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J. Spectr. eory 6 (2016), 215–266 Journal of Spectral eory DOI 10.4171/JST/124 © European Mathematical Society
On the spectrum and numerical range of tridiagonal random operators
Raael Hagger
Abstract. In this paper we derive an explicit formula for the numerical range of (non-self- adjoint) tridiagonal random operators. As a corollary we obtain that the numerical range of such an operator is always the convex hull of its spectrum, this (surprisingly) holding whether or not the random operator is normal. Furthermore, we introduce a method to compute numerical ranges of (not necessarily random) tridiagonal operators that is based on the Schur test. In a somewhat combinatorial approach we use this method to compute the numerical range of the square of the (generalized) Feinberg–Zee random hopping matrix to obtain an improved upper bound to the spectrum. In particular, we show that the spectrum of the Feinberg–Zee random hopping matrix is not convex.
Mathematics Subject Classi cation (2010). Primary: 47B80; Secondary 47A10, 47A12, 47B36.
Keywords. Random operator, spectrum, numerical range, tridiagonal, pseudo-ergodic.
Contents
1 Introduction...... 216 2 enumericalrange ...... 221 3 eFeinberg–Zeerandomhoppingmatrix ...... 239 References...... 265 216 R. Hagger
1. Introduction
Since the introduction of random operators to nuclear physics by Eugene Wigner [24] in 1955, there is an ongoing interest in random quantum systems, the most famous example probably being the Anderson model [1]. In the last twenty years also non-self-adjoint random systems were extensively studied, starting with the work of Hatano and Nelson [14]. Compared to self-adjoint random operators, non-self-adjoint random operators give rise to many new phenomena like complex spectra, (non-trivial) pseudospectra, etc. In return, the study of non-self-adjoint operators requires new techniques as the standard methods from spectral theory are often not available. We start with some limit operator and approximation results for numerical ranges of random operators. We then focus on the physically most relevant case of tridiagonal operators. In particular, we prove an easy formula for the (closure of the) numerical range of tridiagonal random operators (eorem 16). As a corollary we get that the (closure of the) numerical range is equal to the convex hull of the spectrum for these operators, just like for self-adjoint or normal operators. eorem 16 thus provides the best possible convex upper bound to the spectrum of a random tridiagonal operator. In particular, it improves the upper bound given in [4] for a particular class of random tridiagonal operators. e authors of [4] considered the following tridiagonal random operator:
: : :: :: : 0 :: 0 1 1 B c 1 0 1 C B C ; B c0 0 1 C B : C B c 0 ::C B 1 C B : : C B :: ::C B C @ A where .cj /j Z is a sequence of i.i.d. random variables taking values in and 2 ¹˙ º .0; 1. e special case 1 was already considered earlier (e.g. in [2], [3], 2 D [9], and [15]) and is called the Feinberg–Zee random hopping matrix. It is also the main topic of [11]and[12], where the symmetries of the spectrum and the connec- tions to the spectra of nite sections of this operator are studied, respectively. On the spectrum and numerical range of tridiagonal random operators 217
eorem 16 also determines the spectrum completely in some cases. Consider for example the Hatano-Nelson operator : : :: ::
:: g 0 : v 1 e 1 g g B e v0 e C A B C ; D B e g v eg C B 1 : C B e g v ::C B 2 C B : : C B :: ::C B C @ A where .vj /j Z is a sequence of i.i.d. random variables taking values in some 2 bounded set V R and g > 0 is a constant, and assume that V is an interval of length at least 4 cosh.g/. en eorem 16 implies that the spectrum of A is equal to the numerical range, which is given by the union of the ellipses
g i# .g i#/ Ev e C v e C # Œ0; 2/ ; v V: WD ¹ C C W 2 º 2 In Section 2.3 we introduce a method to compute numerical ranges of arbi- trary (not necessarily random) tridiagonal operators that is based on the Schur test. For the (generalized) Feinberg–Zee random hopping matrix as studied in [4] and mentioned above, we use this method to compute the numerical range of the square of the random operator, which will provide an improved upper bound to the spectrum. is is related to the concept of higher order numerical ranges as used in [7] and [21] for example. In the last part we provide explicit formulas for the numerical range and the numerical range of the square in the case of the (generalized) Feinberg–Zee ran- dom hopping matrix in order to show that this new upper bound is indeed a tighter bound to the spectrum than the numerical range. In particular, we con rm and im- prove the numerical results obtained in [3] concerning the question whether the spectrum is equal to the (closure of the) numerical range in the case 1. More D precisely, we show that the spectrum is a proper subset of the (closure of the) numerical range and not convex.
1.1. Notation. roughout this paper we consider the Hilbert space
X `2.Z/ WD and its closed subspace `2.N/. e set of all bounded linear operators X X ! will be denoted by L.X/. e set of all compact operators X X will be denoted ! by K.X/. 218 R. Hagger
We want to think of L.X/ as a space of in nite matrices. Operators in L.X/ are identi ed with in nite matrices in the following way. Let ; be a scalar h i product de ned on X and let ei i Z be a corresponding orthonormal basis, i.e. ¹ º 2 ei ;ej ıi;j for all i;j Z. We will keep this orthonormal basis xed for the h i D 2 rest of the paper. e subsequent notions may depend on the chosen basis. Let A L.X/. entheentry Ai;j is given by Aej ;ei . e matrix .Ai;j /i;j Z, 2 h i 2 in the following again denoted by A, acts on a vector v X in the usual way. 2 If vj is the j -th component of v, then the i-th component of Av is given by Ai;j vj . is identi cation of operators and matrices on X is an isomorphism j Z (seeP2 e.g. [17, Section 1.3.5]). erefore we do not distinguish between operators and matrices. As usual, the vector .Ai;j /j Z X is called the i-th row 2 2 and .Ai;j /i Z X is called the j -th column of A. For k Z the vector 2 2 2 .Ai k;i /i Z X is called the k-th diagonal of A or the diagonal with index k. C 2 2 A is called a band operator if only a nite number of diagonals are non-zero. e set of all band operators will be denoted by BO.X/. Furthermore, we call A tridiagonal if all diagonals with index k 1; 0; 1 vanish. … ¹ º We consider the following subclasses. Let n m be integers and let
Un;:::;Um C be non-empty compact sets. en we de ne
M.Un;:::;Um/ A L.X/ Ai k;i Uk if n k m and D¹ 2 W C 2 Ai k;i 0 otherwise ; C D º i.e. the k-th diagonal only contains elements from Uk. Similarly, we denote the set of all nite square matrices with this property by
M n.Un;:::;Um/:
If A M.Un;:::;Um/ satis es 2
Ai;j Ai p;j p for all i;j Z and some p 1, D C C 2 then A is called p-periodic and the set of all of these operators will be denoted by
Mper;p.Un;:::;Um/:
In the special case p 1 these operators are usually called Laurent operators and D therefore we additionally de ne
L.Un;:::;Um/ Mper;1.Un;:::;Um/: WD On the spectrum and numerical range of tridiagonal random operators 219
e set of all periodic operators will be denoted by
Mper.Un;:::;Um/:
A M.Un;:::;Um/ is called a random operator if for k n;:::;m the 2 2 ¹ º entries along the k-th diagonal of A are chosen randomly (say i.i.d.) with re- spect to some probability measure on Uk. Finally, pseudo-ergodic operators are de ned as follows. Let Pk;l be the orthogonal projection onto span ek;:::;el . ¹ º en A M.Un;:::;Um/ is called pseudo-ergodic if for all " > 0 and all 2 B M n.Un;:::;Um/ there exist k and l such that 2 Pk;l APk;l B ": k k In other words, every nite square matrix of this particular kindcanbe found upto epsilon when moving along the diagonal of a pseudo-ergodic operator. Note that if all ofthe Uk are discrete, one can simply put " 0 in the de nition. At rst sight, it D is not easy to see why one may want to consider operators of this type, but in fact, pseudo-ergodic operators are closely related to random operators. Under some reasonable conditions on the probability measure (see e.g. [18, Section 5.5.3]), one can show that a random operator is pseudo-ergodic almost surely. erefore the de nition of pseudo-ergodic operators is a nice circumvention of probabilistic arguments when dealing with random operators. We will make use of this fact for the rest of the paper and just mention here that every statement that holds for a pseudo-ergodic operator, holds for a random operator almost surely. e set of pseudo-ergodic operators is denoted by
‰E.Un;:::;Um/: e notion of pseudo-ergodic operators goes back to Davies [6].
