Nuclear embeddings of Morrey sequence spaces and smoothness Morrey spaces

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain $\Omega \subset {\mathbb R}^d$. This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of $\ell_r$ type, $1 \leq r \leq\infty$.


Introduction
Let Ω ⊂ R d be a bounded Lipschitz domain and N s u,p,q (Ω) and E s u,p,q (Ω) smoothness Morrey spaces, with s i ∈ R, 0 < p i ≤ u i < ∞, or p i = u i = ∞, 0 < q i ≤ ∞, i = 1, 2. Roughly speaking, these spaces N s u,p,q and E s u,p,q are the counterparts of the well-known Besov and Triebel-Lizorkin function spaces B s p,q and F s p,q , respectively, where the basic L p space in the latter scales are replaced by the Morrey space M u,p , 0 < p ≤ u < ∞: this is the set of all locally p-integrable functions f ∈ L loc p (R d ) such that Consequently M p,p = L p and likewise B s p,q = N s p,p,q , F s p,q = E s p,p,q .For the precise definition and further properties we refer to Section 2 below.
Motivated also by applications to PDE, smoothness Morrey spaces have been studied intensely in the last years opening a wide field of possible applications.The Besov-Morrey spaces N s u,p,q (R d ) were introduced in [22] by Kozono and Yamazaki and used by them and Mazzucato [24] to study Navier-Stokes equations.Corresponding Triebel-Lizorkin-Morrey spaces E s u,p,q (R d ) were introduced in [41] by Tang and Xu, where the authors established the Morrey version of the Fefferman-Stein vector-valued inequality.
Another class of generalisations, the Besov-type space B s,τ p,q (R d ) and the Triebel-Lizorkin-type space F s,τ p,q (R d ) were introduced in [54].They coincide with B s p,q and F s p,q when τ = 0. Their homogeneous versions were originally investigated by El Baraka in [5,6,7] and by Yuan and Yang [50,51].There are also some applications in partial differential equations for spaces of type B s,τ p,q (R d ) and F s,τ p,q (R d ), such as (fractional) Navier-Stokes equations, cf.[23].
Although the above scales are defined in different ways, they are closely connected and share a number of properties.Both approaches can be seen as examples of more general scales of Morrey smoothness spaces, we refer to our recent paper [20] in this context.Further details, definitions, properties and references can be found in Section 2 below.
Parallel to the situation in Besov and Triebel-Lizorkin spaces, characterisations are well-known, when embeddings between spaces in these scales of Morrey smoothness spaces on R d are continuous.But there cannot exist a compact embedding, unless one imposes further assumptions, like weights, certain subspaces, or -as in the case dealt with in the present paper -one considers spaces defined on bounded domains.
The main purpose of the present paper is to study the nuclearity of embeddings of type (1.2).Grothendieck introduced the concept of nuclearity in [11] more than 60 years ago.It provided the basis for many famous developments in functional analysis afterwards.Recall that Enflo used nuclearity in his famous solution [8] of the approximation problem, a long-standing problem of Banach from the Scottish Book.We refer to [30,28], and, in particular, to [31] for further historic details.
Let X, Y be Banach spaces, T ∈ L(X, Y ) a linear and bounded operator.Then T is called nuclear, denoted by T ∈ N (X, Y ), if there exist elements a j ∈ X ′ , the dual space of X, and y j ∈ Y , j ∈ N, such that ∞ j=1 a j X ′ y j Y < ∞ and a nuclear representation T x = ∞ j=1 a j (x)y j for any x ∈ X. Together with the nuclear norm where the infimum is taken over all nuclear representations of T , the space N (X, Y ) becomes a Banach space.It is obvious that nuclear operators are, in particular, compact.
