An Affirmative Answer to a Core Issue on Limit Operators Marko Lindner∗ and Markus Seidel† 4 1 Abstract 0 An operator on an lp-space is called band-dominated if it can be approximated, in the 2 operator norm, by operators with a banded matrix representation. It is known that a rich r band-dominated operator is P-Fredholm (which is a generalization of the classical Fredholm a property) if and only if all of its so-called limit operators are invertible and their inverses are M uniformly bounded. Weshow that thecondition on uniform boundednessis redundantin this statement. 7 1 AMS subject classification: 47A53; 47B07, 46E40, 47B36, 47L80. Keywords: Fredholm theory; Limit operator; Band-dominated operator. ] A F 1 Introduction . h t One of the really great achievements of the limit operator theory is that it provides powerful tools a m for the characterization and description of the Fredholm properties for many important classes of operators, above all the class of band-dominated operators on generalized lp-sequence spaces. [ These results range from the pure characterization of the Fredholmness (or the generalized - 2 P Fredholmness)ofanoperatorAintermsofthe invertibilityofallits limitoperators[21,40,43,25, v 50] to formulas for the Fredholm index of A [39, 46, 37, 26, 51, 52], as well as for the connections 0 of kernel and cokernel dimensions of A to the respective properties of its finite sections [51, 52], 0 3 formulasforessentialspectraandpseudospectra[4,17,35,43,53]andevenformulasfortheessential 1 norm and the essential lower norm [18]. 1. Without goingintothe details here,one canroughlysaythat, givenanoperatorA, one obtains 0 its limit operators by considering sequences of shifted copies of A and by passing to limits of such 4 sequences. This important family of all limit operators of a given A is usually denoted by σ (A) op 1 andreferredtoasitsoperatorspectrum. Thecentralobservationinthatmachineryisthefollowing : v Theorem 1. Let A be a rich band-dominated operator. Then A is -Fredholm if and only if all i P X its limit operators are invertible and their inverses are uniformly bounded. r In [25] one of the authors summarizes a “It is the biggest question of the whole limit operator business whether or not in The- orem 1, and consequently in all its applications, uniform invertibility of σ (A) can be op replaced by elementwise invertibility. In other words: Big question: Is the operator spectrum of a rich operator automatically uniformly invertible if it is elementwise invertible?” ∗TechnischeUniversitätHamburg(TUHH),InstitutfürMathematik,21073Hamburg,Germany,lindner@tuhh.de †Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany, markus.seidel@mathematik.tu-chemnitz.de 1 This problem is on the spot since the very first versions of Theorem 1 have been proved, and in fact it has been the engine for far reaching developments. In 1985, Lange and Rabinovich [21] studied a particular subalgebra of band-dominated operators, the so called Wiener algebra, and initiated a series of works [40, 43, 25, 10, 26] which show that for such operators the question can be answered affirmatively. Another special class where the problem is solved is the set of slowly oscillating band-dominated operators which was treated by Rabinovich, Roch and Silbermann [40] using localizing techniques [36], which lead to a well-engineered theory on local operator spectra. Athirdclassofoperatorsforwhichthe problemwassolvedin[44]arethe