Extreme cases of limit operator theory on metric spaces

The theory of limit operators was developed by Rabinovich, Roch and Silbermann to study the Fredholmness of band-dominated operators on $\ell^p(\mathbb{Z}^N)$ for $p \in \{0\} \cup [1,\infty]$, and recently generalised to discrete metric spaces with property A by \v{S}pakula and Willett for $p \in (1,\infty)$. In this paper, we study the remained extreme cases of $p \in\{0,1,\infty\}$ (in the metric setting) to fill the gaps.


Introduction
As in linear algebra, a linear operator A on ℓ p (Z N ) can be regarded as a Z N -by-Z N matrix. We say that A is a band operator if all non-zero entries in its matrix sit within a fixed distance from the diagonal, and that A is a band-dominated operator if it is a norm-limit of band operators. For each k ∈ Z N , the k-shift operator V k : ℓ p (Z N ) → ℓ p (Z N ) maps (x i ) i∈Z N to (y i ) i∈Z N with y i+k = x i . Given a band-dominated operator A and a sequence (k m ) m∈N in Z N tending to infinity, the sequence (V −k m AV k m ) m∈N of shifts of A always contains an entry-wise convergent subsequence by Bolzano-Weierstrass Theorem and a Cantor diagonal argument, and the limit is called a limit operator of A. The collection of all limit operators of A is called the operator spectrum of A. See the excellent book [10] and the survey paper [3] for relevant references.
Intuitively, the Fredholmness of a band-dominated operator is invariant under compact perturbations such as arbitrary modifications to finitely many entries of the associated matrix, so it should be "encoded" in the asymptotic behaviours of the matrix. This leads to the central problem in limit operator theory: to study the Fredholmness of a band-dominated operator on the discrete domain Z N in terms of its operator spectrum. Major work has been done around it by Lange, Rabinovich, Roch, Roe and Silbermann [6,10,11,12] with several recent contributions by Chandler-Wilde, Lindner, Seidel and others [3,8,9,15]. Theorem 1.1 ([8, 10]). Let A be a band-dominated operator on ℓ p (Z N ), where p ∈ {0} ∪ [1, ∞]