Already in the early years there was a strong interest to study examples of nuclear operators beyond diagonal operators in ℓ p sequence spaces, where a complete answer was obtained in [42].Concentrating on embedding operators in spaces of Sobolev type, first results can be found, for instance, in [32,27].We noticed an increased interest in studies of nuclearity in the last years.Dealing with the Sobolev embedding for spaces on a bounded domain, some of the recent papers we have in mind are [4,3,49,1,2] using quite different techniques however.
There might be several reasons for this.For example, the problem to describe a compact operator outside the Hilbert space setting is a partly open and very important one.It is well known from the remarkable Enflo result [8] that there are compact operators between Banach spaces which cannot be approximated by finite-rank operators.This led to a number of -meanwhile well-established and famous -methods to circumvent this difficulty and find alternative ways to 'measure' the compactness or 'degree' of compactness of an operator.It can be described by the asymptotic behaviour of its approximation or entropy numbers, which are basic tools for many different problems nowadays, e.g.eigenvalue distribution of compact operators in Banach spaces, optimal approximation of Sobolev-type embeddings, but also for numerical questions.In all these problems, the decomposition of a given compact operator into a series is an essential proof technique.It turns out that in many of the recent contributions [49,1,2] studying nuclearity, a key tool in the arguments are new decomposition techniques as well, adapted to the different spaces.Inspired by the nice paper [1] we also used such arguments in our papers [19,12], and intend to follow this strategy here again.
As mentioned above, function spaces of Besov or Triebel-Lizorkin type, as well as their Morrey counterparts, defined on R d never admit a compact, let alone nuclear embedding.But replacing R d by a bounded Lipschitz domain Ω ⊂ R d , then the question of nuclearity in the scale of Besov and Triebel-Lizorkin spaces has already been solved, cf.[27] (with a forerunner in [32]) for the sufficient conditions, and [49] with some forerunner in [27] and partial results in [3,4] for the necessity of the conditions.More precisely, for Besov spaces on bounded Lipschitz domains, B s p,q (Ω), it is well known that id B Ω : B s1 p1,q1 (Ω) ֒→ B s2 p2,q2 (Ω) is nuclear if, and only if, The counterpart for spaces of type N s u,p,q (Ω), reads now as follows, see Theorem 5.1 below: let , where t(r 1 , r 2 ) is defined via Clearly the two above-mentioned results coincide in case of u i = p i , i = 1, 2. We obtained parallel results in the context of spaces E s u,p,q (Ω), B s,τ p,q (Ω) and F s,τ p,q (Ω), see Theorems 5.1 and 5.3 below.As already indicated, we follow here the general ideas presented in [1] which use decomposition techniques and benefit from Tong's result [42] about nuclear diagonal operators acting in sequence spaces of type ℓ p .For that reason we study first appropriate Morrey sequence spaces which are adapted to the wavelet decomposition of our Morrey function spaces.The nuclearity result in this sequence space setting can be found in Theorem 4.9 below.Here we also rely on our earlier results in [18].
Finally, the present paper can also be seen as an answer to a question by our friend and colleague, David E. Edmunds (University of Sussex at Brighton), who asked us some years ago what is known about nuclear embeddings in the setting of Morrey smoothness spaces.We did not know any result, but became sufficiently fascinated by the topic to study this question ourselves.This is the outcome.
The paper is organised as follows.In Section 2 we recall basic facts about the sequence and function spaces we shall work with, Section 3 is devoted to the general concept of nuclear embeddings and some preceding results.In Section 4 we concentrate on the Morrey sequence spaces with the main nuclearity result in Theorem 4.9, while Section 5 deals with the question of nuclear embeddings in Morrey smoothness spaces.Here the main findings are collected in Theorems 5.1 and 5.3.We conclude the paper with a number of examples.