blockdiagonaloperators formed by the sequence of so-called finite sections of another band-dominated operator B. Notice that the story of limit operators actually had begun much earlier in the late 1920’s in Favard’s paper [13] for studying ODEs with almost-periodic coefficients, and in the work of Muhamadiev [29, 30, 31, 32]. A review of this history is, for example, in the introduction of [10]. A comprehensive presentation of these results, further achievements and applications e.g. to convolution and pseudo-differential operators, as well as the required tools, can be found in the 2004 book [43] of Rabinovich, Roch and Silbermann. Startingwith that book,[43], the technicalframeworkwasextended ina waythat now alsothe extremal cases l∞ and l1 [28, 24, 25, 26] as well as vector-valued lp-spaces could be treated (see [50] for the current state of the art). After the extension to l∞ and l1 it turned out that in these two extremal cases the uniform boundedness condition in Theorem 1 was redundant [24, 25] (note that for l∞ this implicitly occurs in [40] already). In l∞, results got even better when previous techniques were combined with the generalized collectively compact operator theory of Chandler- Wilde andZhang[11]. Thisledtoa furthersimplificationoftheFredholmcriteriatojustrequiring the injectivity of the limit operators [9, 27, 10], which is known as Favard’s condition. Another natural playground for the limit operator method, which is not subject of the present text, is the stability analysis for approximation methods applied to band-dominated and related operators(cf. [48,41, 42, 40, 23, 27, 51,52, 53, 49], againthe books [16, 17, 43, 25, 47,10] and the references cited there). Throughout all these works the “big question” has been ubiquitous e.g. in the introductions of the books [43, 25], the open problems section of [10], the open questions in [40, Section 6], the discussionofsufficientorweaklysufficientfamiliesofhomomorphismsin[17],thecollectionofmain results in [47, Section 2], the papers [40, 41, 42, 44] and many others. In his review of the article [36], A. Böttcher writes about the uniform boundedness condition in Theorem 1: “Condition (*) is nasty to work with.” There is nothing to add to this. The aim of this paper is to solve the problem in general and to give an affirmative answer to the bigquestion(cf. Theorem11)by showingthat the “nastycondition” is redundant. For this,we start with recalling the required definitions and basic results in the next section. InSection3themainresultsareproved. Inshort,thegoldenthreadthereisasfollows: Weshow that the lower norm of a band-dominated operator A can be approximated by a sort of restricted lower norms which are defined with respect to elements of uniformly bounded support. Moreover, this approximationturnsoutto extenduniformlytoalloperatorsinσ (A). Finally,forrichband- op dominated A, this permits to identify an element in the operator spectrum σ (A) which realizes op the infimum of the lower norms of all limit operatorsof A. This removes the uniform boundedness condition from Theorem 1 and allows to write the -essential spectrum as in Corollary 12. P ThefinalSection4shinesalightontheimportanceoftheprerequisitesrich andband-dominated in the main results. 