1.2. Limit operator techniques. Limit operators are an important tool in the study of band operators. For k Z de ne the k-th shift operator Vk by 2 .Vkx/j xj k for all x X. D 2 Let A L.X/ and let 2 h .hm/m N WD 2 be a sequence of integers tending to in nity such that the strong limit1
Ah lim V hm AVhm WD m !1 1 Sometimes dierent and more sophisticated notions of convergence are used to de ne limit operators. In the case of band operators on `2.Z/ all these notions coincide (see e.g. [17, Section 1.6.3] or [5, Example 4.6]). 220 R. Hagger exists. en Ah is called a limit operator of A. e set of all limit operators is called the operator spectrum of A and denoted by
op.A/:
Here are some basic properties of limit operators that we will need in the following (see e.g. [17, Proposition 3.4, Corollary 3.24]).
Proposition 1. Let A;B BO.X/ and let h .hm/m N be a sequenceof integers 2 WD 2 tending to in nity. en the following statements hold:
there exists a subsequence g .gm/m N of h such that Ag and Bg exist; WD 2
if Ah and Bh exist, so does .A B/h and .A B/h Ah Bh; C C D C
if Ah and Bh exist, so does .AB/h and .AB/h AhBh; D
if Ah exists, so does .A /h and .A /h .Ah/ ; D
if Ah exists, then Ah A ; k k k k
if A K.X/, then Ah 0. 2 D
We call an operator A Fredholm if ker.A/ and im.A/? are both nite-dimen- sional. As usual we de ne the spectrum
sp.A/ C A I is not invertible WD ¹ 2 W º and the essential spectrum
sp .A/ C A I is not Fredholm : ess WD ¹ 2 W º After introducing all the notation, we can cite the main theorem of limit oper- ator theory (which holds in much more generality than stated and needed here).
eorem 2 (e.g. [18, Corollary 5.26]). Let A BO.X/. en 2 sp .A/ sp.B/: ess D B op.A/ 2[ In order to apply this theorem to pseudo-ergodic operators, we use the follow- ing result that characterizes them in terms of limit operators. On the spectrum and numerical range of tridiagonal random operators 221
Proposition 3. Let Un;:::;Um be non-empty and compact. en
op A ‰E.Un;:::;Um/ .A/ M.Un;:::;Um/: 2 () D Proof. For diagonal operators on `2.Z/ this is Corollary 3.70 in [17]. e proof easily carries over to the case of band operators.
Using this and A M.Un;:::;Um/, we get the following corollary. 2
Corollary 4. Let Un;:::;Um be non-empty and compact and consider and oper- ator A ‰E.Un;:::;Um/. en 2 sp.A/ sp .A/ sp.B/: (1) D ess D B M.Un;:::;Um/ 2 [ In particular, we see that the spectrum of a pseudo-ergodic operator only de- pends on the sets Un;:::;Um. Furthermore, equation (1) provides a somewhat easy method to obtain lower bounds for the spectrum of A ‰E.Un;:::;Um/. 2 Indeed, we can take any operator B M.Un;:::;Um/ with a known spectrum 2 and get a lower bound for the spectrum of A. For example the spectrum of a peri- odic operator B can be computed via the Fourier transform.
eorem 5 (e.g. [8, eorem 4.4.9]). Let Un;:::;Um be non-empty and compact, p p N, B Mper;p.Un;:::;Um/ and let Bk L.C / be de ned by 2 2 2
.Bk/i;j Bi kp;j for all i;j 1; : : : ; p and k Z. D C 2¹ º 2 en ik# sp.B/ sp Bke : (2) D # Œ0;2/ k Z 2[ X2 is brief summary of limit operator theory is su cient for the rest of this paper. We recommend [17] and[22] for more details and further reading.
2. e numerical range
For the reader’s convenience we start with the de nition and some basic properties of the numerical range.
De nition 6. Let A L.X/. en the numerical range is de ned as 2 N.A/ clos Ax;x x X; x 1 : WD ¹h iW 2 k kD º 222 R. Hagger
For ' Œ0; 2/ the (rotated) numerical abscissa is de ned as 2 i' r'.A/ max Re z z N.e A/ : WD ¹ W 2 º Note that the numerical range is usually de ned without the closure (and de- noted by W.A/), but we prefer to consider the numerical range as a compact set here. e following results are well-known and also hold in arbitrary Hilbert spaces.
eorem 7 (Hausdor–Toeplitz). Let A L.X/. en N.A/ is convex. 2 eorem 8. Let A L.X/. It holds 2 conv.sp.A// N.A/ with equality if A is normal. Moreover,
sup Ax;x A x 1 jh ij k k k kD with equality if A is normal.
To determine the numerical range of an operator A, one usually applies the fol- lowing method by Johnson [16]. Since the numerical range is convex by eorem 7, it su cesto compute the numerical abscissae r'.A/ for every angle ' Œ0; 2/. 2 Fix ' Œ0; 2/ and let 2 1 i' i' B .e A e A/: WD 2 C en
i' r'.A/ sup Re e Ax;x D x 1 k kD ˝ ˛ 1 i' i' sup .e A e A/x;x D x 1 2 C k kD ˝ ˛ sup Bx;x D x 1 h i k kD r0.B/: D
Since B is self-adjoint, r'.A/ is exactly equal to the rightmost point of the spec- trum of B. is observation is the starting point for almost every result we prove in this paper. We will also nd it useful to talk about convergence of set sequences. On the spectrum and numerical range of tridiagonal random operators 223
De nition 9. Let .Mn/n N be a sequence of compact subsets of C. en we 2 de ne
lim sup Mn m C m is an accumulation point n WD ¹ 2 W !1 of a sequence .mn/n N; mn Mn ; 2 2 º
lim inf Mn m C m is the limit of a sequence .mn/n N; mn Mn : n WD ¹ 2 W 2 2 º !1 e Hausdor metric for compact sets A;B C is de ned as h.A;B/ max max min a b ; max min a b : WD ¹ a A b B j j b B a A j jº 2 2 2 2
Moreover, we de ne lim Mn as the limit of the sequence .Mn/n N with respect n 2 to the Hausdor metric.!1
ese notions are compatible with each other in the sense that they satisfy the same relations as they do for ordinary sequences:
Proposition 10 ([10, Proposition 3.6]). Let .Mn/n N be a sequence of compact 2 subsets of C. en the limit lim Mn exists if and only if lim sup Mn lim inf Mn n n D n and in this case we have !1 !1 !1
lim Mn lim sup Mn lim inf Mn: n D n D n !1 !1 !1
2.1. Limit operator and approximation results. We will rst prove the fol- lowing limit operator result, which can be proven (without further eort) in much more generality than we state it here.