. Then A is Fredholm if and only if all limit operators of A are invertible and their inverses are uniformly bounded in norm.
It had been a longstanding question whether the requirement of uniform boundedness can be removed in the above theorem, and was recently answered by Lindner and Seidel affirmatively [9]. In some literatures (for example [3,10]), Theorem 1.1 is stated in a Banach space valued version, which makes it possible to deal with continuous spaces as well.
Roe realised that it is essentially the coarse geometry of the underlying space that plays a key role in the limit operator theory. In [13], he generalised limit operators in the case of p = 2 to all discrete groups and proved Theorem 1.1 for exact discrete groups. Following Roe's philosophy,Špakula and Willett [17] introduced and studied limit operator theory for any discrete metric space X with Property A. The notion of Property A was introduced by Yu [20], and it is equivalent to exactness in the case of groups. Having noticed that generally there are no natural shift operators on X, they considered the Stone-Čech boundary of X and constructed a canonical limit space associated to each boundary point ω, denoted by X(ω). Intuitively, X(ω) captures the geometry of X as one 'looks towards infinity in the direction of ω'. Furthermore, for a rich band-dominated operator A on ℓ p (X) where p ∈ (1, ∞), they constructed the limit operator Φ ω (A) as a bounded operator on ℓ p (X(ω)).
Having established the above setup,Špakula and Willett proved the following generalisation of Theorem 1.1: Theorem 1.2 ([17]). Let X be a strongly discrete metric space with bounded geometry, p ∈ (1, ∞) and E be a Banach space. Assume that X has Property A. Let A be a rich band-dominated operator on ℓ p E (X). Then the following are equivalent: (1) A is P-Fredholm; (2) all the limit operators are invertible and their inverses have uniformly bounded norms; (3) all the limit operators are invertible.
In this paper, we study the remained cases of p ∈ {0, 1, ∞} in the above metric setup, and prove that the above theorem still holds in these cases (Theorem 5.1). As we will see in the paper, surprisingly some parts of the proof are quite different from the case of p ∈ (1, ∞), hence new tools and techniques are required. For convenience of the readers, we list the main points as follows.
The first obstacle is to construct limit operators for p = ∞. Analogous construction byŠpakula and Willett in the case of p ∈ (1, ∞) only produces a formal matrix, while we have to verify that it can be realised as the matrix coefficients of a unique bounded operator. This holds trivial when p < ∞, while unfortunately, it is not always the case when p = ∞ due to the fact that the set of finitely supported vectors are no longer dense in ℓ ∞ (X) for infinite X.
We work out by proving a density result for p ∈ {0, 1, ∞}, providing an approach to approximate rich band-dominated operators via rich band operators. Notice that when p ∈ (1, ∞), this has already been shown under the additional assumption of Property A on the underlying space [17,Theorem 6.6], which is not necessary for p ∈ {0, 1, ∞} from our arguments. We would also like to mention that the approach we develop here also plays a role in a very recent work byŠpakula and the author to study quasi-locality and property A [18].
On the other hand, non-density of finitely supported vectors in ℓ ∞ (X) also prevents us from using the same tools in [17] directly to prove some parts of the result. While alternatively, thanks to the imposition of band-domination on operators, we may still consider only finitely supported vectors via a commutant technique.
Another obstacle here is the lack of duality for p = ∞. In the case of p ∈ (1, ∞), as shown in [17], properties of operators on ℓ p (X) can be easily transferred to their adjoints since they still act on ℓ p -type spaces. Unfortunately, this does not apply to the space ℓ ∞ (X). Instead, we borrow dual-space arguments from [3,7]. Roughly speaking, the idea is to consider the double-predual of ℓ ∞ (X) (i.e., c 0 (X)) and properties of operators on ℓ ∞ (X) can be characterised nicely via their restrictions on c 0 (X), which are much easier to handle.
Outline. For completeness, most of the materials here are written for general p rather than just p ∈ {0, 1, ∞}. In Section 2, we recall several classes of operators on ℓ p -spaces including band(-dominated) operators; and study when operators can be uniquely determined by their matrix coefficients. In Sections 3.1 and 3.2, we recall the notion of limit spaces and limit operators in the case of p ∈ {0} ∪ [1, ∞). After stating the density result (Theorem 3.13) whose proof is postponed to Section 4.2, we show in Section 3.3 how to implement limit operators when p = ∞. Section 4.1 contains several technical lemmas providing approximations of an operator via its block cutdowns, which is mainly devoted to proving Theorem 3.13 and constructing parametrices later. Finally in Section 5, we state our main theorem and provide a detailed proof which is divided into several parts.
For a metric space (X, d), denote by B(x, R) the closed ball in X with radius R and centre x. We say that X has bounded geometry if for any R, the number sup x∈X ♯B(x, R) is finite; that X is strongly discrete if the set {d(x, y) : x, y ∈ X} is a discrete subset of R. We say that 'X is a space' as shortened for 'X is a strongly discrete metric space with bounded geometry' from now on.
Throughout the paper, always denote X to be a space and E to be a Banach space.
2.1. Banach space valued ℓ p -space. We start with the following notions of Banach space valued ℓ p -spaces: , which denotes the Banach space of psummable functions from X to E with respect to the counting measure; , which denotes the Banach space of bounded functions from X to E; Denote by F the set of all finite subsets in X, equipped with the order by inclusion. For any F ∈ F , define an operator P F := ρ(χ F ) on ℓ p E (X) where χ F denotes the characteristic function of F, and set Q F := I − P F . Clearly, the net P := {P F } F∈F satisfies the following conditions for p ∈ {0} ∪ [1, ∞]: Furthermore, denote by ℓ p E,0 (X) the closure of F∈F P F (ℓ p E (X)) in ℓ p E (X). We point out the following elementary observation: Notice that K p E (X), L p E (X) and S p E (X) are also denoted by K(Y, P), L(Y, P) and S(Y) for Y = ℓ p E (X) respectively in some literatures, for example [3,8,10]. Clearly ). Furthermore, we have: Proof. It suffices to show the closeness under multiplication, which is obvious for K p E (X). We only prove for S p E (X), and the argument is similar for L p E (X). Given S, T ∈ S p E (X), for any F ′ ∈ F and ǫ > 0, there exists G ∈ F such that P F ′ SQ G < ǫ 2 T . For such G, there exists some F ∈ F such that P G TQ F < ǫ 2 S . Hence we have: and we finish the proof.
On the other hand, for any ǫ > 0, there exists G ∈ F such that SQ G < ǫ 2 T . For such G, there exists some F ∈ F such that P G TQ F < ǫ 2 S . Hence, So ST ∈ K p E (X), and similarly TS ∈ K p E (X). For the second statement, given T ∈ B(ℓ p E (X)) satisfying that ST and TS sit in K p E (X) for any S ∈ K p E (X), then T ∈ L p E (X) since P ⊆ K p E (X).
Combining them together, we have Taking ǫ → 0, we have Q G Tv → 0, which implies that Tv ∈ ℓ p E,0 (X).
The notions of P-Fredholmness and invertibility at infinity are fundamental in limit operator theory, whose definitions are recalled as follows: and ) and x, y ∈ X, define its xy-matrix coefficient to be T xy = R x TE y ∈ B(E).
A natural question is that to what extent do matrix coefficients determine the operator itself? It is obvious that for ξ ∈ F∈F P F (ℓ p E (X)), Tξ can be reformulated as for each x ∈ X. Passing to the closure and by Lemma 2.1, we obtain that for p ∈ {0} ∪ [1, ∞), an operator on ℓ p E (X) can be determined by its matrix coefficients. Furthermore, in this case we have that for any T ∈ B(ℓ p E (X)), Conversely, we have the following result, whose proof is straightforward. Unfortunately, since finitely supported vectors are no longer dense in ℓ ∞ E (X) generally, things become complicated when p = ∞. However, we still have: x ∈ X, the series y∈X T xy ξ(y) converges to (Tξ)(x), where T xy is the xy-matrix coefficient of T. Furthermore, T = sup F∈F P F TP F .
The proof follows directly from the definition, hence omitted.   (1) Let f : X → B(E) be a bounded function. Then the diagonal matrix defined by is a band operator of propagation 0, called a multiplication operator. (2) Let D, R be subsets of X, and t : D → R be a bijection such that sup x∈D d(x, t(x)) is finite. Define a matrix V by Then V is a band operator, called a partial translation operator.
The above two classes of operators indeed generate C[X; E] as an algebra. The following lemma is very well-known: ([17]). Let A ∈ C[X; E] have propagation at most r and N = sup x∈X ♯B(x, r). Then there exist multiplication operators f 1 , . . . , f N with f k ≤ sup x,y A xy , and partial translation operators V 1 , . . . , V N of propagation at most r such that: Consequently, a band operator can always be realised as a bounded linear operator on any ℓ p -space: Conversely, an operator A ∈ L p E (X) is a band operator if there exists some r > 0 such that A xy = 0 for d(x, y) > r. Taking norm closures, we have: Combined with Lemma 2.4, we end up with the following inclusions:

