 by d in our estimates. We consider T 1 = T * K • T K and T 2 = T K • T * K , which are non-negative and with kernels K 1 and K 2 in H s (R d 1 +d 1 ) and H s (R d 2 +d 2 ), respectively. Hence, by the first part of the proof we get 12) (4.2) now follows from (4.11) and (4.12).
(3) By Theorem 3.3 and Remark 3.5, we get where the corresponding kernels satisfy K 0,j ∈ H 1/2 (R d j +d j ), K j ∈ H ♭ 2σ (R d j +d j ), and K 0 ∈ H 1/2 (R d 2 +d 1 ), j = 1, 2. Furthermore, all kernels except K 0 to the operators in (4.13) are positive semi-definite Hermite diagonal operators. It follows that and are continuous. By similar arguments as in the proof of (1), we now get Hence, in view of (4.5)-(4.6). This gives (3). (5) By [1,9,16,22] we get where the kernels K 1 , K 2 and K 3 of the operators T K 1 , T K 2 and T K 3 belong to S (R d 0 +d 1 ), S (R d 0 +d 0 ) and S (R d 2 +d 0 ), respectively. Furthermore, we may assume that T K 2 is a positive semi-definite Hermite diagonal operator (cf. e. g. [21]). It follows that T K 1 is continuous from B 1 to L 2 (R d 0 ), and T K 3 is continuous from L 2 (R d 0 ) to B 2 . Hence, by (4.5) and (4.6) it suffices to prove that for every N > 0 there is a constant C > 0 such that (4.15) By the constructions we have K 2 is given by (4.9), where c α fulfills for every N > 0. By the same arguments as in the first part of the proof we now get for every N, and (5) follows. Finally, by (4.2)-(4.4) it also follows that {σ k (T, B 1 , B 2 )} belongs to ℓ p for every p > 0. This gives the second parts of (1)-(5).

Discrete characterizations of kernels to smoothing operators
In this section we show that any operators with kernels in Gelfand-Shilov, Pilipović or Schwartz spaces can be characterized by convenient expansions of the form Here OG(L 2 (R d )) is the set of all orthogonal sequences {f j } ∞ j=1 in L 2 (R d ), i. e., 0 = f j ∈ L 2 (R d ), and f j 1 ⊥f j 2 when j 1 = j 2 . Note that we do not require that f j should be normalized with respect to L 2 (R d ).
For future references we also let ON(L 2 (R d )) be the set of all orthonormal sequences in L 2 (R d ).
Theorem 5.1. Let p ∈ [1, ∞] and T be a linear and continuous oper- . Then the following is true:
The corresponding characterizations of operators with Pilipović kernels are given in the following theorem.
. Then the following is true: 4) for k = 1, 2 and some h > 0 and r > 0 (every h > 0 and r > 0), where the latter supremum is taken over all j ≥ 0 and N ≥ 0; (2) on the other hand, if instead K ∈ C ∞ (R d 2 × R d 1 ) and satisfies (5.1) for some {λ j } ∞ j=1 ⊆ R + , and (5.4) holds for k = 1, 2 and some r > 0 (every r > 0), We need some preparations for the proof. First we observe that H p possess the expected interpolation properties. Lemma 5.3. Let θ ∈ [0, 1], ϑ, ϑ 1 and ϑ 2 be weights on N d , and let p, p 1 , p 2 ∈ [1, ∞] be such that Then Proof. The result follows from the fact that the map We also need the following result on powers of non-negative selfadjoint operators on L 2 (R d ).
Proposition 5.4. Let s ≥ 0, r > 0 and let T be a self-adjoint and non-negative operator on Then the following is true: (1) the kernel of T r belongs to H s (R d × R d ); (2) T r is continuous from H ′ s (R d ) to H s (R d ). The same holds true if the H s and H ′ s spaces are replaced by H 0,s and H ′ 0,s spaces, respectively, or by S and S ′ spaces, respectively. Proof. We only prove the result when K ∈ H s (R d × R d ). The other cases follows by similar arguments and is left for the reader.
Let Ω = { z ∈ C ; 0 < Re(z) < 1 } and T 0 (z) = T z when z ∈ Ω. Then the map z → T (z) with values in L (L 2 (R d )) is continuous on Ω and analytic on Ω.
Furthermore, by writing T z = T x •T iy when z = x+iy, and using that T iy is bounded on L 2 (R d ) when y ∈ R, it follows from the assumptions that for some c > 0, where ϑ c (α) = e c|α| It now follows from Lemma 5.3 and Calderon-Lion's interpolation theorem (cf. Theorem IX.20 in [15]) that T r is continuous from Hence, by duality it follows that T r : and T r : are continuous. Hence, by interpolation we obtain that We also need the following characterization of Pilipović kernels.
The equivalence bewteen (3) and (4) now follows from these invariance properties and the fact that The proof is complete.
Proofs of Theorems 5.1 and 5.2. We only prove Theorem 5.2 and then in the Roumieu case. The other cases follow by similar arguments and are left for the reader.
(1) Assume that K ∈ H s (R d 2 × R d 1 ). By polar decomposition we have