eorem 11. Let A BO.X/. en 2
N.A K/ conv N.B/ : (3) C D K K.X/ B op.A/ 2\ 2[
To prove this, we need the following lemma that we will then apply to se- K quences .V hn .A K/Vhn /n N, where K .X/ and .hn/n N is a sequence of C 2 2 2 integers tending to in nity. 224 R. Hagger
Lemma 12. Let A L.X/ and let .An/n N be a sequence in L.X/ that converges 2 2 to A in weak operator topology. en N.A/ lim inf N.An/. n !1 Proof. An A in the weak operator topology implies .An A/x;x 0 for ! h i ! all x X as n . Let z N.A/. Choose x1 X with x1 1 such 2 ! 1 2 2 k k D that z Ax1; x1 < 1 and n1 such that .An A/x1; x1 < 1 for all n n1. j N h ij jh ij 1 For j choose xj 1 X with xj 1 1 such that z Axj 1; xj 1 < j 1 2 C 2 k C kD 1 j h C C ij C and nj 1 > nj such that .An A/xj 1; xj 1 < j 1 for all n nj 1. Ofcourse C jh 2 C C ij C C this implies z Anxj ; xj < j for all n nj . Now de ne a sequence .zn/n N j h ij 2 of complex numbers as follows. For n < n1 choose zn N.An/ arbitrarily. N 2 2 For j and nj n < nj 1 choose zn N.An/ such that z zn < j . 2 C 2 j j We get z zn 0 as n . us N.A/ lim inf N.An/. j j! !1 n !1 Proof of eorem 11. Let B op.A/ and K K.X/. To prove “ ” it su ces to 2 2 show N.B/ N.A K/ because the intersection of convex sets is again con- C vex. So let h be a sequence of integers tending to in nity such that Ah B. D By Proposition 1, B is also a limit operator of A K: C .A K/h Ah Kh Ah 0 Ah B: C D C D C D D
Applying Lemma 12 to the sequence .V hn .A K/Vhn /n N and using that the C 2 numerical range is invariant under unitary transformations, we get
N.B/ lim inf N.V hn .A K/Vhn / lim inf N.A K/ N.A K/: n C D n C D C !1 !1 To prove the other inclusion, recall that it su ces to compare numerical ab- scissae, i.e. to show op inf r'.A K/ max r'.B/ B .A/ : K K.X/ C W 2 2 i'® ¯ for all ' Œ0; 2/. Since r'.A/ r0.e A/ for all A L.X/ and ' Œ0; 2/, 2 D 2 2 it even su ces to consider ' 0. Set z0 A . en D WD k k r0.A K/ sup Re .A K/x;x C D x 1 h C i k kD sup Re .A K z0I/x;x z0 D x 1 h C C i k kD sup Re .A K z0I/x;x z0 x 1 j h C C ij k kD 1 sup .A K .A K/ 2z0I/x;x z0 D 2h C C C C i x 1 ˇ ˇ k kD ˇ ˇ 1 ˇ ˇ A ˇ K .A K/ 2z0I z0; ˇ D 2 k C C C C k On the spectrum and numerical range of tridiagonal random operators 225 where we applied eorem 8 to the self-adjoint (hence normal) operator
A K .A K/ 2z0I: C C C C Taking the in mum, we arrive at 1 inf r0.A K/ inf A K .A K/ 2z0I z0 K K.X/ C 2 K K.X/ k C C C C k 2 2 1 inf A A K 2z0I z0: D 2 K K.X/ k C C C k K2 K D For a self-adjoint operator C L.X/, the norm C K is minimized by a self- 2 k C k adjoint operator K K.X/. is can be seen as follows: 2 C K sup .C K/x;x k C k x 1 jh C ij k kD K K K K sup C C x; x C x; x D C 2 C C 2 x 1 ˇ ˇ k kD ˇD E D Eˇ ˇ K K ˇ sup ˇ C C x; x ˇ C 2 x 1 ˇ ˇ k kD ˇD Eˇ ˇK K ˇ C ˇ C ; ˇ D C 2
where we used eorem 8 and the fact that K K K K C C x; x R and C x; x iR C 2 2 C 2 2 D E D E for all x X. Moreover, we have 2 inf A K max B K K.X/ k C k D B op.A/ k k 2 2 for all A BO.X/ by [13, eorem 3.2]. Combining these results and using Propo- 2 sition 1, we get 1 inf r0.A K/ inf A A K 2z0I z0 K K.X/ C 2 K K.X/ k C C C k 2 2 1 op max B B .A A 2z0I/ z0 D 2 ¹k k W 2 C C º 1 op max B B 2z0I B .A/ z0: D 2 ¹k C C k W 2 º 226 R. Hagger
Since r'.B/ B A for all ' Œ0; 2/ by eorem 8 and Proposition 1, k k k k 2 op N.B z0I/ is contained in the right half plane for every B .A/. is implies C 2
r0.B/ sup Re Bx;x D x 1 h i k kD sup Re .B z0I/x;x z0 D x 1 h C i k kD sup Re .B z0I/x;x z0 D x 1 j h C ij k kD 1 sup .B B 2z0I/x;x z0 D 2 h C C i x 1 ˇ ˇ k kD ˇ ˇ 1 ˇ ˇ B ˇ B 2z0I z0: ˇ D 2 k C C k We conclude
op inf r0.A K/ max r0.B/ B .A/ : K K.X/ C ¹ W 2 º 2 If we apply this result to pseudo-ergodic operators, we get the following corol- lary:
Corollary 13. Let Un;:::;Um be non-empty and compact. It holds
N.A/ N.B/ D B M.Un;:::;Um/ 2 [ for all A ‰E.Un;:::;Um/. 2
Note that taking the convex hull is obviously not necessary here. In fact, it su ces to consider periodic operators on the right-hand side:
Corollary 14. Let Un;:::;Um be non-empty and compact. It holds
N.A/ clos N.B/ D B Mper.Un;:::;Um/ 2 [ for all A ‰E.Un;:::;Um/. 2 On the spectrum and numerical range of tridiagonal random operators 227
Proof. Let A ‰E.Un;:::;Um/. It is not di cult to nd a sequence 2
.Ak/k N Mper.Un;:::;Um/ 2 that converges weakly to A (even strongly). us by Lemma 12 and Corollary 13, we have
N.A/ lim inf N.Ak/ clos N.B/ N.A/: k !1 B Mper.Un;:::;Um/ 2 [
In the next section we will see that in the case of a tridiagonal pseudo-ergodic operator A, it even su ces to consider the Laurent operators contained in op.A/. So far we only considered numerical ranges of operators A BO.`2.Z//. 2 However, it is sometimes more convenient to work with operators A L.`2.N//. 2 We will thus nd the following well-known proposition useful.