Limit spaces and limit operators
Throughout this section, X is a space, βX is the associated Stone-Čech compactification and ∂X = X \ βX is the boundary. We use the classical identification for βX in terms of ultrafilters on X. More details can be found in literatures, for example [2, Appendix A] or [17, Appendix A].
3.1. Limit spaces. The notion of limit spaces for general metric spaces were introduced byŠpakula and Willett. In the case of groups, all limit spaces are isometric to the group itself due to homogeneity. Here we only collect the notions and results required later, while more details and illuminating examples can be found in [17,Section 3].
Recall from Example 2.11 that a function t : D → R with D, R ⊆ X is called a partial translation if t is a bijection from D to R, and sup x∈X d(x, t(x)) is finite.
In this case, regarding t as a function from D to βX, define An ultrafilter α ∈ βX is compatible with ω if there exists a partial translation t which is compatible with ω and t(ω) = α. Remark 3.2. As pointed out in [17], an ultrafilter α ∈ βX is compatible with ω ∈ βX if and only if there exists a partial translation t : D → R such that ω(D) = 1 and α(S) = 1 iff ω(t −1 (S ∩ R)) = 1 for any S ⊆ X. Therefore compatibility is an equivalence relation.
Let us recall the following uniqueness statement: Let ω be an ultrafilter on X, and t : D t → R t and s : D s → R s be two partial translations compatible with ω such that s(ω) = t(ω). Then if we have that ω(D) = 1.

Definition 3.4 ([17]
). Fix an ultrafilter ω on X. Write X(ω) for the collection of all ultrafilters on X compatible with ω. A compatible family for ω is a collection of partial translations {t α } α∈X(ω) such that each t α is compatible with ω and t α (ω) = α.
A metric can be imposed on X(ω) as follows:

Proposition 3.5 ([17]). Fix an ultrafilter ω on X, and a compatible family
Then d ω is a metric on X(ω) that does not depend on the choice of {t α }. Moreover,

Definition 3.6 ([17]
). For each non-principal ultrafilter ω on X, the metric space (X(ω), d ω ) is called the limit space of X at ω.
It is shown in [17,Proposition 3.9] that X(ω) does not depend on the choice of ω in the sense that for any α ∈ X(ω), X(α) = X(ω). Now we recall the following result stating that the local geometry of X can be captured by that of its limit space.