2 2 -Fredholmness and limit operators P In all what follows we let X stand for a (generalized) sequence space lp(ZN,X) with parameters p 0 [1, ], N N and a Banach space X. These (generalized) sequences are of the form ∈ { }∪ ∞ ∈ lx0(=ZN(x,iX)i)∈sZtNanwdisthforaltlhxeicl∈osXur.eiTnhle∞s(pZaNce,sXa)roefetqhueispepteodfawllitsheqtuheenucessua(xlip)-i∈nZoNrmw.itIhnfionuitrensoutpaptioornt., Notice that this model in particularcovers the spaces Lp(RN) by a natural isometric identification (see (2) below). For a subset F ZN we denote its characteristic function by χ . F ⊂ Operators and convergence The following definitions and results are taken from e.g. [43, 25, 51, 50]. Starting with the sequence =(P ) of the canonical projections n P Pn :X→X, (xi)i∈ZN 7→(χ{−n,...,n}N(i)xi)i∈ZN one says that a bounded linear operator K on X is -compact if P (I P )K + K(I P ) 0 as n . n n k − k k − k→ →∞ The set of all -compact operators is denoted by (X, ). Moreover, we introduce the subset P K P (X, ) of (X), the set of all bounded linear operators on X, as follows: L P L (X, ):= A (X):AK,KA (X, ) for all K (X, ) . L P { ∈L ∈K P ∈K P } One may say that those operators are compatible with (X, ) and, indeed, (X, ) is a closed K P L P subalgebra of (X) and (X, ) forms a closed two sided ideal in (X, ). L K P L P An operator A (X, ) is said to be -Fredholm if there is an operator B (X) such that ∈L P P ∈L AB I and BA I are -compact. By [52, Theorem 1.16] B belongs to (X, ) in this case, i.e. − − P L P -Fredholmness of A is equivalent to the invertibility of A+ (X, ) in (X, )/ (X, ). P K P L P K P Say that a bounded sequence (A ) (X, ) converges -stronglyto an operatorA (X) if n ⊂L P P ∈L P (A A) + (A A)P 0 as n m n n m k − k k − k→ →∞ for every m N. We shortly write A P A in that case and note that A (X, ) n ∈ → ∈L P Remark 2. Notice that for all p (1, ) and dimX < these -notions coincide with the ∈ ∞ ∞ P classical ones: (X, ) = (X) the set of compact operators, (X, ) = (X), an operator is K P K L P L -Fredholm if and only if it is Fredholm, and a sequence (A ) converges -strongly to A if and n P P only if A A and A∗ A∗ strongly. n → n → The reason for the definition of the -notions is simply to extend the concepts, tools and P connectionsbetweentheclassicalnotions,whicharewellknownfromstandardFunctionalAnalysis. Band-dominated operators Every sequence a = (a ) l∞(ZN, (X)) gives rise to an n operator aI (X), a so-called multiplication operator, via th∈e rule (ax)L = a x , i ZN. For i i i every α ZN∈, wLe define the shift operator V :X X, (x ) (x ). ∈ α i i−α ∈ → 7→ A band operator is a finite sum of the form a V , where a I are multiplication operators. α α α In terms of the generalized matrix-vector multiplPication (aij)i,j∈ZN (xj)j∈ZN =(yi)i∈ZN with yi = aijxj, i∈ZN, where aij ∈L(X), jX∈ZN 3 band operators A act on X = lp(ZN,X) via multiplication by band matrices (a ), that means ij a =0 if i j exceeds the so-called band-width of A. ij Typica|l −exa|mples are discretizations of differential operators on RN, such as discrete Schrö- dinger operators. (There is large interest in spectral properties of discrete Schrödinger and more general Jacobi operators. Here are some references that involve limit-operator-type arguments: [6, 33, 12, 22, 45, 38].) In many physical models however, interaction a between data at locations i and j decreases ij in a certainwayas i j rather than suddenly stop at a prescribeddistance of i and j. This | − |→∞ is one of the reasons for introducing the following notion: An operator is called band-dominated if it is the limit, w.r.t. the operator norm, of a sequence of band