A class of tridiagonal operators associated to some subshifts

We consider a class of tridiagonal operators induced by not necessary pseudoergodic biinfinite sequences. Using only elementary techniques we prove that the numerical range of such operators is contained in the convex hull of the union of the numerical ranges of the operators corresponding to the constant biinfinite sequences; whilst the other inclusion is shown to hold when the constant sequences belong to the subshift generated by the given biinfinite sequence. Applying recent results by S. N. Chandler-Wilde et al. and R. Hagger, which rely on limit operator techniques, we are able to provide more general results although the closure of the numerical range needs to be taken.


Introduction
Let b D .b i / i2Z be a biinfinite sequence in A Z where A is a finite set, called an alphabet. In this paper we study the operator A b W`2.Z/ !`2.Z/ defined as the tridiagonal operator : : : : : : where the rectangle marks the matrix entry at .0; 0/. When the alphabet is the set f 1; 1g, the corresponding operators are related to the so called "hopping sign model" introduced in [4] and subsequently studied in [1][2][3] and [5,6]. We remark that results found in the literature focus on the case when b is a pseudoergodic sequence, that is to say, when every finite sequence of˙1 appears somewhere in b. In this paper we aim to investigate the case when b is not necessary pseudergodic and although some authors have in fact pointed out that some of their results hold if the pseudoergodic condition is dropped, we try to formalize this approach. For this reason, we begin by thinking that b is an element in a full shift space and as such it is more usual to consider the alphabet A to be the set f0; 1g rather that f 1; 1g. By means of elementary methods, we are able to establish similar results to the ones found in the literature and slightly generalize others. More concretely, we show that the numerical range of A b has an inclusion-wise maximal, but that such bound is sharp not only when b is pseudoergodic but whenever the subshift generated by b (the closure of the orbit of b) contains the constant biinfinite sequences. With this motivation we employ recent limit operator techniques to extend our results to more general alphabets; however, for these generalizations the closure of the numerical range must be taken. We feel our approach might be of interest as it employs elementary mathematics to obtain some results that are the motivation to state more general results. We divided this work into three sections. In the first section we review some of the fundamental results needed in the rest of the paper, both operator theoretical concepts and symbolic dynamics notions. In the second section we apply elementary techniques to bound the numerical range of A b , for any b 2 f0; 1g Z and prove that such upper bound is sharp when the subshift generated by b contains the constant sequences. In the last section we generalize our results to more general alphabets by means of recent results which employ limit operator techniques.

Background
We review in this section some of the fundamental results and notation to be used throughout in the paper.

Operators
We are considering bounded linear operators on the Hilbert space`2.Z/. Hence, every time we refer to an operator we assume it is a bounded linear operator on`2.Z/. The inner product on`2.Z/ is denoted by h ; i. For an operator A we define the spectrum of A as The numerical range of A is defined by For easy reference, we state the following well known results (see e.g. [8,Chapter 1]) which are true in general Hilbert spaces.

Symbolic dynamics
For details about symbolic dynamics we refer the reader to [7]. An alphabet A is nothing but a finite set. We refer to the elements of A as symbols. The set A Z of all biinfinite sequences of symbols from A is termed the full shift on A. When writing down an element in A Z it is customary to distinguish the 0th coordinate with a dot. For example a biinfinite sequence b in f0; 1g Z expressed as b D .: : : 0 1 1 1 0 P 1 0 0 0 1 0 : : :/ is such that its coordinates are b 2 D 1, b 1 D 0, b 0 D 1, b 1 D 0 and so on. Given a biinfinite sequence b D .b i / i2Z in A Z , a block (or word) of b is defined to be a finite subsequence of b. The length of a block of b is the number of symbols it contains. If b belongs to A Z and i < j , then we will denote the block of coordinates in b from position i to position j by b OEi;j . When a finite sequence u of symbols of A satisfies that u D b OEi;j for some i; j , we will say that u occurs in b. The blocks of the form b OE k;k are called central blocks of b. Note then an equivalent way to say that b 2 A is pseudoergodic is to require that every block of every size of symbols in A occur in b.
The full shift A Z is actually a metric space with a metric given by Hence, we may say that two biinfinite sequences are close to each other when their central blocks agree. We denote the shift map 'W A Z ! A Z as the map which moves the zeroth coordinate one slot to the right, that is A subshift X is a subspace of A Z which is closed and invariant under '. The orbit of a point b 2 A Z is the set of iterates f' n .b/g n2Z and we will denote it as orb .b/. Given b 2 A Z , the subshift generated by b is defined to be orb .b/ and denoted by X b . Notice that when b is pseudoergodic, it follows that so that there is always an element in orb .b/ as close as desired to c, proving that c belongs to X b , as wanted.
For a 2 A, we denote its corresponding biinfinite sequence in A Z with boldface a D . a a P a a /. The tridiagonal operators A a corresponding to the constant sequences a are known as Laurent operators.