Proposition 3.7 ([17]). Let ω be a non-principal ultrafilter on X, and {t
is a surjective isometry. Such a collection { f y } y∈Y is called a local coordinate systerm for F, and the maps f y are called local coordinates.
Interesting examples of limits spaces can be found in [17, Example 3.14].
3.2. Limit operators. Now we introduce the notion of limit operators on metric spaces. The case of p ∈ (1, ∞) was studied byŠpakula and Willett [17], while for the convenience to the readers, they are stated here as well. We start with the following condition of richness for an operator, which is designed to ensure that limit operators always exist.
is said to be rich at ω, if for any pair of partial translations t, s compatible with ω, the limit exists under the norm topology on B(E). Denote by A p E (X) $,ω the collection of all band-dominated operators rich at ω.
If A is rich at ω for all ω ∈ ∂X, it is said to be rich. Denote by A p E (X) $ the collection of all rich band-dominated operators.
From Lemma 3.3, we know the above definition is proper: Up till now, Φ ω (A) is only an abstractly defined infinite matrix, rather than an operator on any space. We will follow the way in [17] to make it concrete in the case of p ∈ {0} ∪ [1, ∞) as follows, while leave the case of p = ∞ to the next subsection where more technical tools are developed. Proposition 3.11. Let ω be a non-principal ultrafilter on X, p ∈ {0} ∪ [1, ∞] and A a band-dominated operator on ℓ p E (X) rich at ω. Let F be a finite subset of X(ω) and ǫ > 0. Let {t α } be a compatible family of partial translations for ω. Then there exists a local ) is the linear isometry induced by f y , then we have: The proof follows directly from Proposition 3.7 and is the same as that of [17,Proposition 4.6], hence omitted. Consequently, we have the following concrete implementation for limit operators in the case of p ∈ {0} ∪ [1, ∞). The proof is just a combination of Lemma 2.8 and the above proposition, hence omitted.
Therefore in the case of p ∈ {0}∪[1, ∞), we may regard limit operators as concrete bounded linear operators on ℓ p -type spaces. See [17,Section 4] for illuminating examples, and [17,Appendix B] for the comparisons with the classical limit operators in the case of groups. 3.3. Implementing limit operators for p = ∞. As we point out in Section 2.3, operators on ℓ ∞ E (X) may not be uniquely determined by their matrix coefficients in general. Hence, to realise limit operators via matrices in this case, we need an extra density result as follows. It is stated for p ∈ {0, 1, ∞} due to further uses.
The proof is postponed to Section 4.2 after we develop the tool to construct operators via blocks. We point out that the density result also holds for p ∈ (1, ∞) under the extra assumption of Property A on X [17, Theorem 6.6], while surprisingly as we will see, it holds generally for p ∈ {0, 1, ∞}. Now we use Theorem 3.13 to show that Φ ω (A) can be implemented as a bounded operator when p = ∞. We start with the following lemma.
Lemma 3.14. Let ω be a non-principal ultrafilter on X, and A ∈ C[X; E] be a band operator on ℓ ∞ E (X) rich at ω. Then Φ ω (A) is a band operator whose propagation does not exceed that of A, and Φ ω (A) ≤ A .
Proof. The claim on propagation follows directly from that of [17,Theorem 4.10(2)], hence omitted. Concerning norms, note that Φ ω (A) ∈ C[X(ω); E] ⊆ S ∞ E (X(ω)). Hence by Lemma 2.9, Φ ω (A) = sup F∈F P F Φ ω (A)P F . On the other hand, by lemma 3.11, P F Φ ω (A)P F ≤ A for any finite F ⊆ X. So the lemma holds. Proposition 3.15. Let X be a space and ω a non-principal ultrafilter on X. Suppose and norm at most A . Furthermore, this operator belongs to A ∞ E (X(ω)).
Proof. By Theorem 3.13, there exist band operators {A n } n∈N on ℓ ∞ E (X) rich at ω and converging to A. By Lemma 3.14, each Φ ω (A n ) is a band operator and {Φ ω (A n )} n∈N is Cauchy, hence it converges to some T in A ∞ E (X(ω)). For any α, β ∈ X(ω) and for any ǫ, which implies that So T is the required operator and it is unique by Lemma 2.9.   Later in Theorem 5.1, we will detect the properties of P-Fredholmness and invertibility at infinity for a rich band-dominated operator in terms of its operator spectrum for p ∈ {0} ∪ [1, ∞].

Partition of unity and the density theorem
In this section, as promised, we develop the technique to approximate operators via block cutdowns, especially in the case of p ∈ {0, 1, ∞}. They are not only crucial to prove the density result (Theorem 3.13) above, but also useful to construct parametrices in later proof of our main theorem.
Throughout this section, X is a space and E is a Banach space. For p ∈ [1, ∞), set q ∈ (1, ∞] to be the conjugate exponent of p, i.e., 1/p + 1/q = 1. [20], and since then it has been shown to be equivalent to a lot of other properties. Here we shall use the formulation in terms of the existence of partitions of unity with small variation [19,Theorem 1.2.4], which helps us to cut an operator into blocks. (1) There exists N ∈ N such that for each x ∈ X, at most N of φ i (x) are non-zero.

Property A, partition of unity, and constructing operators. Property A was first introduced by Yu
(2) {φ i } i∈I have uniformly bounded supports, i.e., A space X is said to have Property A if for any r, ǫ > 0, there exists a metric p-partition of unity with (r, ǫ)-variation.
A lot of interesting spaces and groups are known to have Property A. For example, amenable groups [20], hyperbolic spaces [14], CAT(0) cube complexes with finite dimension [1] and all linear groups over any field [5].
To deal with the extreme cases that p ∈ {0, 1, ∞}, we introduce the following notion of dual family.

Definition 4.2.
Let {φ i } i∈I be a metric 1-partition of unity on X. A dual family of {φ i } i∈I is defined to be a collection {ψ i : X → [0, 1]} i∈I satisfying ψ i | supp(φ i ) ≡ 1 and there exists some R > 0 such that supp(ψ i ) ⊆ N R (supp(φ i )) for any i ∈ I. A dual Clearly, for any metric 1-partition of unity on X and any L > 0, L-Lipschitz dual family always exists. And there exists N ′ ∈ N such that for each x ∈ X, at most N ′ of ψ i (x) are non-zero since X has bounded geometry.
Furthermore, to deal with the case of p = ∞, we also need the following topology on B(ℓ ∞ E (X)). Definition 4.3. Let {B i } i∈I be a collection of bounded linear operators on ℓ ∞ E (X). If for any v ∈ ℓ ∞ E (X), the series i∈I B i v converges point-wise to a vector in ℓ ∞ E (X), and the map is a bounded linear operator, then we say i∈I B i converges point-wise strongly.
Clearly by definition, for a metric 1-partition of unity {φ i } i∈I , i∈I φ i converges point-wise strongly to the identity on ℓ ∞ E (X). For compositions, we have the following elementary observation:

Lemma 4.4. Let {B i } i∈I be a collection of linear bounded operators on ℓ ∞
E (X) such that i∈I B i converges point-wise strongly, and A ∈ B(ℓ ∞ E (X)). Then: (1) i∈I B i A converges point-wise strongly, and we have i∈I B i A = i∈I B i A. (2) If A is a band operator, then i∈I AB i converges point-wise strongly, and we have A i∈I B i = i∈I AB i .
(1) holds directly by definition and for (2), we only need to verify in the case that A is a multiplication operator or a partial translation by Lemma 2.12. Both of them are straightforward, hence we omit the proof. Now we are in the position to construct operators via blocks. To unify the statements, we take the liberty of calling a metric p-partition of unity for p ∈ {0, ∞} instead of a metric 1-partition of unity. (1) When p < ∞, consider the following: Each of them converges strongly to a band operator of norm at most M on ℓ p E (X).
(2) When p = ∞, consider i∈J φ i B i ψ i . It converges point-wise strongly to a band operator of norm at most M on ℓ ∞ E (X).
Proof. Case (1)(a) is from [17, Lemma 6.3], hence omitted. We start with Case (1)(b). For any v ∈ ℓ 0 E (X) with finite support, only finitely many of the terms in i∈J φ i B i ψ i v is non-zero, hence produces a well-defined vector in ℓ 0 E (X). For any x ∈ X, we have Due to density of finitely supported vectors in ℓ 0 E (X), (4.1) holds for all v ∈ ℓ 0 E (X). We move on to Case (1)(c): Again, for any v ∈ ℓ 1 E (X) with finite support, Due to density of finitely supported vectors in ℓ 1 E (X), the above estimates hold for all v ∈ ℓ 1 E (X). Finally we deal with Case (2). For any v ∈ ℓ ∞ E (X) and any x ∈ X, we have which is a finite sum since φ i (x) is non-zero for only finitely many i ∈ J. Furthermore, (4.1) still holds for such v, hence the result holds. (1) When p < ∞, consider the following:

Each of them converges strongly to a band operator of norm
converges point-wise strongly to a band operator of norm ≤ ǫNM A on ℓ ∞ E (X).
Proof. By Lemma 2.12, A has the form of N k=1 f k V k where each f k : X → B(E) is a bounded function with norm at most A , and V k is a partial translation operator in C[X; E] defined by t k : D k → R k with propagation at most r. For any function ϕ : X → R, v ∈ ℓ p E (X) and x ∈ X, we have otherwise.

Case (1)(a) is from [17, Lemma 6.4], hence omitted. Starting with Case (1)(b): for any finitely supported
Due to density of finitely supported vectors in ℓ 0 E (X), (4.4) holds for all v ∈ ℓ 0 E (X). We move on to Case (1)(c): Again, for any v ∈ ℓ 1 E (X) with finite support, Note that {φ i } i∈J has (r, ǫ)-variation and d(x, t −1 k (x)) ≤ r, hence by (4.2): for each k. Combining with (4.5), we have Due to density of finitely supported vectors in ℓ 1 E (X), (4.6) holds for all v ∈ ℓ 1 E (X). Finally we deal with Case (2). For any v ∈ ℓ ∞ E (X) and any x ∈ X, we have which is a finite sum since φ i (x) is non-zero for only finitely many i ∈ J. Furthermore, (4.3) and (4.4) still hold for such v, hence the result holds.
On the other hand, we have the following lemma, whose proof is quite similar to the above, hence omitted. (1) When p < ∞, consider the following: Each of them converges strongly to a band operator of norm ≤ ǫNM A on ℓ p E (X).
(2) When p = ∞, suppose {φ i } i∈I has (r, ǫ)-variation and consider i∈J [φ i , A]B i ψ i . It converges point-wise strongly to a band operator of norm ≤ ǫNM A on ℓ ∞ E (X).

Density of rich band operators.
Having established the technical lemmas above, we are now ready to prove Theorem 3.13. First let us fix a metric 1-partition of unity {φ i } i∈I on X. For example, one may take an arbitrary disjoint bounded cover {U i } i∈I of X, and φ i to be the characteristic function of U i . For each n ∈ N, take {ψ (n) i } i∈I to be a 1/n-Lipschitz dual family of {φ i } i∈I .