Function spaces of Morrey type
First we fix some notation.By N we denote the set of natural numbers, by N 0 the set N ∪ {0}, and by Z d the set of all lattice points in R d having integer components.For a ∈ R, let ⌊a⌋ := max{k ∈ Z : k ≤ a} and a + := max{a, 0}.All unimportant positive constants will be denoted by C, occasionally with subscripts.By the notation A B, we mean that there exists a positive constant C such that A ≤ C B, whereas the symbol A ∼ B stands for A B A. We denote by B(x, r) := {y ∈ R d : |x − y| < r} the ball centred at x ∈ R d with radius r > 0, and | • | denotes the Lebesgue measure when applied to measurable subsets of R d .
Given two (quasi-)Banach spaces X and Y , we write X ֒→ Y if X ⊂ Y and the natural embedding of X into Y is continuous.

Smoothness spaces of Morrey type on R d
Let S(R d ) be the set of all Schwartz functions on R d , endowed with the usual topology, and denote by S ′ (R d ) its topological dual, namely, the space of all bounded linear functionals on S(R d ) endowed with the weak * -topology.For all f ∈ S(R d ) or S ′ (R d ), we use f to denote its Fourier transform, and f ∨ for its inverse.Let Q be the collection of all dyadic cubes in R d , namely, The symbol ℓ(Q) denotes the side-length of the cube Q and j Q := − log 2 ℓ(Q).
(i) Let p ∈ (0, ∞].The Besov-type space B s,τ p,q (R d ) is defined to be the collection of all f ∈ S ′ (R d ) such that f | B s,τ p,q (R d ) := sup with the usual modifications made in case of p = ∞ and/or q = ∞.
(ii) Let p ∈ (0, ∞).The Triebel-Lizorkin-type space F s,τ p,q (R d ) is defined to be the collection of all f ∈ S ′ (R d ) such that with the usual modification made in case of q = ∞.Remark 2.2.These spaces were introduced in [54] and proved therein to be quasi-Banach spaces.In the Banach case the scale of Nikol'skij-Besov type spaces B s,τ p,q (R d ) had already been introduced and investigated in [5,6,7].It is easy to see that, when τ = 0, then B s,τ p,q (R d ) and F s,τ p,q (R d ) coincide with the classical Besov space B s p,q (R d ) and Triebel-Lizorkin space F s p,q (R d ), respectively.There exists extensive literature on such spaces; we refer, in particular, to the series of monographs [44,45,46] for a comprehensive treatment.In case of τ < 0 the spaces are not very interesting, , respectively, when both scales of spaces are meant simultaneously in some context.
We have elementary embeddings within this scale of spaces (see [54, Proposition 2.1]), and as well as which directly extends the well-known classical case from (2.6) The following remarkable feature was proved in [52].

Now we come to smoothness spaces of Morrey type
Remark 2.4.The spaces M u,p (R d ) are quasi-Banach spaces (Banach spaces for p ≥ 1).They originated from Morrey's study on PDE (see [25]) and are part of the wider class of Morrey-Campanato spaces; cf.[26].They can be considered as a complement to L p spaces.As a matter of fact, for u < p, and that for 0 In an analogous way, one can define the spaces M ∞,p (R d ), p ∈ (0, ∞), but using the Lebesgue differentiation theorem, one can easily prove that We refer to the recent monographs [35,36] for a detailed treatment and, in particular, application of the concept in the study of PDE.
with the usual modification made in case of q = ∞.
with the usual modification made in case of q = ∞.
Convention.Again we adopt the usual custom to write A s u,p,q instead of N s u,p,q or E s u,p,q , when both scales of spaces are meant simultaneously in some context.Remark 2.6.The spaces A s u,p,q (R d ) are independent of the particular choices of ϕ 0 , ϕ appearing in their definitions.They are quasi-Banach spaces (Banach spaces for p, q ≥ 1), and S(R d ) ֒→ A s u,p,q (R d ) ֒→ S ′ (R d ).Moreover, for u = p we re-obtain the usual Besov and Triebel-Lizorkin spaces, (2.10) Besov-Morrey spaces were introduced by Kozono and Yamazaki in [22].They studied semi-linear heat equations and Navier-Stokes equations with initial data belonging to Besov-Morrey spaces.The investigations were continued by Mazzucato [24], where one can find the atomic decomposition of some spaces.The Triebel-Lizorkin-Morrey spaces were later introduced by Tang and Xu [41].We follow the ideas of Tang and Xu [41], where a somewhat different definition is proposed.The ideas were further developed by Sawano and Tanaka [38,37,34,33].The most systematic and general approach to the spaces of this type can be found in the book [54] or in the survey papers by Sickel [39,40].It turned out that many of the results from the classical situation have their counterparts for the spaces A s u,p,q (R d ), e. g., and However, there also exist some differences.Sawano proved in [33] that, for s ∈ R and 0 where, for the latter embedding, r = ∞ cannot be improved -unlike in case of u = p (see (2.5) with τ = 0).More precisely, Remark 2.7.Let s, u, p and q be as in Definition 2.5 and Moreover, the above embedding is proper if τ > 0 and q < ∞.If τ = 0 or q = ∞, then both spaces coincide with each other, in particular, As for the F -spaces, if 0 ≤ τ < 1/p, then where Q appearing in the above definition runs over all cubes in R d , and f Q denotes the mean value of f with respect to Q, namely, cf. [39, Propositions 3.4 and 3.5].
Remark 2.9.In contrast to this approach, Triebel followed the original Morrey-Campanato ideas to develop local spaces L r A s p,q (R d ) in [47], and so-called 'hybrid' spaces L r A s p,q (R d ) in [48], where 0 < p < ∞, 0 < q ≤ ∞, s ∈ R, and − d p ≤ r < ∞.This construction is based on wavelet decompositions and also combines local and global elements as in Definitions 2.1 and 2.5.However, Triebel proved in [48,Thm. 3.38] that in all admitted cases.We return to this coincidence below.
Remark 2.10.The most recent approach to Morrey smoothness spaces can be found in [20]: ) and Λ ̺ A s p,q (R d ) were introduced there, which satisfy that and As many interesting properties of the spaces Λ ̺ A s p,q (R d ) and Λ ̺ A s p,q (R d ) are similar for the same parameter ̺, the authors introduced in [20] so called ̺-clans ̺-A s p,q (R d ) of spaces, −d < ̺ < 0, which share such important features.We shall return to this generalisation and concept below.