Tridiagonal operators for biinfinite sequences of zeroes and ones
In this section we discuss tridiagonal operators A b where b is a biinfinite sequence of symbols in the alphabet A D f0; 1g. Similar arguments can be given for A D f 1; 1g and we will comment about this at the end of this section. We begin this section by computing the numerical range of the Laurent operators A 0 and A 1 ; although it is a well known result, we provide a sketch of the proof since we believe it sheds some light in the understanding of the computation of W .A b /, for more general b, which is one of the main results of this section.
Proof. For the proof of the inclusions W .A 0 / D and W .A 1 / . 2; 2/, we will show that for each 2 D there is x in the unit ball of`2.Z/ such that hA 0 x; xi D and hA 1 x; xi D 2Re. /. Indeed, we define It follows that To prove the other inclusions, first notice that since A 1 is self-adjoint then W .A 1 / R by Theorem 1.1. Let x 2`2.Z/ such that kxk D 1. Since x and A 0 x are linearly independent, Cauchy-Schwarz inequality gives jhA 0 x; xij < kA 0 xk kxk D 1 and so jhA 1 x; xij D j2Re. /j < 2 so that W .A 0 / D and W .A 1 / . 2; 2/, as was to be proved.
Proof. A direct computation shows On the other hand we have and so Substituting this last equality in (1) we complete the proof. Proof. Since x is in the unit ball of`2.Z/ then the element y in`2.Z/ defined by y k D jx k j is also in the unit ball. Therefore, by Proposition 2.1, we obtain hA 0 y; yi < 1. This together with Lemma 2.2 give us jx k x kC1 j D hA 0 y; yi < 1: as was to be proved.  Proof. For convenience, let us denote by the right hand side of the inclusion in the theorem, see Figure 1. Using Proposition 2.3, it will suffice to show that, for each x D .x j / j 2Z in the unit ball of`2.Z/, the corresponding ellipse E x is contained in . This is achieved by showing that the boundary of lies outside of each E x . By symmetry, we may only prove the case when hA 0 x; xi lies in the first quadrant. Since the case hA 0 x; xi D 0 implies E x is the open unit ball, satisfying the desired property, we may assume that D hA 0 x; xi 6 D 0, say D re iÂ with 0 < r < 1 and 0 Ä Â Ä =2. Since the distance from 2Re. / to the closest boundary line is 1 2 .2 2Re. // D 1 r cos Â , by Heron's Problem, the minimum sum of the distances from the boundary line of to the points and 2Re. / iš 2 . This proves that the sum of the distances from the boundary line to the points and 2Re. / has minimal value 1. Hence it lies outside E x , as desired. It remains to prove that the sum of the distances from a point in the curved part of the boundary of to and 2Re. / is also at least 1. To this purpose, let z D e i' with =3 < ' Ä =2 and let 0 D 1 r . We consider two cases, namely, either 0 Ä Â Ä ' or ' < Â Ä =2. In case 0 Ä Â Ä ', we look at the triangle with vertices z, ,and 0 and the triangle with vertices z, , and 2Re. /, see Figure 2. Since 0 is, by its definition, the closest point to in the unit circle, we obtain jz j j 0 j. On the other hand, by letting z D a C ib and D c C id and using the assumption on ' and Â we have 0 Ä a Ä c and 0 Ä d Ä b. A straightforward computation then shows that jz 2Re. /j 1 j j D j 2Re. /j. We obtain jz j C jz 2Re. /j j 0 j C j j D 1; as wanted. To complete the proof we now assume ' < Â Ä =2. We use Ptolemy's inequality (see Figure 3) on the quadrilateral with vertices , z, 2Re. / and the origin to obtain jz j j2Re. /j C j j jz 2Re. /j j 2Re. /j: Using j2Re. /j D j2r cos Â j Ä j2r cos. =3/j D r D j j and again that j 2Re. /j D j j , we get jz j j j C j j jz 2Re. /j j zj j2Re. /j C j j jz 2Re. /j j j; and so jz j C jz 2Re. /j 1; as was to be proved.
As mentioned in the introduction, the following result might be derived from limit operator techniques when the closure of the numerical range is considered, see e.g. [5]. However, our approach might be of interest since it is elementary and provides the construction of the unitary elements in`2.Z/ corresponding to a given element in the numerical range.
Proof. To prove the proposition, it will suffice to show that for each 2 D we can provide elements x and y in the unit ball of`2.Z/ such that hA b x; xi D and hA b y; yi D 2Re. /. We first observe that the case D 0 is taken care of by choosing x 2`2.Z/ with just one component equal to 1 and all others equal to zero, since then hA b x; xi D 0. So we may assume in what follows that 6 D 0. Let us assume we have 2 D, say D re iÂ . Since 0 < r < 1, then 0 < rC1 2 < 1. Let k 0 be an integer such that Consider the polynomial function f .t / D t t 2k 0 C1 r. Note that (2) implies f .r/ D r 2k 0 C1 < 0 and The Intermediate Value Theorem gives us then a r < t 0 < rC1 Now, since the constant sequence 0 (resp. 1) is a limit point of the orbit of b, there exists an integer j 0 (resp. l 0 ) such that b OEj 0 C1;j 0 Ck 0 (resp. b OEl 0 C1;l 0 Ck 0 ) is a sub-word of zeros (resp. of ones) of b with length k 0 . We define x D .x k / k2Z in`2.Z/ in the following way. For each k 2 Z and define y k similarly by replacing j 0 with l 0 in the definition of x k . It follows that and similarly kyk D 1. Thus x and y are elements in the unit ball of`2.Z/. Using b j 0 C1 D D b j 0 Ck 0 D 0 and (3) we obtain as was to be proved.
Part (i) of the following corollary is analogous to [1, Lemma 3.1] for b pseudoergodic and the alphabet A D f 1; 1g. Part (ii) may also be derived from [5,Theorem 16]. However, notice we do not require the operator A b to be pseudoergodic and, thus far, we do not have employed limit operator techniques. In case we replace the numerical range with its closure, our result is analogous to [5,Corollary 14], where the operator is taken to be pseudoergodic and the union is taken over all periodic operators.
Proof. To prove (i), observe that since W .A b / is a convex set, it follows from Proposition 2.6 that W .
The other inclusion is an application of Theorem 2.5. For equality (ii), since A 0 and A 1 are normal operators, Theorem 1.1 gives us that conv . .A 0 // D W .A 0 / and conv . .A 1 // D W .A 1 /. Furthermore, since the numerical range of an operator is a convex set, by applying the closure to both sides of the equality in (i), a straightforward computation gives the desired result.
Notice that in the proof of Theorem 2.6, by replacing 0 with 1, it is easy to see that actually hA 1 x; xi D . Hence, it might be possible to use our techniques to recover some known results for the hopping sign model operators.