Then each M n is a well-defined linear operator of norm at most one. Moreover, M n (A) is a band operator and M n (A) → A in norm as n → ∞ for each A ∈ A
p E (X).
Proof. By Lemma 4.5, M n is well-defined and has norm at most one. Clearly, M n (A) is a band operator for all A ∈ A p E (X). For the convergence statement, we treat them separately.
• p = 0 or ∞: for each n and any band operator A, where all sums converge strongly when p = 0, and point-wise strongly when p = ∞ by Lemma 4.5 and 4.6. Since {ψ (n) i } i∈I is a dual family of {φ i } i∈I , we have i∈I φ i ψ (n) i A = i∈I φ i A, which converges strongly to A when p = 0, and converges point-wise strongly to A when p = ∞ by Lemma 4.4. For the second term, it has norm at most A N/n for some fixed N by Lemma 4.6, hence tends to 0. Finally note that M n ≤ 1 for all n, so the result also holds for band-dominated operators.
• p = 1: for each n and any band operator A, where all sums converge strongly by Lemma 4.5 and 4.7. Again we have i∈I Aψ (n) For the second term, it has norm at most A N/n for some fixed N by Lemma 4.7, hence tends to 0. Finally note that M n ≤ 1 for all n, so the result holds for band-dominated operators as well.
Now the proof of Theorem 3.13 follows from the same argument as that of [17,Theorem 6.6], while replacing [17,Corollary 6.5] with Lemma 4.8 we just established above, hence omitted. Also notice that [17,Corollary 6.5] is the only place where Property A is used in proving [17, Theorem 6.6], hence it is not necessary for Theorem 3.13 since Lemma 4.8 holds generally.

The Main Theorem
We are now in the position to state our main theorem, which characterises the properties of P-Fredholmness and invertibility at infinity for a rich banddominated operator in terms of its operator spectrum. Note that for p ∈ (1, ∞), the above theorem is exactly [17, Theorem 5.1]. Our major work here is to fill the gaps of p ∈ {0, 1, ∞}. Meanwhile, it is somewhat surprising that in these extreme cases, some parts of the theorem holds without the assumption of Property A, for example "(4) ⇔ (5)".

Theorem 5.1. Let X be a space with Property A, E a Banach space and p ∈ {0} ∪ [1, ∞]. Let A be a rich band-dominated operator on ℓ p E (X). Then the following are equivalent: (1) A is invertible at infinity; (2) A is P-Fredholm; (3) A is invertible in A
The proof occupies the rest of the paper, which is divided into several parts. In Section 5.1, we deal with the equivalence between (1) and (2); from Section 5.2 to 5.4, we establish the equivalence between (2), (3) and (4); and finally, in Section 5.5, we prove "(4) ⇔ (5)".

Commutant technique.
We prove "(1) ⇔ (2)" using a commutant technique, inspired by the classical limit operator theory for Z n and part of the results in [16]. Let us start with the following class of operators on ℓ p E (X). We remark that elements in C are exactly quasi-local operators (see [16]). The following lemma is direct from definition, hence the proof is omitted.
Remark 5.4. Conversely, we would like to mention that very recently after the paper is finished,Špakula and the author [18] showed that A p E (X) = C either if X has Property A and p ∈ (1, ∞), or without any assumption on X when p ∈ {0, 1, ∞}, partially using the tools developed in Section 4 to approach the latter. While we do not need appeal to that result in this paper.

Lemma 5.5. For any p
Proof. Fix T ∈ C and a finite subset F ⊆ X. For any ǫ > 0, there exists some L > 0 such that [T, f ] < ǫ for any L-Lipschitz function f ∈ C b (X) 1 . Now for any finite G ⊆ X with d(F, G c ) > 1/L, there exists an L-Lipschitz function g ∈ C b (X) with range in [0, 1] and satisfying g| F ≡ 1 and g| G c ≡ 0. Hence [T, g] < ǫ. Consequently, We finish the proof.
Proof of Theorem 5.1, "(1) ⇔ (2)". Clearly, (2) implies (1). For the other direction, assume that there exists some B ∈ B(ℓ p E (X)) such that AB = I + K 1 and BA = I + K 2 for some K 1 , K 2 ∈ K p E (X). By Lemma 5.3, A ∈ A p E (X) ⊆ C. We claim that B ∈ C as well, hence B ∈ L p E (X) by Lemma 5.5. Indeed for any ǫ > 0, there exists some L 1 > 0 such that [A, f ] < ǫ/(3 B 2 ) for any L 1 -Lipschitz function f ∈ C b (X) 1 . On the other hand, since K 1 , K 2 ∈ K p E (X), there exists some finite subset F 0 ⊆ X such that and Choose a point x 0 ∈ X, and take For any L-Lipschitz function f ∈ C b (X) 1 , we takef = f − f (x 0 ). Then For the above F 0 ⊆ X, we have By (5.1), we obtain (5.4) Bf Q F 0 K 1 < ǫ/6 and K 2 Q F 0f B < ǫ/6. Furthermore, since x 0 ∈ F 0 , we have , which implies that Hence combining (5.2), (5.3), (5.4) and (5.5), we have: Hence we finish the proof.
From the above proof, we obtain a bonus result as follows.