Wavelet decomposition
We briefly recall the wavelet characterisation of Besov-Morrey spaces proved in [33].It will be essential in our approach.For m ∈ Z d and ν ∈ Z we define a d-dimensional dyadic cube with sides parallel to the axes of coordinates by Let φ be a scaling function on R with compact support and of sufficiently high regularity.Let ψ be an associated wavelet.Then the tensor-product ansatz yields a scaling function φ and associated wavelets ψ 1 , . . ., ψ 2 d −1 , all defined now on R d .We suppose φ ∈ C N1 (R) and supp φ ⊂ [−N 2 , N 2 ] for certain natural numbers N 1 and N 2 .This implies for i = 1, . . ., 2 d − 1.We use the standard abbreviations To formulate the result we introduce some sequence spaces.For 0 However, in many situations the following equivalent norm in the space n σ u,p,q is more useful cf. [13].The following theorem was proved in [33].

Spaces on domains
Let Ω denote a bounded Lipschitz domain in R d .We consider smoothness Morrey spaces on Ω defined by restriction.Let D(Ω) be the set of all infinitely differentiable functions supported in Ω and denote by D ′ (Ω) its dual.
endowed with the quasi-norm Remark 2.14.The spaces A s,τ p,q (Ω) are defined in the same way by restriction.They are as well as the spaces A s u,p,q (Ω) quasi-Banach spaces (Banach spaces for p, q ≥ 1).When p = u or τ = 0 we obtain the usual Besov and Triebel-Lizorkin spaces defined on domains.In [9] we studied the extension operator of spaces A s,τ p,q (Ω) and studied limiting embeddings.We obtained, for instance, that -in addition to the monotonicity in the smoothness parameter s and the fine index q, as recalled in (2.3) and (2.4), respectively, -there is some monotonicity in τ , too: we proved that A s,τ1 p,q (Ω) ֒→ A s,τ2 p,q (Ω) when 0 ≤ τ 2 ≤ τ 1 , cf. [9, Proposition 3.9].The sufficient and necessary conditions for compactness of embeddings of the Besov-Morrey and Triebel-Lizorkin spaces were proved in [14] and [15] with a small contribution in the case p = u in [10].In the last paper one can also find the corresponding results for Besov-type and Triebel-Lizorkin type spaces.The above conditions read as follows.
is compact if, and only if, Remark 2.16.We refer to [14] for further details.In [17] we studied entropy numbers of such compact embeddings, see also [16] for some first results on corresponding approximation numbers.
The counterpart for spaces A s,τ p,q (Ω) can be found in [10].Let us introduce the notation (2.25) Theorem 2.17.
3 Nuclear operators Definition 3.1.An operator T ∈ L(X, Y ) is called nuclear if it can be written in the form where The nuclear norm is given by where the infimum is taken over all representations (3.1) Remark 3.2.The concept of nuclear operators has been introduced by Grothendieck [11] and was intensively studied afterwards, cf.[29,28] and also [31] for some history.
One can easily see that if T is a nuclear operator, then the infinite series of the terms x * k ⊗ y k : x → x * k (x)y k is convergent in L(X, Y ).So any nuclear operator can be approximated by finite-rank operators.It is wellknown that N (X, Y ) possesses the ideal property.In Hilbert spaces H 1 , H 2 , the nuclear operators N (H 1 , H 2 ) coincide with the trace class S 1 (H 1 , H 2 ), consisting of those T with singular numbers (s n (T )) n ∈ ℓ 1 .
We collect some further properties for later use and for convenience.
(ii) For any Banach spaces X and any bounded linear operator T : ℓ n ∞ → X we have T e i |X .
(iii) If T ∈ L(X, Y ) is a nuclear operator and S ∈ L(X 0 , X) and R ∈ L(Y, Y 0 ), then RT S is a nuclear operator and ν(RT S) ≤ R ν(T ) S .
(i) Then D τ is nuclear if, and only if, τ = (τ j ) j ∈ ℓ t(r1,r2) , with ℓ t(r1,r2 Example 3.5.In the special case of τ ≡ 1, i.e., D τ = id, (i) is not applicable and (ii) reads as In particular, ν(id : ℓ n 1 → ℓ n ∞ ) = 1.Remark 3.6.We refer also to [28] for the case is nuclear if, and only if, Remark 3.8.The proposition is stated in [49] for the B-case only, but due to the independence of (3.8) of the fine parameters q i , i = 1, 2, and in view of (the corresponding counterpart of) (2.5) (with τ = 0) it can be extended immediately to F -spaces.The if-part of the above result is essentially covered by [27] (with a forerunner in [32]).Also part of the necessity of (3.8) for the nuclearity of id Ω was proved by Pietsch in [27] such that only the limiting case ) + was open for many decades.Edmunds, Gurka and Lang in [3] (with a forerunner in [4]) obtained some answer in the limiting case which was then completely solved in [49].Note that in [27] some endpoint cases (with p i , q i ∈ {1, ∞}) were already discussed for embeddings of Sobolev and certain Besov spaces (with p = q) into Lebesgue spaces.In our paper [19] we were able to further extend Theorem 3.7 in view of the borderline cases.Here we essentially benefited from the strategy of the proof presented in [1] which studies nuclear embeddings of spaces with modified smoothness.
For better comparison one can reformulate the compactness and nuclearity characterisations of id Ω in (3.7) as follows, involving the number t(p 1 , p 2 ) defined in (3.4) .
Hence apart from the extremal cases {p 1 , p 2 } = {1, ∞} (when t(p 1 , p 2 ) = p * ) nuclearity is indeed stronger than compactness.We observed similar phenomena -including the replacement of p * and q * (for compactness assertions) by t(p 1 , p 2 ), t(q 1 , q 2 ) (for their nuclearity counterparts) -in the weighted setting in [19] as well as for vector-valued sequence spaces and function spaces on quasi-bounded domains in [12].