operators. Here typical examples are integral operators A:f k(,t)f(t)dt (1) 7→ZRN · onLp(RN),where the kernelfunction k(s,t)decayssufficiently fastas s t . Suchoperators | − |→∞ are often connected with boundary integral problems (see e.g. [19, 2, 34, 1, 11, 3, 43, 37, 8]). Via the isometric isomorphism Lp(RN) lp(ZN,Lp([0,1]N)) that sends f to → the sequence (fi)i∈ZN of restrictions fi :=f|i+[0,1]N ∈Lp(i+[0,1]N)∼=Lp([0,1]N)=:X, (2) the integraloperator(1)correspondsto aninfinite matrix(aij)i,j∈ZN withintegraloperatorentries a :Lp(j+[0,1]N)=X Lp(i+[0,1]N)=X. ij We denote the cl∼ass o→f all band-domina∼ted operators on X = lp(ZN,X) by . In contrast to p A the set of all band operators (which is an algebra but not closed in (X)), the set is a Banach p L A algebra, for which the inclusions (X, ) (X, ) (X) hold. p K P ⊂A ⊂L P ⊂L Thefirststudiesofparticularsubclassesofband-dominatedoperatorsandtheirFredholmprop- ertieswereforthecaseofconstantmatrixdiagonals,thatiswhenthematrixentriesa onlydepend ij on the difference i j, so that A is a convolution operator (a.k.a. Laurent or bi-infinite Toeplitz − matrix,thestationarycase)[15,54,14,16,5]. Subsequently,thefocuswenttomoregeneralclasses, such as convergent, periodic and almost periodic matrix diagonals, until at the current point arbitrary matrix diagonalscan be studied – as long as they are bounded. This possibility is due to the notion of limit operators that enables evaluation of the asymptotic behavior of an operator A even for merely bounded diagonals in the matrix (a ). ij Limitoperators Saythatasequenceh=(h ) ZN tendstoinfinityif h asn . n n If h=(h ) ZN tends to infinity and A (X, ) t⊂hen | |→∞ →∞ n ⊂ ∈L P A := -lim V AV , h Pn→∞ −hn hn if it exists, is called the limit operator of A w.r.t. the sequence h. The set σ (A):= A :h=(h ) ZN tends to infinity op h n { ⊂ } of all limit operators of A is called its operator spectrum. We say that A (X, ) has a rich operator spectrum (or we simply call A a rich operator) if every sequence h∈ LZN tPending to infinity has a subsequence g h such that the limit operator ⊂ ⊂ A of A w.r.t. g exists. The set of all rich operators is denoted by $(X, ) and the set of all rich g L P band-dominated operators by $. Recall from [43, Corollary 2.1.17] or [25, Corollary 3.24] that Ap $ = whenever dimX < . Ap Ap ∞ 4 Proposition 3. ([24, Proposition 3.7] or [43, Corollary 1.2.7]; and [50, Corollary 17]) Let again X=lp(ZN,X), where p 0 [1, ], N N and X is a Banach space. ∈{ }∪ ∞ ∈ For every A $(X, ), σ (A) is sequentially compact with respect to -strongconvergence. op • ∈L P P The set $(X, ) forms a closed subalgebra of (X, ) and contains (X, ) as a closed • L P L P K P two-sided ideal. If p< and A $(X, ), then A∗ $(X∗, ) and • ∞ ∈L P ∈L P σ (A∗)=(σ (A))∗ := B∗ :B σ (A) . (3) op op op { ∈ } NotethatProposition3translatesto $aswellandthatσ (A) ifA . Finallyrecallfrom Ap op ⊂Ap ∈Ap the introductionthat, for operatorsin $, Theorem1 is in force andprovidesa characterizationof Ap their -Fredholm property. P 3 The main results Definition 4. Given two Banach spaces Y,Z the lower norm of an operator A (Y,Z) is ∈L ν(A):=inf Ay :y Y, y =1 . {k k ∈ k k } For operators A (X) and subsets F ZN we shortly write ν(A ) for the lower norm of the F ∈ L ⊂ | restricted operator A :=Aχ I :imχ I X. F F F | → If A is invertible then ν(A) = 1/ A−1 . So our big question translates: If every B σ (A) is op k k ∈ invertible, are then the lower norms