"(2) ⇒ (4)" for finite p.
We move on to prove "(2) ⇒ (4)" without the assumption of Property A. First, in this subsection, we deal with the case of finite p. The idea follows partially from [17] together with the commutant technique developed above. However, things become complicated when p = ∞ due to lack of characterisation of the dual space of ℓ ∞ E (X). Hence we leave it to the next subsection after new tools are introduced.
Let us start with the following lemma. The proof is almost the same as that of [17,Lemma 5.3], together with Theorem 3.13 and Proposition 3.17, hence omitted.
Proof. Since A is invertible at infinity, there exists a bounded operator B on ℓ p E (X) such that K 1 := AB − I and K 2 := BA − I are in K p E (X). We claim that for any ω ∈ ∂X and finitely supported v ∈ ℓ p E (X(ω)), To prove the claim, we fix a ω ∈ ∂X, a finitely supported v ∈ ℓ p E (X(ω)) and an ǫ > 0. Since K 2 ∈ K p E (X), there exists some finite G ⊆ X such that K 2 Q G < ǫ. By Lemma 5.7, there exists a unit vector w ∈ ℓ p E (X) such that Aw − Φ ω (A)v < ǫ and supp(w) ∩ G = ∅. Hence we have K 2 w < ǫ, and Combining them together, we have Letting ǫ → 0, we obtain (5.6) as required and the claim holds. Now for p < ∞, (5.6) holds for all vectors in ℓ p E (X(ω)) since finitely supported vectors are dense in ℓ p E (X(ω)). For p = ∞, we fix a vector v ∈ ℓ ∞ E (X(ω)) and an ǫ > 0. There exists some α ∈ X(ω) such that v(α) E > v ∞ − ǫ. Applying Proposition 3.17 and Lemma 5.3, Φ ω (A) ∈ A ∞ E (X(ω)) ⊆ C. Hence there exists some L > 0 such that for any L-Lipschitz function f ∈ C b (X(ω)) 1 , we have [Φ ω (A), f ] < ǫ. Take an L-Lipschitz function f with range in [0, 1] and supported in the 1/L-neighbourhood of α with f (α) = 1. Note that f v is finitely supported, hence (5.6) holds for f v. Furthermore, Combining them together, we have Taking ǫ → 0, (5.6) holds for any v ∈ ℓ ∞ E (X(ω)) as required. We finish the proof.
We need the following auxiliary lemma concerning adjoints of limit operators. The proof is straightforward, hence omitted. Notice that for p = 0, we set q = 1 as its conjugate exponent; and for p ∈ [1, ∞), we set its conjugate exponent to be q ∈ (1, ∞] satisfying 1/p + 1/q = 1. On the other hand, since p ∈ {0} ∪ [1, ∞), we take q ∈ [1, ∞] to be its conjugate exponent. Consider the adjoint A * , which is a rich band-dominated operator on ℓ q E * (X) by Lemma 5.9. Since A is invertible at infinity, there exists a bounded operator B on ℓ p E (X) such that K 1 := AB − I and K 2 := BA − I are in K p E (X), which implies that K * 1 = B * A * − I and K * 2 = A * B * − I are in K q E * (X). Hence, A * is invertible at infinity as well. Applying Proposition 5.8 to A * , the operator spectrum σ op (A * ) is uniformly bounded below, which implies {Φ ω (A) * } ω∈∂X is uniformly bounded below by Lemma 5.9. Therefore, condition (4) holds and we finish the proof.
Remark 5.10. The above proof does not work for p = ∞, since the dual of ℓ ∞ E (X) is no longer an ℓ p -type space. Hence we cannot refer to Proposition 5.8 any more. New techniques are required, which are introduced in the next subsection. 5.3. Dual space arguments for p = ∞. Now we focus on the case of p = ∞ and finish the proof of "(2) ⇒ (4)" completely. The key ingredient here is the dual space argument, which showed its power in the classical limit operator theory. The idea is that properties of operators on ℓ ∞ -spaces can be characterised via those of its "double predual" on c 0 -spaces.
Recall that from Lemma 2.1, ℓ 0 . We have the following elementary observation.
. The following proposition is taken from [3]. Although the setting there is X = Z N , the proof applies to any general space X, hence omitted. Proposition 5.12 (Corollary 6.20, [3]). For A ∈ L ∞ E (X), it holds that A is invertible if and only if A 0 is invertible. In this case, we have A −1 0 = (A −1 )| ℓ 0 E (X) . Consequently, we have the following result for operator spectra under restriction.