Nuclear embedding of Morrey sequence spaces
Our main goal is to find a counterpart of Theorem 3.7 when id Ω in (3.7) is replaced by id A in (2.24) or id τ in (2.26), respectively.We follow the strategy introduced in [1] and use wavelet decomposition arguments, based on Section 2.2, together with related nuclearity results for appropriate sequence spaces, extending Proposition 3.4 to our setting.

Finite-dimensional Morrey sequence spaces
First we deal with finite-dimensional sequence spaces of Morrey type.
Definition 4.1.Let 0 < p ≤ u ≤ ∞, j ∈ N 0 be fixed and where the supremum is taken over all ν ∈ N 0 and m Remark 4.2.Similarly one can define spaces related to any cube Q −j,m , m ∈ Z d , but they are isometrically isomorphic to m 2 jd u,p , so we restrict our attention to the last space.Clearly, for u = p this space coincides with the usual 2 jd -dimensional space ℓ 2 jd p , that is, m 2 jd p,p = ℓ 2 jd p .Moreover m 2 jd ∞,p = ℓ 2 jd ∞ for any p ≤ ∞, cf.[18].In the sequel it will be convenient to denote this space by ∞, and j ∈ N 0 be given.Then the norm of the compact identity operator satisfies and in the remaining case, there is a constant c, 0 < c ≤ 1, independent of j, such that Remark 4.4.The above result can be found in [18].
Corollary 4.5.Let 0 < p ≤ u < ∞ and j ∈ N 0 be given.Then the norm of the identity operator ) , whereas the norm of the operator ) Proof.The value of the norm of the operator (4.5) follows directly from the definition of the spaces.The upper estimate of the norm of the second operator can be proved in a similar way.The estimate from below follows from (4.3) and the simple factorisation Now we can give its counterpart for the nuclear norm ν(id j ) which also extends Tong's result, Proposition 3.3(ii), from spaces ℓ 2 jd p to m 2 jd u,p .
Proposition 4.6.Let 1 ≤ p i ≤ u i < ∞, or p i = u i = ∞, i = 1, 2, j ∈ N 0 , and id j be given by (4.2).Then the nuclear norm of id j satisfies and in the remaining case, there is a constant c ≥ 1, independent of j, such that Proof.
Step 1.First we deal with all the estimates from below and benefit from Lemma 4.3 and Corollary 4.5.Note that by Proposition 3.3(i) and (iii), Now in the first three cases of (4.7) we have id j : m 2 jd u2,p2 ֒→ m 2 jd u1,p1 = 1, such that (4.9) leads to ν(id j ) ≥ 2 jd as desired.In the last case of (4.7), Lemma 4.3 and Corollary 4.5 provide id j : ) and thus (4.9) implies ν(id j ) ≥ 2 jd(1− 1 u 1 ) .Finally, in case of (4.8), then in view of Lemma 4.3 again, especially (4.4), which together with (4.9) completes the lower estimate in that case.
Step 2. Now we show the estimates from above in the first three cases of (4.7).We have the following commutative diagram 3) and (3.6) with n = 2 jd .
Step 3. Next we deal with the estimate from above in the last case of (4.7).Here we use the following commutative diagram ) .