ν(B) uniformly bounded away from zero? In short: ? ( B σ (A): B is invertible) = inf ν(B):B σ (A) >0 (BQ) op op ∀ ∈ ⇒ { ∈ } Weprovethattheansweris‘yes’. Ourapproachusesthesequentialcompactnesspropertyofσ (A) op as mentioned in Proposition 3. This property motivates the following strategy of proof: Suppose all B σ (A) were invertible but there was a sequence B ,B ,... σ (A) with ν(B ) 0. By op 1 2 op n ∈ ∈ → our compactness, there is a subsequence (B ) with limit B σ (A). Can B be invertible? nk ∈ op Incase B B 0onecanconcludeν(B ) ν(B)andhenceν(B)=0,whichcontradicts the invertibkilitnyk−of Bk→(proof finished). Unfortunnakte→ly, all we have, by Proposition 3, is B P B, which does not imply ν(B ) ν(B). As an example, look at B := P + 1(I P ) P nIk=→: B, where the “region”, ZN nkn,.→..,n N, that is responsible for the smnall valnues onf ν(−B )n=→1, moves \{− } n n away from the origin as n whence it has no impact on -limB and its lower norm. n →∞ P To get back on track with our proof, we will passto translatesV B V ofB suchthat the “badregion” ofB shifts to the originand • −j nk j nk nk hencecontributestothelowernormofthe -limitB. (NotethatobviouslyV BV σ (A) −j j op if B σ (A) and j ZN.) P ∈ op ∈ ∈ study the lower norms ν(B ) from Definition 4, in particular for bounded sets F ZN. F • (Note that B P B implies| (B B)χ I 0 and hence ν(B ) ν(B ) ν(⊂B) for all bounded snekts→F ZN.) k nk− F k→ nk|F → |F ≥ ⊂ 5 inordertofindappropriateshiftsV andboundedsetsF,studythelowernormsofoperators, j • restricted to elements x X with support in any set of a prescribed diameter D (i.e. in an ∈ arbitrary translate of the cube [ D,D]N) – see Definition 5 below. −2 2 Definition 5. The support of a sequence x = (x ) X is the set suppx := n ZN : x = 0 . n n The diameter of a subset U ZN is defined as diam∈U := sup u v : u,v{ ∈U , where6 } ∞ ∞ denotes the maximum norm ⊂on RN. Moreover,put dist(U,V):={|m−in |u v :∈u }U,v V|·f|or ∞ two nonempty sets U,V ZN. For A (X), F ZN and D N, w{e|no−w p|ut ∈ ∈ } ⊂ ∈L ⊂ ∈ ν (A ):=inf A x :x imχ I, x =1,diamsuppx D . D F F F | {k | k ∈ k k ≤ } We first show that ν (A ) ν(A ) as D , uniformly in a surprisingly general sense. D F F We say A has band-width les|s th→an w |N if χ A→χ ∞I =0 for all U,V ZN with dist(U,V)>w. U V ∈ ⊂ Proposition 6. Let δ > 0, r > 0 and w N. Then there is a D N such that for all band ∈ ∈ operators A with A (2rδ−1)p and D :=mR=4mw. If N >1, then we define functions χ :=χ , χ :=χ , ..., χ :=χ . 1 [−1,1)×RN−1 2 R×[−1,1)×RN−2 N RN−1×[−1,1) 2 2 2 2 2 2 For x χ I, applying the above estimates N times we get F ∈ m−1 Ax (cid:13) Aχj1+l01+n1,Rx(cid:13) rm−1/p x k k≥(cid:13) 1 (cid:13)− k k (cid:13)(cid:13)jX1=1n1X∈mZ (cid:13)(cid:13) (cid:13) (cid:13) .(cid:13).. (cid:13) ≥ (cid:13) Aχj1+l01+n1,R χjN+l0N+nN,Rx(cid:13) Nrm−1/p x ≥(cid:13) 1 ··· N (cid:13)− k k (cid:13)(cid:13)(j1,...,jN)∈X{1,...,m−1}N(n1,...,nXN)∈(mZ)N (cid:13)(cid:13) (cid:13) (cid:13) and with ϕ(cid:13) := χk1+l01+1/2+n1,R χkN+l0N+1/2+(cid:13)nN,R the first sum- (n1,...,nN) (k1,...,kN)∈{0,...,m−1}N 1 ··· N mand again turns out toPbe p ν (A )p(cid:13) χj1+l01+n1,R χjN+l0N+nN,Rx(cid:13) , ≥ mR |F (cid:13) 1 ··· N (cid:13) (cid:13)(cid:13)(j1,...,jN)∈X{1,...,m−1}N(n1,...,nXN)∈(mZ)N (cid:13)(cid:13) (cid:13) (cid:13) hence the assertion aga(cid:13)in follows by (cid:13) Ax ν (A ) x 2Nrm−1/p x . mR F k k≥ | k k− k k Finally the case p 0, : This time put χ := χ and ψ(t) := max(1 t ,0) for ∈ { ∞} [−1,1]N − | |∞ t