2 2 Proposition 15. Let A BO.` .Z// and let A PNAPN im PN L.` .N//, 2 C WD j 2 where PN denotes the projection onto span e1;e2;::: . If there exists a sequence ¹ º .hm/m N of integers tending such that Ah exists and is equal to A, then 2 C1 N.A/ N.A /. D C
Proof. Clearly, A x; x Ax;x for all x im PN and thus N.A / N.A/. h C i D h i 2 C Conversely, let c N.A /, QN I PN and consider 2 C WD
A PNAPN im PN cQN im QN : z WD j C j en
Ax;x A PNx;PNx c QNx;QNx h z iD h C i C h i PNx PNx 2 2 A ; PNx c QNx : D C PNx PNx k k C k k k k k k
2 2 2 Since PNx QNx x and N.A / is convex, we get N.A/ N.A /. k k C k k D k k C z C Moreover, A is a limit operator of A and thus N.A/ N.A/ by eorem 11. z z We conclude N.A/ N.A /. D C 228 R. Hagger
2.2. Tridiagonal pseudo-ergodic operators. In this section we focus on the case of tridiagonal pseudo-ergodic operators. Here the following simpli cation of Corollary 14 can be achieved:
eorem 16. Let U 1, U0 and U1 be non-empty and compact. en, given an operator A ‰E.U 1; U0; U1/, the following formula holds: 2 (i) N.A/ conv sp.B/ D B L.U 1;U0;U1/ 2 [ (ii) i# i# conv u 1e u0 u1e # Œ0; 2/ : D ¹ C C W 2 º uk Uk ; k [21;0;1 D In particular, sp.A/ N.A/ if sp.B/ is convex. D B L.U 1;U0;U1/ 2 S Proof. e last assertion follows from (i) since
sp.B/ sp.A/ N.A/ B L.U 1;U0;U1/ 2 [ by Corollary 4 and eorem 8. Moreover, (ii) follows immediately from eo- rem 5. We thus focus on the proof of (i). “ ”. eorem 8 and Corollary 4 imply N.A/ sp.A/ sp.B/ B L.U 1;U0;U1/ 2 [ as above. Taking the convex hull on both sides yields
N.A/ conv sp.B/ : B L.U 1;U0;U1/ 2 [ by eorem 7.
“ ”. Asin the proofof eorem 11,itsu cestocompare max r0.B/ B L.U 1;U0;U1/ 2 with r0.A/. is then implies
N.A/ conv N.B/ B L.U 1;U0;U1/ 2 [ and hence
N.A/ conv conv.sp.B// conv sp.B/ D B L.U 1;U0;U1/ B L.U 1;U0;U1/ 2 [ 2 [ On the spectrum and numerical range of tridiagonal random operators 229 because Laurent operators are normal. We also set z0 A again, which implies D k k N.B z0I/ CRe 0 for all B M.U 1; U0; U1/. It follows C 2
r0.A/ sup Re Ax;x D x 1 h i k kD sup Re .A z0I/x;x z0 D x 1 h C i k kD sup Re .A z0I/x;x z0 D x 1 j h C ij k kD 1 sup .A A 2z0I/x;x z0 D 2 x 1 jh C C ij k kD 1 A A 2z0I z0; D 2 k C C k where we used eorem 8 in the last line. Using that the norm of an operator is bounded by the sum of the maximal elements of its diagonals (also called Wiener estimate, see e.g. [17, p. 25]), we arrive at
1 r0.A/ A A 2z0I z0 D 2 k C C k 1 max u1 u 1 max u0 u0 2z0 z0: (4) u 1 U 1 j C j C 2 u0 U0 j C C j u12U1 2 2
Fix w 1 U 1, w0 U0 and w1 U1 such that the maximum in (4) is attained, 2 2 2 i.e.
i' i' i' i' max e u1 e u 1 e w1 e w 1 u 1 U 1 j C j D j C j u12U1 2 and i' i' i' i' max e u0 e u0 2z0 e w0 e w0 2z0 : u0 U0 j C C j D j C C j 2 It is not hard to see that the spectrum of a tridiagonal Laurent operator L.v 1;v0;v1/ (to simplify the notation we identify the set
L.v 1;v0;v1/ L. v 1 ; v0 ; v1 / WD ¹ º ¹ º ¹ º with its only element) is given by an ellipse with center v0 and half-axes v 1 v1 (see e.g. [20]). If in addition j j j ˙ j j j
C L.v 1;v0;v1/ WD 230 R. Hagger is self-adjoint, then its spectrum is given by the interval
sp.C/ Œv0 v 1 v1 ;v0 v 1 v1 D j j j j C j j C j j and thus C v0 v 1 v1 . In our case, if we put k k D j j C j j C j j
B L.w 1;w0;w1/; WD we get
B B 2z0I 2 w1 w 1 w0 w0 2z0 k C C kD j C j C j C C j and therefore 1 r0.A/ B B 2z0I z0: 2k C C k From here we can go all the way back to nish the proof:
1 r0.A/ B B 2z0I z0 2 k C C k 1 sup .B B 2z0I/x;x z0 D 2 x 1 jh C C ij k kD sup Re .B z0I/x;x z0 D x 1 j h C ij k kD sup Re .B z0I/x;x z0 D x 1 h C i k kD r0.B/: D Combining Corollary 4, eorem 8 and eorem 16 we also get the following corollary.
Corollary 17. Let U 1, U0 and U1 be non-empty and compact and consider the operator A ‰E.U 1; U0; U1/. en A has the following property: 2 N.A/ conv.sp.A//: D is corollary is quite remarkable becauseone can usually not expectthis prop- erty from non-normal operators. As a consequence, any tridiagonal random oper- ator has this property almost surely. We do not know if pseudo-ergodic operators with more than three diagonals share this property, but we do know that eo- rem 16 is wrong if the tridiagonality assumption is dropped. On the spectrum and numerical range of tridiagonal random operators 231
Example 18. Let U 2 1 , U 1 1 , U0 0 , U1 1 and U2 1 . D ¹ º D ¹˙ º D ¹ º D ¹ º D ¹ º Consider the 3-periodic operator : : : :: :: :: : 0 :: 0 1 1 1 B:: C B : 1 0 1 1 C A B C Mper;3.U 2;:::;U2/: D B :: C 2 B 1 1 0 1 :C B C B : C B 1 1 0 ::C B C B : : : C B :: :: ::C B C en @ A : : : :: :: :: : 0 :: 0 1 1 1 B:: C 1 B : 1 0 1 1 C B .A A/ B C : WD 2 C D B :: C B 1 1 0 0 :C B C B : C B 1 0 0 ::C B C B : : : C B :: :: ::C B C By eorem 5 we get @ A sp.B/ sp.b.#//; D # Œ0;2/ 2[ where 0 1 e i# 1 e i# C C b.#/ 1 ei# 0 e i# : WD 0 C 1 1 ei# ei# 0 @ C A e spectrum of b.0/ is given by 1 p33 ; 1; 1 p33 . So in particular, we get 2 2 2 C 2 1 p33 sp.B/ and thus 2 2 2 ® ¯ p33 1 9 r .A/ r0. A/ > : D 2 2 4 Let us denote the two operators in L.U 2;:::;U2/ by C1 and C2. We get 2i# i# i# 2i# min Re z min Re.e e e e / z sp.C1/ D # Œ0;2/ C C C 2 2 min 2.cos.2#/ cos.#// D # Œ0;2/ C 2 9 D 4 232 R. Hagger and 2i# i# i# 2i# min Re z min Re.e e e e / z sp.C2/ D # Œ0;2/ C C 2 2 min 2 cos.2#/ D # Œ0;2/ 2 2 D by eorem 5 again. is implies that the numerical range of A exceeds the convex hull of the spectra of Laurent operators in the direction of the negative real axis, i.e. N.A/ conv sp.B/ : 6 B L.U 1;U0;U1/ 2 [ So in particular, in view of Corollary 13, eorem 16 is not valid for ve diagonals.
