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 ℓp(ZN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^p(\mathbb {Z}^N)$$\end{document} for p∈{0}∪[1,∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in \{0\} \cup [1,\infty ]$$\end{document}, and recently generalised to discrete metric spaces with Property A by Špakula and Willett for p∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in (1,\infty )$$\end{document}. In this paper, we study the remaining extreme cases of p∈{0,1,∞}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in \{0,1,\infty \}$$\end{document} (in the metric setting) to fill the gaps.


Introduction
As in linear algebra, a linear operator A on 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 banddominated operator if it is a norm-limit of band operators. For each k ∈ Z N , the k-shift operator V k : p (Z N ) → 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 −km AV km ) 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 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 ∞ (X) for infinite X.
We proceed 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 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 ∞ (X) also prevents us from using the same tools in [17] directly to prove some parts of the result. However, 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 p (X) can be easily transferred to their adjoints since they still act on p -type spaces. Unfortunately, this does not apply to the space ∞ (X). Instead, we borrow dual-space arguments from [3,7]. Roughly speaking, the idea is to consider the doublepredual of ∞ (X) (i.e., c 0 (X)) and properties of operators on ∞ (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 Sect. 2, we recall several classes of operators on p -spaces including band(-dominated) operators; and study when operators can be uniquely determined by their matrix coefficients. In Sects. 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 Sect. 4.2, we show in Sect. 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 Sect. 5, we state our main theorem and provide a detailed proof which is divided into several parts.

Preliminaries
We collect several background notions in operator algebra theory and establish our settings in this section. See [3,10] for more information. For a metric space (X, d) and R > 0, denote by B(x, R) the closed ball in X with radius R and centre x; and for a subset A ⊆ X, denote by N R (A) the R-neighbourhood of A in 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.

Banach Space Valued p -Space
We start with the following notions of Banach space valued p -spaces: • p E (X) := p (X; E) for p ∈ [1, ∞), which denotes the Banach space of psummable functions from X to E with respect to the counting measure;  [1, ∞], ρ : C b (X) → B( p E (X)) is a representation defined by point-wise multiplication. To simplify notations, we write fξ instead of ρ(f )(ξ) for f ∈ C b (X) and ξ ∈ p E (X). 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 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, ∞]: . We point out the following elementary observation:

Classes of Operators
Here we recall several classes of operators on p E (X  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: 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.

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. 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. Let ω be a non-principal ultrafilter on X, p ∈ {0} ∪ [1, ∞] and A be a band-dominated operator on p E (X) rich at ω. Fix a compatible family {t α } α∈X(ω) for ω. The limit operator of A at ω, denoted by Φ ω (A), is an X(ω)-by-X(ω) indexed matrix with entries in B(E) defined by 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.
) is the linear isometry induced by f y , then we have: , acting on p E (F ) by matrix multiplication. 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 2.9 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 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.

Implementing Limit Operators for p = ∞
As we point out in Sect. 2.3, operators on ∞ 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. Theorem 3.13. Let X be a space, ω be a non-principal ultrafilter on X and p ∈ {0, 1, ∞}. The set of band operators on p E (X) rich at ω is dense in A p E (X) $,ω ; and the set of rich band operators on p E (X) is dense in A p E (X) $ . The proof is postponed to Sect. 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 Proof. The claim on propagation follows directly from that of [17, Theorem 4.10(2)], hence omitted. Concerning norms, note that So the lemma holds. (ω) 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 ∞ 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 > 0, there exists some N such that

Proposition 3.15. Let X be a space and ω a non-principal ultrafilter on
Therefore, for any , which implies that So T is the required operator and it is unique by Lemma 2.10. .

Φ ω takes band operators to band operators and does not increase prop-
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.

Property A, Partition of Unity, and Constructing Operators
Property A was first introduced by Yu [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. Let r, > 0. A metric p-partition of unity {φ i } i∈I is said to have (r, )variation if for any x, y ∈ X with d(x, y) ≤ r, we have A space X is said to have Property A if for any r, > 0, there exists a metric p-partition of unity with (r, )-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.
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( ∞ E (X)). Definition 4.3. Let {B i } i∈I be a collection of bounded linear operators on ∞ E (X). If for any v ∈ ∞ E (X), the series i∈I B i v converges point-wise to a vector in ∞ 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 ∞ E (X). For compositions, we have the following elementary observation: 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 2.13. 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: Due to density of finitely supported vectors in 0 E (X), (4.1) holds for all v ∈ 0 E (X). We move on to Case (1)(c): Again, for any v ∈ 1 E (X) with finite support, Due to density of finitely supported vectors in 1 E (X), the above estimates hold for all v ∈ 1 E (X). Finally we deal with Case (2). For any v ∈ ∞ 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.
, if p = 1 and suppose {φ i } i∈I has (r, )-variation. Each of them converges strongly to a band operator of norm ≤ N M A on p E (X).

When
It converges point-wise strongly to a band operator of norm Proof. By Lemma 2.13, A has the form of 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 : otherwise.
Due to density of finitely supported vectors in 0 E (X), (4.4) holds for all v ∈ 0 E (X). We move on to Case (1)(c): Again, for any v ∈ 1 E (X) with finite support, i∈J Note that {φ i } i∈J has (r, )-variation and d(x, t −1 k (x)) ≤ r, hence by (4.2): Due to density of finitely supported vectors in 1 E (X), (4.6) holds for all v ∈ 1 E (X). Finally we deal with Case (2). For any v ∈ ∞ 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.
Each of them converges strongly to a band operator of norm ≤ N M A on p 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.

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)".