RN. For α,β ZN and k N we note that ∈ ∈ ∈ w χβ/w,wψα,k =ψα,k(β)χβ/w,w+ϕ with ϕ =sup ϕ(t) : t β w ∞ ∞ k k {| | | − | ≤ }≤ k and χ A=χ Aχβ/w,w since A has band-width less than w. We conclude, for all x X, {β} {β} ∈ χ Aψα,kx = χ Aχβ/w,wψα,kx ψα,k(β) χ Aχβ/w,wx + χ Aϕx {β} {β} {β} {β} k k k k≤ k k k k w wr 1 χ Ax + A x Ax + x ≤ ·k {β} k kk kk k≤k k k k k since A < r. Taking the supremum over all β ZN yields Aψα,kx Ax + wr x , and k k ∈ k k ≤ k k k k k hence, forall x imχ I with x =1: ν (A ) ψα,kx Ax +wr. Now takingthe supremum over all α ZN∈yieldsFν (A k) k Ax +2kwr|.FFkinallykta≤kekthekinfimkum over all x imχ I with ∈ 2k |F ≤ k k k ∈ F x =1 to get ν (A ) ν(A )+ wr and make k sufficiently large for wr <δ. k k 2k |F ≤ |F k k This result is going to be a main ingredient of our plot. The basic idea in the proof presented above for Proposition 6, namely to split the domain of a band operator A into separated regions andto exploit thatthe actionofA is limited by its band-width, has a longand successfultradition andhasleadtodeepresultsoninverseclosednessandontheexistenceof -regularizersin (X, ) P L P or (see [20,5, 34, 43,49, 52]). We alsopoint out that the proofrevealsexplicit formulas for the p A choice of D depending on δ, r and w. In addition, we would like to show an alternative proof for the case p [1, ) that is based on an idea of [7], where it is used to bound and approximate the ∈ ∞ pseudo-spectrum of A by a union of pseudo-spectra of (moderately sized) finite sections of A. 7 Proof. Let A be a band operator of the norm A 0, one has (a+b)p (1+ϕ)p−1ap+(1+ϕ−1)p−1bp, with equality iff ϕ= b. ≤ a (This is a simple calculus exercise: minimize the right-hand side as a function of ϕ.) Now,to ourarbitraryδ >0, choosen N largeenoughthat dn <( δ )p for alln n . Moreover, 0 ∈ cn 4r ≥ 0 given an arbitrary F ZN, fix x imχ I such that Ax <(ν(A )+ δ) x . Then we conclude ⊂ ∈ F k k |F 2 k k as follows: AP x p ( P Ax + [P ,A]x )p n,k n,k n,k k k ≤ k k k k kX∈ZN kX∈ZN (c) (1+ϕ)p−1 P Ax p+(1+ϕ−1)p−1 [P ,A]x p, ϕ>0 n,k n,k ≤ k k k k ∀ kX∈ZN kX∈ZN (a),(b) (1+ϕ)p−1c Ax p+(1+ϕ−1)p−12prpd x p, ϕ>0. n n ≤ k k k k ∀ Minimizing the latter term over all ϕ>0 we get by the fact (c) that p AP x p c1/p Ax +d1/p2r x kX∈ZNk n,k k ≤(cid:16) n k k n k k(cid:17) 1/p δ d p ν(A )+ + n 2r c x p <(ν(A )+δ)p P x p F n F n,k ≤(cid:16) | 2 (cid:18)cn(cid:19) (cid:17) k k | kX∈ZNk k use (a) <δ2 if n≥n0 | {z } | {z } for all n n . The last inequality shows that there must be some k ZN for which 0 ≥ ∈ AP x AP x p <(ν(A )+δ)p P x p, and hence ν (A ) k n,k k <ν(A )+δ n,k F n,k 2n F F k k | k k | ≤ P x | n,k k k for all n n since x imχ I. This finishes the proof. 0 F ≥ ∈ Next we observe that the previous result extends to band-dominated operators and, actually, even to their operator spectra in a uniform manner. 8 Corollary 7. Let A and δ >0. Then there is a D N such that p ∈A ∈ ν(B )+δ ν (B ) ν(B ) for all D˜ D, F ZN and B A σ (A). |F ≥ D˜ |F ≥ |F ≥ ⊂ ∈{ }∪ op Proof. Note that the algebra of band operators is dense in the Banach algebra of band-dominated operators. Thus, given A and δ, there is a band operator A′ such that A A′ δ/3, hence k − k ≤ ν(A ) ν(A′ ), ν (A ) ν (A′ ) A A′ δ/3 as well. By the previous proposition F F D F D F t|here| ex−ists a |D |>| 0 suc|h t−hat ν(A|′ )| ≤+kδ/3− νk ≤(A′ ) ν(A′ ) for all F ZN. Since F D F F ν () ν() always holds, we arrive at|ν(A )+δ≥ ν (A| ) ≥ν(A |) for all F Z⊂N. D F D F F N· o≥w, le·t A σ (A), F ZN and ǫ >| 