2.3. A method to compute numerical ranges for general tridiagonal oper- ators. In this section we introduce a method to compute numerical ranges for tridiagonal operators. As explained at the beginning of Section 2, it su ces to compute the numerical abscissae r' for ' Œ0; 2/. Fix ' Œ0; 2/ and recall 2 2 that we have r'.A/ r0.B/ for D
1 i' i' B .e A e A/: WD 2 C In case A is a tridiagonal in nite matrix acting on `2.N/ or `2.Z/, the non-zero entries of B are given by
1 i' i' Bj;j 1 .e Aj;j 1 e Aj 1;j /; D 2 C
1 i' i' i' Bj;j .e Aj;j e Aj;j / Re.e Aj;j /; D 2 C D
1 i' i' Bj;j 1 .e Aj;j 1 e Aj 1;j / C D 2 C C C for all j in the respective index set. B can now be transformed to a real symmetric matrix by applying the unitary diagonal transformation that is de ned recursively as follows:
T1;1 1; D
Tj 1;j 1 sign.Bj;j 1/Tj;j C C D C On the spectrum and numerical range of tridiagonal random operators 233 for all j in the respective index set, where sign C T is de ned as W ! z if z 0; sign.z/ z ¤ WD 8j j <ˆ1 if z 0: D ˆ e matrix : C TBT WD is then real and symmetric with r0.C/ r0.B/ r'.A/ and D D
i' Cj;j Re.e Aj;j / R; D 2
1 i' i' Cj;j 1 e Aj;j 1 e Aj 1;j 0: (5) C D 2 C C C ˇ ˇ ˇ ˇ us the computation of r'.A/ is reduced to the computation of r0.C/, which is also the rightmost point in the spectrum of C . In the following we can also assume that Cj;j 1 > 0 for all j because if Cj;j 1 0 for some j , then C can be divided C C D into blocks and the spectrum of C is then given by the closure of the union of the spectra of these blocks. Moreover, shifting C by I for some R only shifts 2 the spectrum of C by . us we can also assume that C only has positive entries on its main diagonal. is matrix C now satis es the requirements of the following lemma by Szwarc2 that is basically a reformulation of the Schur test.
Lemma 19 ([23, Proposition 1]). Let C L.`2.N// be real, symmetric and tridi- 2 agonal with Cj;j ; Cj;j 1 > 0 for all j N and N > sup Cj;j . If there is a C 2 j N 2 sequence .gj /j N that satis es gj Œ0; 1 and 2 2 2 Cj;j 1 C gj 1.1 gj / (6) .N Cj;j /.N Cj 1;j 1/ C C C for all j N, then r0.C/ N . 2
2 Szwarc [23] actually proved it for C L.`2.Z//, but the proof is very similar for C 2 2 L.`2.N//. 234 R. Hagger
In fact, also the converse is true:
Lemma 20 ([23, Proposition 2]). Let C L.`2.N// be real, symmetric and tridi- 2 agonal with Cj;j ; Cj;j 1 > 0 for all j N. en there exists a sequence .gj /j N C 2 2 with the following properties:
gj Œ0; 1/ for all j N; 2 2 gj 0 if and only if j 1; D D for all j N, 2 2 Cj;j 1 C gj 1.1 gj /: (7) .r0.C/ Cj;j /.r0.C/ Cj 1;j 1/ D C C C To demonstrate the procedure, we prove the following proposition that we need later on. Note that this proposition can also be shown using eorem 5 applied to 1 B .A A /: WD 2 C is results in the computation of the eigenvalues of a 2 2 matrix and one obtains Corollary 22 directly.
Proposition 21. Let I N; Z , let A L.`2.I// be tridiagonal and 2-periodic 2 ¹ º 2 and let N > sup Re Ai;i . Further assume that A A is not diagonal. De ne i I C 2 2 A1;2 A2;1 1.A/ j C j WD 4.N Re A1;1/.N Re A2;2/ and 2 A2;3 A3;2 2.A/ j C j : WD 4.N Re A2;2/.N Re A3;3/ en 1.A/ 2.A/ 1 N r0.A/: C D () D p 2 p 2 Proof. Clearly, A L.` .Z// and PNAPN P L.` .N// have the same numer- 2 j N 2 ical range by Proposition 15. It thus su ces to consider the case A L.`2.N//. 2 Let C beasin (5) with ' 0 so that r0.A/ r0.C/. We can assume that Cj;j > 0 D D for all j N (shifting by R does not change anything). 2 2 If A1;2 A2;1 0. en 1.A/ 0 and an easy computation shows 2.A/ 1 C D D D if and only if N r0.A/. e case A2;3 A3;2 0 is similar. So let us assume D C D Aj;j 1 Aj 1;j 0 for all j N for the rest of the proof. Clearly, this implies C C C ¤ 2 1.A/; 2.A/ > 0 and Cj;j 1 > 0 for all j N. C 2 On the spectrum and numerical range of tridiagonal random operators 235
Let N r0.A/. Lemma 20 applied to C yields a sequence .gj /j N with the D 2 properties
gj Œ0; 1/ for all j N, 2 2 gj 0 if and only if j 1, D D for all j N, 2 2 Aj;j 1 Aj 1;j C C C gj 1.1 gj /: 4.r0.A/ Reˇ Aj;j /.r0.A/ Reˇ Aj 1;j 1/ D C ˇ ˇ C C Since A is 2-periodic, we have
2 A1;2 A2;1 1.A/ C g2; D 4.r0.A/ Reˇ A1;1/.r0.A/ˇ Re A2;2/ D ˇ ˇ 2 A2;3 A3;2 2.A/ C g3.1 g2/; D 4.r0.A/ Reˇ A2;2/.r0.A/ˇ Re A1;1/ D ˇ ˇ 2 A1;2 A2;1 1.A/ C g4.1 g3/; D 4.r0.A/ Reˇ A1;1/.r0.A/ˇ Re A2;2/ D ˇ ˇ : :
We observe 1.A/ g2 .0; 1/ and 2.A/ g3.1 g2/ .0; 1/. If j is odd, we D 2 D 2 deduce the following recursion:
.A/ .A/ .1 g / .A/ g 2 2 j 2 : (8) j 2 .A/ C D 1 gj 1 D 1 1 D 1 gj 1.A/ C 1 gj e corresponding iteration function
f .0; 1 1.A// R; W ! .1 x/2.A/ (9) x ; 7 ! 1 x 1.A/ has a positive derivative
d .1 x/2.A/ 1.A/2.A/ 2 > 0 (10) dx 1 x 1.A/ D .1 x 1.A// since 1.A/; 2.A/ > 0. us f is strictly increasing. Since .gj /j 2N 1 is 2 a sequence in Œ0; 1/, it is in fact a sequence in Œ0; 1 1.A//. Indeed, if 236 R. Hagger gj 1 1.A/, then by equation (8), gj 2 is either not de ned or negative, a C contradiction. Moreover, we have
2.A/ g3 > 0 g1 D 1 1.A/ D since 1.A/; 2.A/ .0; 1/. We conclude that .gj /j 2N 1 is strictly increasing, 2 2 hence convergent. Denote the limit of this sequence by x. By the xed-point theorem, x has to be a xed point of the iteration function f . After some rear- ranging, we get two possible candidates for a xed point:
.1 x/2.A/ x 1 x 1.A/ D .1 x/2.A/ x.1 x 1.A// () D (11) 2 .x / .1 2.A/ 1.A//x 2.A/ 0 () C C D 2 1 2.A/ 1.A/ .1 2.A/ 1.A// 42.A/ x C ˙ C : () D p 2 Of course the xed point we are looking for has to be real and thus
2 .1 2.A/ 1.A// 42.A/ C has to be non-negative. It follows
2 0 .1 2.A/ 1.A// 42.A/ C 2 2 1 2.A/ 1.A/ 22.A/ 21.A/ 21.A/2.A/ 42.A/ D C C C 2 2 2.A/ 2.1 1.A//2.A/ .1 1.A// : D C C
Solving for 2.A/ yields
2 2 2.A/ 1 1.A/ .1 1.A// .1 1.A// C C 1 1.A/ p2 1.A/ D C 2 .1 1.A//p; D p since 2.A/ < 1. is inequality now implies 1.A/ 2.A/ 1. As we will C prove later, this inequality is actually an equality. p p On the spectrum and numerical range of tridiagonal random operators 237
Conversely, let 1.A/ 2.A/ 1. Of course, we can again assume that C D I N. De ne the sequence .gj /j N as follows: D p p 2
g1 0; WD 1.A/ gj 1 if j is odd; C WD 1 gj 2.A/ gj 1 if j is even: C WD 1 gj
In order to apply Lemma 19, we have to check gj Œ0; 1 for all j N. Let us 2 2 rst consider .gj /j 2N 1 and its iteration function (9). As seen in (11) the xed 2 points of f are given by
2 1 2.A/ 1.A/ .1 2.A/ 1.A// 42.A/ x C ˙ C : D p 2
Plugging our assumption 1.A/ 2.A/ 1 into this equation, we get C D p p 2 2 2 1 2.A/ .1 2.A// .1 2.A/ .1 2.A// / 42.A/ x C ˙ C D p q 2 p
42.A/ 42.A/ 2.A/ D ˙ p 2 p 2.A/: D p us there is only one xed point and x < 1. By (10), the iteration function f is strictly increasing in .0; 1 1.A//, while 2 1 1.A/ 1 .1 2.A// 2 2.A/ 2.A/ > 2.A/ D D p p p since 2.A/ x < 1. Furthermore, g1 0 and thus 0 gj x < 1 D D for all j 2N 1. We conclude gj Œ0; 1 for odd j . Similarly (exchanging 2 2 1.A/ and 2.A/ and using the starting point 1.A/ < 1.A/ < 1), we also get gj Œ0; 1 for even j . Furthermore, Condition (6) is ful lled by de nition. 2 p us .gj /j N meets all the requirements and we can apply Lemma 19 to C , which 2 implies r0.C/ r0.A/ N . So let us summarize what we have so far. We have D (i) 1.A/ 2.A/ 1 if N r0.A/ and C D
(ii) Np r0.A/pif 1.A/ 2.A/ 1. C D p p 238 R. Hagger
Now let 1.A/ 2.A/ 1 and assume r0.A/ < N . en C D p p 2 A1;2 A2;1 1.A/ < C 1.A/; 4.r0.A/ Reˇ A1;1/.r0.A/ˇ Re A2;2/ DW Q ˇ ˇ 2 A2;3 A3;2 2.A/ < C 2.A/ 4.r0.A/ Reˇ A2;2/.r0.A/ˇ Re A3;3/ DW Q ˇ ˇ and thus 1.A/ 2.A/ > 1. But this is a contradiction to (i). us Q C Q p p 1.A/ 2.A/ 1 r0.A/ N: C D H) D p p Conversely, let N r0.A/ and assume 1.A/ 2.A/ < 1. en by D C continuity there exists an " > 0 such that p p
2 A1;2 A2;1 v C u4.N " Re A1;1/.N " Re A2;2/ u ˇ ˇ t ˇ ˇ 2 A2;3 A3;2 v C 1: C u4.N " Re A2;2/.N " Re A3;3/ D u ˇ ˇ t ˇ ˇ is is a contradiction to (ii) since N " Corollary 22. Let I N; Z and let A L.`2.I// be tridiagonal and 2-periodic. 2 ¹ º 2 en 1 2 2 r0.A/ .a b .a b/ .c d/ max a; b (12) D 2 C C C C ¹ º p with equality if and only if c d 0, where D D a Re A1;1 Re A3;3; b Re A2;2; D D D 1 1 c A1;2 A2;1 ; d A2;3 A3;2 : D 2 C D 2j C j ˇ ˇ ˇ ˇ On the spectrum and numerical range of tridiagonal random operators 239 3. e Feinberg–Zee random hopping matrix In this section we consider a generalization of the Feinberg–Zee random hopping matrix that was considered in [4]: : : :: :: : 0 :: 0 1 1 B c 1 0 1 C A B C ; WD B c 0 1 C B 0 : C B c 0 ::C B 1 C B : : C B :: ::C B C @ A where .cj /j Z is a sequence of i.i.d. random variables taking values in and 2 ¹˙ º .0; 1. e authors of [4] showed 2 2 sp.A / x iy x y 2.1 / : ¹ C W j j C j j C º p In the case 1 this square is (almost surely) exactly the numerical range of A D as shown in [3] by an explicit computation. For < 1 the square is tangential to the ellipses in eorem 16 and thus a proper superset of the numerical range of A (see Proposition 28 for an explicit formula of N.A /). We try to further improve 2 this bound obtained in eorem 16 by computing the numerical range of N.A /. e idea is the following: sp.A / z C z sp.A / D ¹ 2 W 2 º z C z2 sp.A2 / ¹ 2 W 2 º z C z2 N.A2 / ¹ 2 W 2 º N.A2 /: DW q We thus obtain another upper bound to the spectrum. As we will see in Sec- 2 tion 3.2, we indeed have N.A / N.A /, thus improving the upper bound to the spectrum for all .0; 1, in particular improving the upper bound of [3] that 2 p was obtained by a massive numerical computation in the case 1. To compute 2 2 D N.A / we will observe that, although A is not tridiagonal itself, it can be decom- posed into tridiagonal matrices and thus the method introduced in Section 2.3 can 2 2 be applied. Explicit formulas for N.A /, N.A / and N.A / are postponed to Section 3.2. To simplify the notation, we x here and drop the index. 240 R. Hagger 3.1. Computation of N.A2/. We will prove the following theorem at the end of 2 2 2 this section. e sets N.B1 /, N.B2 / and N.B2 / are lled ellipses/disks and can be computed explicitly (see Proposition 24). eorem 23 thus provides an explicit formula for the (almost sure) numerical range of A2. eorem 23. Let .0; 1, U 1 1 , U0 0 , and U1 . Take an 2 D ¹ º D ¹ º D ¹˙ º operator A M.U 1; U0; U1/. en 2 N.A2/ conv.N.B2/ N.B2/ N.B2//; 1 [ 2 [ 3 where B1 Mper;4.U 1; U0; U1/ is the operator with period .; ; ; /, 2 B2 Mper;4.U 1; U0; U1/ is the operator with period . ; ; ; /, and 2 B3 Mper;4.U 1; U0; U1/ is the operator with period . ; ; ; /. 2 If A ‰E.U 1; U0; U1/, then equality holds. 