Corollary 5.13. Let A ∈ A ∞
E (X) be a rich band-dominated operator, and A 0 ∈ A 0 E (X) be its restriction on ℓ 0 E (X). Then: : ω ∈ ∂X . In particular, the invertibility of all limit operators of A 0 with uniform boundedness of their inverses is equivalent to the same property for the limit operators of A.
Proof of Theorem 5.1, "(2) ⇒ (4), p = ∞". Since A is P-Fredholm, there exists some B ∈ L ∞ E (X) such that K 1 := AB − I and K 2 := BA − I are in K ∞ E (X). By Lemma 2.6, ℓ 0 E (X) is invariant under B. Denote its restriction on ℓ 0 E (X) by B 0 . By Lemma 5.11, K 1 | ℓ 0 E (X) and K 2 | ℓ 0 E (X) are in K 0 E (X). Hence A 0 is P-Fredholm as well. Since we already proved "(2) ⇒ (4)" for p = 0 in Section 5.2, all limit operators of A 0 are invertible and their inverses are uniformly bounded. Finally, by Corollary 5.13, the same holds for A. So we finish the proof. 5.4. Constructing parametrices. Now we move on to prove "(4) ⇒ (3)", following the ideas in [8] and [17] and using the tools we developed in Section 4 instead. First we recall the following result to construct parametrices.
The proof is the same as that of [17, Lemma 6.8], hence omitted. Now we prove "(4) ⇒ (3)" first for rich band operators. Proof. The case of p ∈ (1, ∞) follows from [17, Proposition 6.7], hence omitted. We focus on p ∈ {0, 1, ∞}. prop(A)). Let {φ i } i∈I be a metric 1partition of unity with (prop(A), ǫ)-variation, and {ψ i } i∈I be an ǫ/prop(A)-Lipschitz dual family for {φ i } i∈I . Applying Lemma 5.14 to the cover {supp(ψ i )} i∈I of X, we obtain a finite subset K in I, and operators B i , C i satisfying (5.8) for i ∈ I \ K. Set P i := P supp(ψ i ) . Now we divide the proof into two cases.
• p = 0 or ∞: by Lemma 4.5, the sum i∈I\K φ i B i ψ i converges strongly or pointwise strongly to an operator on ℓ p E (X) with norm at most M. Consider where the first equality follows from Lemma 4.4 when p = ∞. All the series above converge strongly or point-wise strongly and by Lemma 4.6, T 0 ≤ 1/2. Hence I + T 0 is invertible and (I + T 0 ) −1 ≤ 2. Furthermore, (I + T 0 ) −1 is given by a Neumann series and thus still in A p E (X). Hence the operator Similarly, we have that A · A R − I ∈ K p E (X). • p = 1: Setting and Similarly, by (1)

Uniform boundedness.
Finally we deal with the equivalence between (4) and (5). The case of p ∈ (1, ∞) was proved in [17,Theorem 7.3] under the assumption of Property A. Here we focus on p ∈ {0, 1, ∞} and show that Property A is not necessary for "(4) ⇔ (5)" in this case. Let us start with the following notion.
Furthermore, if F ⊆ X and A ∈ B(ℓ p E (X)), denote the restriction of A to ℓ First we deal with the case of p = ∞, which is clearly a corollary of the following theorem. The proof is divided into several pieces. First recall that the class C of operators on ℓ ∞ E (X) is defined in Section 5.1, such that all band operators sit inside C. Furthermore, we have the following uniform version. Proof. Let N = sup x∈X ♯B(x, r). By Lemma 2.12, there exist multiplication operators f 1 , . . . , f N with f k ≤ A ≤ M, and partial translation operators V 1 , . . . , V N of propagation at most r defined by partial translations t k : D k → R k such that A = N k=1 f k V k . Note that for any k = 1, . . . , N, f ∈ C b (X) and v ∈ ℓ ∞ E (X), we have otherwise.
Hence, taking L = ǫ/(rMN) and for any L-Lipschitz function f ∈ C b (X) 1 , we have So we finish the proof.
The following result asserts the phenomenon of "lower norm localisation", which should be regarded as a dual version of operator norm localisation introduced in [4]. Proof. The first inequality is trivial. For the second, fix δ > 0, M, r ≥ 0 and F ⊆ X. By Lemma 5.19, there exists L = L(δ, r, M) > 0 satisfying the requirement thereby for ǫ = δ/4, M and r. Set s = 2/L and take a unit vector v ∈ ℓ ∞ E (F) such that Av ≤ ν(A| F ) + δ/4, and x 0 ∈ F such that v(x 0 ) > 1 − δ/(2ν s (A| F )). Also take an L-Lipschitz function f ∈ C b (X) 1 , supported in B(x 0 , 1/L) and f (x 0 ) = 1. Hence, and f v ∈ ℓ ∞ E (F) has support in B(x 0 , 1/L) ∩ F. Furthermore, we have Combine them together, we have So we finish the proof.
Corollary 5.21. Let X be a space and A be a band-dominated operator on ℓ ∞ E (X). Then for any δ > 0, there exists s > 0 such that for all ω ∈ ∂X such that A is rich at ω, and all F ⊆ X(ω). Consequently, we obtain Theorem 5.1, "(5) ⇔ (4), p = ∞". Next we deal with the remained case of p ∈ {0, 1}, and we need the following extension result: Lemma 5.22 (Lemma 3.18, [3]). Every A ∈ S 0 E (X) has a unique extension to an operatorÂ ∈ S ∞ E (X). It holds that Â = A , and if A ∈ L 0 For p = 0, let A ∈ A 0 E (X) be a rich operator satisfying condition (5). By Lemma 5.22 and Proposition 5.12, we may consider its extensionÂ ∈ A ∞ E (X) and condition (5) still holds forÂ. Since we already proved the result for p = ∞, condition (4) holds forÂ. By Corollary 5.13, we know that (4) also holds for A as well. So we finish the proof.