and for any cube
One can easily verify that ε|m 2 jd u2,p2 = 1 for any ε ∈ E: for smaller cubes Q −ν,m ′ ⊂ Q −ν0,m , that is, with 0 ≤ ν ≤ ν 0 , there is at most one non-vanishing coefficient inside, thus there are exactly 2 (ν−ν0)d non-vanishing coefficients in the corresponding sum, Here we used (4.10).Thus (4.1) leads to ε|m 2 jd u2,p2 = 1.(4.11) Let us now fix for a moment a certain cube Q −ν0,m0 ⊂ Q −j,0 .For any Q 0,k ⊂ Q −ν0,m0 we shall denote by E k the subset of E that consists of all sequences with ε k = 0. Then E k1 ∩ E k2 = ∅ if k 1 = k 2 and all the sets E k have the same number of elements Let λ ∈ m 2 jd u1,p1 and denote by ( ε∈E ε(λ)ε) k the k-th coordinate of the image of λ in the linear mapping ε∈E ε ⊗ ε.One can easily check that Hence the family E gives the following representation of id j where we also applied (4.12).Thus (3. ) λ|m 2 jd u1,p1 k∈Kj ) , where we used λ|m ) and (4.13) results in as desired.This concludes the proof.Remark 4.7.In case of p i = u i , i = 1, 2, Proposition 4.6 coincides with Tong's result (3.6).Note that the situation (4.8) cannot appear in that setting, but refers to a 'true' Morrey situation for the target space m 2 jd u2,p2 .

The main result for Morrey sequence spaces
Let Q be a unit cube, 0 < p ≤ u < ∞ or p = u = ∞, σ ∈ R, 0 < q ≤ ∞.We define a sequence space n σ u,p,q (Q) putting with the usual modification when q = ∞.Moreover, for fixed j ∈ N 0 , we put where Then the embedding is nuclear if, and only if, the following condition holds . (4.17) Proof.
Step 1.First we prove the sufficiency of the condition (4.17) for the nuclearity of Id.We decompose Id into the sum Id = ∞ j=0 Id j where the operators Id j : n σ1 u1,p1,q1 ֒→ n σ2 u2,p2,q2 are defined in the following way Now we factorise the operator Id j through the embedding of finite-dimensional spaces id j : m 2 jd u1,p1 → m 2 jd u2,p2 .Namely we have the following commutative diagram, where P j denotes the projection of n σ1 u1,p1,q1 onto the j-level, and S j is the natural injection of the finitedimensional j-level spaces into n σ2 u2,p2,q2 .It is easy to check that P j : n σ1 u1,p1,q1 → m ) and S j : m Step 2. We come to the necessity of (4.17 where the operators P j and S j are defined in a similar way to P j and S j , but now for the spaces m (j) u2,p2 and n σ2 u2,p2,q2 , or m (j) u1,p1 and n σ1 u1,p1,q1 , respectively.We have P j : n σ2 u2,p2,q2 → m ) and S j : m  p2 }u 2 ) and we are left to disprove equality of the terms above to complete the proof of (4.17).
) and assume that the embedding (4.16) is nuclear.We consider the following diagram m