2 at in the case A ‰E.U 1; U0; U1/ the right-hand side is a subset of 2 the left-hand side is clear by eorem 11 and the fact that op.B2/ op.B/2 D (see Proposition 1). Moreover, it is su cient to prove N.A2/ conv.N.B2/ N.B2/ N.B2// 1 [ 2 [ 3 for A ‰E.U 1; U0; U1/ by the same reason. To do so, we need to compute 2 N.B2/ for i 1; 2; 3 rst. i 2 ¹ º Proposition 24. Let B1, B2 and B3 be as above. en 2 2 2 2 2 2 2 r'.B / 2 cos.'/ .1 / cos.'/ .1 / sin.'/ ; 1 D C C C 2 2 p r'.B / 1 ; 2 D C 2 2 2 2 2 2 2 r'.B / 2 cos.'/ .1 / cos.'/ .1 / sin.'/ 3 D C C C 2 p 2 and the boundaries of N.B1 / and N.B2 / are given by the following parametriza- tions: @N.B2/ z.t/ 2 .1 2/ cos.t/ i.1 2/ sin.t/; 1 W D C C C @N.B2/ z.t/ .1 2/eit ; 2 W D C @N.B2/ z.t/ 2 .1 2/ cos.t/ i.1 2/ sin.t/: 3 W D C C C On the spectrum and numerical range of tridiagonal random operators 241 Proof. B1 is a Laurent operator with diagonals .1/i Z, .0/i Z and ./i Z and 2 2 2 2 therefore B1 is a Laurent operator with diagonals .1/i Z, .0/i Z, .2/i Z, .0/i Z 2 2 2 2 2 2 and . /i Z. erefore the spectrum of B1 is given by the ellipse 2 E t Œ0; 2/ 2 .1 2/ cos.t/ i.1 2/ sin.t/ WD ¹ 2 W C C C º (see e.g. [20] or use eorem 5). Since Laurent operators are normal, E is equal 2 to the boundary of the numerical range of B1 . An elementary computation yields 2 2 2 2 2 2 2 r'.B / 2 cos.'/ .1 / cos.'/ .1 / sin.t/ : 1 D C C C 2 B2 is a 4-periodic operator thatp looks like this: : : : : : :: :: :: :: :: 0 2 0 0 0 1 1 2 2 B 0 2 0 1 C B2 B 2 C : (13) D B 0 0 0 1 C B 2 C B 0 2 0 1 C B C B :: :: :: :: :: C B : : : : :C B C It can be decomposed@ into an even and an odd part as follows. Let A Xe x X x2j 1 0 for all j Z WD 2 W C D 2 and ® ¯ Xo x X x2j 0 for all j Z : WD 2 W D 2 2 2 en B .Xe/ Xe and B ®.Xo/ Xo. us we can consider¯ 2 2 2 2 C2 A X and D2 A X WD j e WD j o 2 and get A C D with respect to this decomposition of X, where C2 and D2 D ˚ are tridiagonal operators given by : : : :: :: :: 0 2 0 1 1 C2 D 2 0 1 B C B : : : C B :: :: ::C B C and @ A : : : :: :: :: 0 2 2 1 1 D2 : D 2 2 1 B C B : : : C B :: :: ::C B C @ A 242 R. Hagger We see that C2 is a Laurent operator and similarly as before we conclude that the boundary of the numerical range of C2 is given by the ellipse t Œ0; 2/ .1 2/ cos.t/ i.1 2/ sin.t/ : ¹ 2 W C C º D2 is a 2-periodic operator, hence we can apply Proposition 21. Let i' 2 D2;' e D2; N 1 ; WD WD C and let us exclude the cases .;'/ .1; 0/ and .;'/ .1; / for the moment so D D that D2;' D is not diagonal. In the notation of Proposition 21 1.D2;'/ and C 2;' 2.D2;'/ are given by i' 2 i' 2 e e 1.D2;'/ j j D 4.1 2 2 cos.'//.1 2 2 cos.'// C C C .1 2/2 4 2 cos.'/2 C D 4..1 2/2 4 2 cos.'/2/ C 1 ; D 4 1 2.D2;'/ 1.D2;'/ : D D 4 us 1.D2;'/ 2.D2;'/ 1 C D and, by Proposition 21, q q 2 r'.D2/ 1 for all ' Œ0; 2/ (.;'/ .1; 0/; .1; / ). D C 2 …¹ º 1 In the remaining two cases .D2;' D / is a diagonal matrix and thus it is 2 C 2;' easily seen that r'.D2/ 2 holds. erefore we have D 2 r'.D2/ 1 for all ' Œ0; 2/. D C 2 Now obviously N.C2/ N.D2/ holds and thus we get 2 2 r'.B / 1 for all ' Œ0; 2/. 2 D C 2 2 A parametrization of @N.B2 / is then of course given by z.t/ .1 2/eit ; t Œ0; 2/: D C 2 B3 is the same as B1 just with replaced by . On the spectrum and numerical range of tridiagonal random operators 243 Next we have to compute 2 N.'/ max r'.Bj / (14) WD j 1;2;3 2¹ º for every ' Œ0; 2/. 2 Proposition 25. Let B1, B2 and B3 be as above, ' arccos. / and let N WD 1 2 be given by (14). en N takes the following values: C for 0 ' ' , N.'/ 2 cos.'/ .1 2/2 cos.'/2 .1 2/2 sin.'/2 D C C C I p for ' ' ' ; N.'/ 1 2 D C I for ' ' ' , C N.'/ 2 cos.'/ .1 2/2 cos.'/2 .1 2/2 sin.'/2 D C C C I p for ' ' 2 ' , C N.'/ 1 2 D C I for 2 ' ' 2, N.'/ 2 cos.'/ .1 2/2 cos.'/2 .1 2/2 sin.'/2: D C C C p Proof. Since all of these functions are continuous, we only have to check where 2 2 2 2 the graphs of r'.B1 /, r'.B2 / and r'.B3 / intersect. Let us have a look at r'.B1 / 2 and r'.B2 / rst: 2 2 r'.B / r'.B / 1 D 2 2 cos.'/ .1 2/2 cos.'/2 .1 2/2 sin.'/2 1 2 () C C C D C .1 2/2 cosp.'/2 .1 2/2.1 cos.'/2/ .1 2 2 cos.'//2 () C C D C cos.'/ : () D 1 2 C 2 2 us the graphs of r'.B1 / and r'.B2 / only intersect at ' arccos. 1 2 / 2 2 D C and 2 '. Similarly, the graphs of r'.B2 / and r'.B3 / only intersect at 2 2 ' arccos. 1 2 / and '. Finally, r'.B1 / and r'.B3 / obviously only D 3C C intersect at 2 and 2 . Plugging in some angles and using (14), one easily deduces the assertion. 244 R. Hagger Now let us focus on A2. Let us denote the rst subdiagonal of an operator 2 A ‰E.U 1; U0; U1/ by .hj /j Z, i.e. hj Aj 1;j for all j Z. en A has 2 2 WD C 2 the following entries: 2 .A /j;j 2 Aj;j 1Aj 1;j 2 1; C D C C C D 2 .A /j;j 1 Aj;j 1Aj 1;j 1 Aj;j Aj;j 1 0; C D C C C C C D 2 .A /j;j Aj;j 1Aj 1;j Aj;j Aj;j Aj;j 1Aj 1;j hj hj 1; D C C C C D C 2 .A /j;j 1 Aj;j Aj;j 1 Aj;j 1Aj 1;j 1 0; D C D 2 .A /j;j 2 Aj;j 1Aj 1;j 2 hj 1hj 2; D D and can be decomposed as A2 C D as in the proof of Proposition 24. D ˚ e matrices C and D are given by Cj;j 1 1; C D Cj;j h2j h2j 1; D C Cj;j 1 h2j 1h2j 2; D and Dj;j 1 1; C D Dj;j h2j 1 h2j ; D C C Dj;j 1 h2j h2j 1; D for j Z, respectively. We will focus on the computation of the numerical range 2 of C . e computation of the numerical range of D is exactly the same so that we obtain N.C/ N.D/. Since the numerical range of a direct sum is just D the convex hull of the union of the numerical ranges of its components, we get N.A2/ N.C/ N.D/. D D By Proposition 3, we