0. By≥the pre|viou≥s obse|rvation, app⊂lied to A , g op g p ∈ ⊂ ∈ A there is a Dˆ such that ν(A ) + ǫ/2 ν (A ), and moreover there exists an x imχ I, g|F ≥ Dˆ g|F ∈ F x = 1, diamsuppx Dˆ with ν (A ) A x ǫ/2. Choosing k sufficiently large such that k k ≤ Dˆ g|F ≥ k g k− K := k,...,k N suppx we find ν(A )+ǫ A x ν(A ). Then g F g g F∩K {− } ⊃ | ≥k k≥ | ν(A )+ǫ ν(A ) ν(V AV P ) (A V AV )P g|F ≥ g|F∩K ≥ −gn gn k|F∩K −k g − −gn gn kk ν (V AV P ) δ (A V AV )P , ≥ D −gn gn k|F∩K − −k g− −gn gn kk where the latter tends to ν (A P ) δ ν (A ) δ as n . Since ǫ is arbitrary, and D g k F∩K D g F | − ≥ | − → ∞ obviously ν (B ) ν (B ) ν(B ) holds whenever D˜ D, the assertion follows. D |F ≥ D˜ |F ≥ |F ≥ Now we are ready to prove (BQ). Theorem 8. Let A $. Then there exists a C σ (A) with ν(C)=min ν(B):B σ (A) . ∈Ap ∈ op { ∈ op } Proof. We consider the numbers δ := 2−k and define r := ∞ δ = 2−l+1. Then (r ) forms a k l k=l k l strictly decreasing sequence of positive numbers which tendsPto 0. From Corollary 7 we obtain a sequence (D ) N of even numbers such that, for every k N, k ⊂ ∈ D >2D and ν(B )+δ /2>ν (B ) for every B A σ (A) and every F N. k+1 k |F k Dk |F ∈{ }∪ op ⊂ Choose a sequence (B ) σ (A) such that ν(B ) inf ν(B) : B σ (A) as n . For n op n op every n N we are going⊂to construct a suitably shift→ed cop{y C σ ∈(A) of B}such t→hat∞ n op n ∈ ∈ ν(C )<ν(B )+r for all l n, (4) n|F3Dl n l ≤ where we put F := s/2,...,s/2 N for every even integer s. For this we firstly note that there s {− } existsanx0 X, x0 =1,diamsuppx0 D suchthat B x0 <ν (B )+δ /2<ν(B )+δ . Choose a snhi∈ft j0 k ZnkN in order to centnra≤lize ny0 := V x0k, wnhinchkmeaDnns thnat supnpy0 Fn , annd n ∈ n jn0 n n ⊂ Dn set C0 :=V B V σ (A). Then n jn0 n −jn0 ∈ op ν(B )=ν(C0) ν(C0 ) C0y0 = B x0 <ν(B )+δ . n n ≤ n|FDn ≤k n nk k n nk n n Next, for k =1,...,n, we graduallyfind xk, xk =1, suppxk F , diamsuppxk D n k nk n ⊂ Dn−(k−1) n ≤ n−k such that Ck−1xk <ν (Ck−1 )+δ /2<ν(Ck−1 )+δ , k n nk Dn−k n |FDn−(k−1) n−k n |FDn−(k−1) n−k pass to a centralized copy yk := V xk of xk via a suitable shift jk F and then define n jnk n n n ∈ Dn−(k−1) Ck :=V Ck−1V σ (A). For this operator we observe n jnk n −jnk ∈ op ν(Ck ) Ckyk = Ck−1xk <ν(Ck−1 )+δ . n|FDn−k ≤k n nk k n nk n |FDn−(k−1) n−k 9 jk+F n Dn−k suppxk 0 n jk suppyk n n F Dn−k F Dn−(k−1) Figure1: Anillustrationoftheinclusionsuppxkn⊂jnk+FDn−k ⊂FDn−(k−1) indimensionN=1. In particular, for all for l =0,...,n this yields ν(Cn−l )<ν(B )+δ +δ +...+δ <ν(B )+r . (5) n |FDl n l l+1 n n l Finally, we define Cn := Cnn and take into account that Cn = Vjnn+...+jnn−l+1Cnn−lV−(jnn+...+jnn−l+1), where jn+...+jn−l+1 F by construction. Thus, we get n n ∈ Dl ν(C ) ν(Cn−l ), (6) n|F3Dl ≤ n |FDl which, together with (5), implies (4). By the first result in Proposition 3 we can pass to a subsequence (C ) of (C ) with -strong hn n P limit C σ (A). Then, by sending n , and since (C C ) = (C C )P ∈ op → ∞ − hn |F3Dl − hn 3Dl|F3Dl converges to zero in the norm, we get ν(C) ν(C )= lim ν(C ) lim ν(B )+r =inf ν(B):B σ (A) +r ≤ |F3Dl n→∞ hn|F3Dl ≤n→∞ hn l { ∈ op } l for every l. Since r 0 as l we arrive at the assertion. l → →∞ Remark 9. Here is one (but not the only) way to digest this proof: For every n N, we look at ∈ n+1 translates, C0,...,Cn, of the operator B . The lower norm of these translates is studied on n n n nested sets with decreasing diameters D ,...,D . n 0 Bn ∆=s=νh=

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