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 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.  [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 Sect. 4 to approach the latter. While we do not need appeal to that result in this paper.
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( p E (X)) such that AB = I + K 1 and BA = I + K 2 for some K 1 , We claim that B ∈ C as well, hence B ∈ L p E (X) by Lemma 5.5. Indeed for any > 0, there exists some L 1 > 0 such that [A, f ] < /(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 Choose a point x 0 ∈ F 0 , and take 2) For the above F 0 ⊆ X, we have (5.3) By (5.1), we obtain 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.
and A ∈ C. If AB − I and BA − I belong to K p E (X), then B ∈ C as well. In particular, C is closed under taking inverses.

"(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 ∞ 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.

Lemma 5.7. For p ∈ {0} ∪ [1, ∞], let A be a band-dominated operator on
p E (X) rich at ω ∈ ∂X. For any finitely supported unit vector v ∈ p E (X(ω)), finite subset G ⊆ X and > 0, there exists a unit vector w ∈ p E (X) such that [1, ∞], and A be a rich band-dominated operator on p E (X) which is invertible at infinity. Then the operator spectrum σ op (A) is uniformly bounded below, i.e., there exists some M > 0 such that for any Proof. Since A is invertible at infinity, there exists a bounded operator B on 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 ∈ p E (X(ω)), To prove the claim, we fix a ω ∈ ∂X, a finitely supported v ∈ p E (X(ω)) and an > 0. Since K 2 ∈ K p E (X), there exists some finite G ⊆ X such that K 2 Q G < . By Lemma 5.7, there exists a unit vector w ∈ p E (X) such that Aw − Φ ω (A)v < and supp(w) ∩ G = ∅. Hence we have K 2 w < , and Combining them together, we have Letting → 0, we obtain (5.6) as required and the claim holds. Now for p < ∞, (5.6) holds for all vectors in p E (X(ω)) since finitely supported vectors are dense in p E (X(ω)). For p = ∞, we fix a vector v ∈ ∞ E (X(ω)) and an > 0. There exists some α ∈ X(ω) such that v(α) E > v ∞ − . Applying Proposition 3.17 and Take an L-Lipschitz function f with range in [0, 1] and supported in the 1/Lneighbourhood of α with f (α) = 1. Note that fv is finitely supported, hence (5.6) holds for fv. Furthermore, Combining them together, we have Taking → 0, (5.6) holds for any v ∈ ∞ 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.

Lemma 5.9. Let p ∈ {0} ∪ [1, ∞) and q be the conjugate exponent. Let A be a band-dominated operator on
which implies that if A is rich, then Proof of Theorem 5.1, "(2) ⇒ (4), p < ∞". Let A be a P-Fredholm rich operator in A p E (X) for p < ∞, then trivially A is invertible at infinity. By Proposition 5.8, the operator spectrum σ op (A) is uniformly bounded below.
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 q E * (X) by Lemma 5.9. Since A is invertible at infinity, there exists a bounded operator B on p E (X) such that K 1 := AB−I and K 2 := BA−I 73 Page 22 of 28 J. Zhang IEOT 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 ∞ E (X) is no longer an p -type space. Hence we cannot refer to Proposition 5.8 any more. New techniques are required, which are introduced in the next subsection.

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 ∞ -spaces can be characterised via those of its "double predual " on c 0 -spaces.
Recall that from Lemma 2.2, 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.
. 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
(5.7) 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. The proof of (5.7) follows from Lemma 5.11. For the second statement, we know that Φ ω (A) ∈ A ∞ E (X(ω)) for all ω ∈ ∂X by Proposition 3.17. Hence by Proposition 5.12, Φ ω (A) is invertible if and only if Φ ω (A)| 0 E (X(ω)) is, and we have IEOT Extreme Cases of Limit Operator Theory Page 23 of 28 73 By Corollary 5.6, Φ ω (A) −1 ∈ L ∞ E (X(ω)). Hence by Lemma 5.11 again, we have So we finish the proof.
Proof of Theorem 5.1, " . Hence A 0 is P-Fredholm as well. Since we already proved "(2) ⇒ (4)" for p = 0 in Sect. 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.

Constructing Parametrices
Now we move on to prove "(4) ⇒ (3)", following the ideas in [8] and [17] and using the tools we developed in Sect. 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.  , prop(A)). Let {φ i } i∈I be a metric 1-partition of unity with (prop(A), )-variation, and {ψ i } i∈I be an /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(ψi) . Now we divide the proof into two cases.

J. Zhang IEOT
• p = 0 or ∞: by Lemma 4.5, the sum i∈I\K φ i B i ψ i converges strongly or point-wise strongly to an operator on 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 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.  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 ∞ E (X) is defined in Sect. 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 2.13, 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 ∈ ∞ E (X), we have otherwise.
Hence, taking L = /(rM N ) 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]. for any A ∈ B( ∞ E (X)) with propagation at most r, norm at most M , and any F ⊆ X.
Consequently, we have the following corollary by the same proof as [17, Corollary 7.10], replacing [17,Proposition 7.6] with Proposition 5.20 above, hence omitted. for all ω ∈ ∂X such that A is rich at ω, and all F ⊆ X(ω).
Using Corollary 5.21 instead of [17,Corollary 7.10], now the rest of the proof for Theorem 5.18 follows exactly the same as that of [17,Theorem 7.3], occupying [17,Section 7.2]. Hence we omit the rest of the proof. Also notice that [17,Corollary 7.10] is the only place where Property A is used to prove [17,Theorem 7.3], so it is unnecessary for Theorem 5.18.
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 E (X), 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 extension A ∈ 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. Finally, we deal with the case of p = 1. Assume A is a rich banddominated operator on 1 E (X), and that all B ∈ σ op (A) are invertible. Hence all their adjoints are invertible as well, which compose σ op (A * ) by Lemma 5.9. From the result for p = ∞ proved above, we have that So we finish the proof.