Tridiagonal operators associated to subshifts
For this section, A denotes a finite alphabet. Proof. Since d 2 orb .b/, then exist k 0 2 Z such that d D ' k .b/. We define S D A k 0 . Note S is an unitary operator. For j 2 Z it follows that Lemma 3.2. If A and B are unitarily equivalent, then they have the same numerical range and spectrum, furthermore, they have the same eigenvalues.
Proof. Since A and B Proof. Follows from Lemma 3.2 and Proposition 3.1.
The following may be derived from [5, Lemma 12]; however, we provide a simplified proof here for the sake of completeness.
Since m is arbitrary, this yields˝A ' n k .b/ x; y˛to be as close as wanted to hA d x; yi for k sufficiently large, as desired.
Proof. Since d 2 orb.b/, there exists a sequence of integers fn k g k2Z such that ' n k .b/ ! d as k ! 1. Using Lemma 3.4, we also have A ' n k .b/ ! A d in the weak operator topology. To prove the first inclusion, we let z 2 W .A d /. Then there is x 2`2.Z/ with kxk D 1 such that z D hA d x; xi. Hence˝A ' n k .b/ x; x˛ z D .A ' n k .b/ A d /x; x˛! 0 as k ! 1 and so˝A ' n k .b/ x; x˛! z. Using Theorem 3.3, we obtain that˝A ' n k .b/ x; xb elongs to W A ' n k .b/ D W .A b /, this proves z 2 W .A b /, as wanted. The second inclusion is immediate from Theorem 1.1.
Proof. The nontrivial inclusion is an application of Proposition 3.5 Part (i) of the following corollary is analogous to [5,Theorem 16], in fact, we rely on its proof. This will allow us to argue Part (ii) which is somewhat more general than a particular case of [5,Corollary 17], as it applies to not necessarily pseudoergodic sequences. Proof. The equality in the right hand side of (i) follows from Theorem 1.1 as the Laurent operator A a is normal. For the left hand side equality in (i), the inclusion " " follows from an application of Proposition 3.5 by taking closures and using that the closure of a convex set is convex. The inclusion " " follows from the proof of [5,Theorem 16] by taking U 1 D A, U 0 D f0g and U 1 D f1g and noticing that the hypothesis on A D A b to be pseudoergodic is not required in that part of the proof. For equality (ii) we use Theorem 1.1 to conclude the " " inclusion. For the other inclusion we observe that the hypothesis ensures that Laurent operators A a are limit operators of A b (see e.g. [2,Section 2] for an introduction of this notion). We may apply now [2, Theorem 2.1] to obtain S a2A .A a / .A b /: This together with an application of (i) completes the proof.