have A op.A/ and thus there exists a sequence of inte- 2 gers .gn/n N tending to in nity such that Ag exists and is equal to A. 2 Without loss of generality we may assume that this sequence tends to . en C1 2 2 2 .A /g .Ag/ A C D D D D ˚ by Proposition 1. Observe that V gn .C D/Vgn V gn=2CVgn=2 V gn=2DVgn=2 ˚ D ˚ On the spectrum and numerical range of tridiagonal random operators 245 if gn is even and V gn .C D/Vgn V .gn 1/=2DV.gn 1/=2 V .gn 1/=2CV.gn 1/=2 ˚ D ˚ C C if gn is odd. Clearly either n N gn is even or n N gn is odd is an in- ¹ 2 W º ¹ 2 W º nite set. Let us rst assume that n N gn is even is in nite and denote the ¹ 2 W º e e e sequence of even elements in g by g . en by construction V gn=2CVgn=2 con- e e verges strongly to C and V gn=2DVgn=2 converges strongly to D as n . us op op !1 C .C/ and D .D/. Similarly, assume that n N gn is odd is in - 2 2 ¹ 2 W º nite and denote the sequence of odd elements in g by go. en by construction, o o o o V .gn 1/=2CV.gn 1/=2 converges strongly to D and V .gn 1/=2DV.gn 1/=2 con- verges strongly to C as n . us D op.C/ and C C op.D/ inC this case. ! 1 2 2 Since limit operators of limit operators are again limit operators of the original operator (see e.g. [17, Corollary 3.97]), we also get C op.C/ and D op.D/ 2 2 in this case. Since ge and go tend to , we can apply Proposition 15 to get C1 N.A2/ N.C/ N.C /; D D C where 2 C PNCPN im PN L.` .N//: C WD j 2 Fix ' Œ0; 2/ and let E.'/ be the real symmetric tridiagonal operator that 2 satis es i' Ej;j .'/ Re.e .C /j;j /; D C 1 i' i' Ej;j 1.'/ e .C /j;j 1 e .C /j 1;j C D 2j C C C C C j and 2 r'.A / r'.C / r0.E.'// D C D (cf. (5)). Now for every angle ' there are 16 dierent combinations for .h2j 1;h2j ;h2j 1;h2j 2/ C C in (6). De ne 2 Ej;j 1.'/ j .'/ C (15) WD .N.'/ Ej;j .'//.N.'/ Ej 1;j 1.'// C C for all j N, where N.'/ is given by Proposition 25. Let us consider ' Œ' ; 2 2 2 rst. For these angles, we have the following table. For later reference we num- bered the 16 cases lexicographically. 246 R. Hagger Table 1 tj .h2j 1;h2j ;h2j 1;h2j 2/ j .'/ C C 2 2 2 2 1 .; ; ; / .1 / 4 cos.'/ 4.1 2C2 cos.'//2 C .1 2/2 42 cos.'/2 2 .; ; ; / C 4.1 2 2 cos.'//.1 2/ C 2 2 2 C2 3 .; ; ;/ .1 / 4 cos.'/ 4.1 C2 2 cos.'//.1 2/ C C 2 2 2 2 4 .; ; ; / .1 / 4 cos.'/ 4.1 2 2C cos.'//.1 2 2 cos.'// C C C 2 2 2 2 5 .; ; ; / .1 / 4 cos.'/ 4.1 C2 2 cos.'//.1 2/ C C .1 2/2 42 cos.'/2 6 .; ; ; / C 4.1 2/2 C .1 2/2 42 cos.'/2 7 .; ; ;/ 4.1C 2/2 C 2 2 2 2 8 .; ; ; / .1 / 4 cos.'/ 4.1 2 2Ccos.'//.1 2/ C C C 2 2 2 2 9 . ; ; ; / .1 / 4 cos.'/ 4.1 2 2Ccos.'//.1 2/ C C .1 2/2 42 cos.'/2 10 . ; ; ; / 4.1C 2/2 C .1 2/2 42 cos.'/2 11 . ; ; ;/ C 4.1 2/2 C .1 2/2 42 cos.'/2 12 . ; ; ; / C 4.1 2 2 cos.'//.1 2/ C C2 2 2 C2 13 . ; ; ; / .1 / 4 cos.'/ 4.1 2 2C cos.'//.1 2 2 cos.'// C C C .1 2/2 42 cos.'/2 14 . ; ; ; / C 4.1 2 2 cos.'//.1 2/ C 2C2 2 C2 15 . ; ; ;/ .1 / 4 cos.'/ 4.1 2 2Ccos.'//.1 2/ C C C 2 2 2 2 16 . ; ; ; / .1 / 4 cos.'/ 4.1 2 C2 cos.'//2 C C is table has to be read as follows. e sequence .hj /j N induces a sequence 2 .tj /j N. For example if the sequence .hj /j N starts with 2 2 .; ; ; ; ; ; ; ; ; ;:::/; the sequence .tj /j N starts with .7; 9; 2; 6; : : :/. e numbers tj are used to refer 2 to the respective j , which are computed via Formula (15). So if, for example, 2 2 2 2 t 6 .'/ .1 / 4 cos.'/ j , then j C 4.1 2/2 . D D C On the spectrum and numerical range of tridiagonal random operators 247 We will nd the following equalities and inequalities useful: 0 cos.'/ < 1 (16) 1 2 C 4 4 .1 4/2 .1 2/2 4 2 cos.'/2 .1 2/2 C (17) C C .1 2/2 D .1 2/2 C C 2 2 1 4 1 2 2 cos.'/ 1 2 C (18) C C 1 2 D 1 2 C C .1 2/2 4 2 cos.'/2 .1 2 cos.'//.1 2 cos.'// (19) C D C C C 1 Using these, it is not di cult to see that j .'/ 2 for all ' Œ'; 2 and N 2 1 j (i.e. for all possible values of j .'/ in Table 1). We even have j .'/ 4 2 for all ' Œ' ; and j N with tj 3; 5 . is observation is very useful to 2 2 2 … ¹ º nally construct the sequence needed for Lemma 19. Proposition 26. Let .0; 1, U 1 1 , U0 0 , U1 and let A 2 D ¹ º D ¹ º D ¹˙Nº 2 ‰E.U 1; U0; U1/. Let ' Œ'; 2 , j j .'/ and tj for all j be de ned as 2 WD 2 above. en the sequence .gj /j N, de ned by the following prescription, satis es 2 gj Œ0; 1 and j gj 1.1 gj / for all j N: 2 C 2 1 1 2 2 cos.'/ if t1 5, choose g1 2 C 1 2 ; D D C if there is some k N such that t1 ::: tk 6 and tk 1 5, choose 1 1 2 2 cos2.'/ D D D C D g1 2 C 1 2 ; D C 1 if neither is true, choose g1 ; D 2 1 1 2 2 cos.'/ if tj 2; 6; 10; 14 and tj 1 5, choose gj 1 2 C 1 2 ; 2 ¹ º C D C D C if tj 2; 6; 10; 14 , there is some k>j such that tj 1 ::: tk 6 and C 2 ¹ º 1 1 2 2 cos.'/ D D D tk 1 5, choose gj 1 2 C 1 2 ; C D C D C 2 t 3 g 1 1 2 cos.'/ if j , choose j 1 2 C 1C 2 ; D C D C if tj 11, there is some k j such that tk ::: tj 11 and tk 1 3, D 1 1 2 2 cos.'/ D D D D choose gj 1 2 C 1C 2 ; C D C 1 if none of the above is true, choose gj 1 2 . C D 248 R. Hagger Proof. at gj Œ0; 1 holds for all j N follows from (16). So it remains 2 2 1 to prove that j gj 1.1 gj / holds. Above we observed that j 4 unless C tj 3; 5 . So if tj 3; 5 for all j N, then j gj 1.1 gj / is obviously 2 ¹ º … ¹ º 2 C satis ed. It remains to investigate what happens if tj 3; 5 for some j N. 2 ¹ º 2 Roughly speaking, the idea is that the cases tj 3 and tj 5 aect the sequence ND 1 D .gk/k N only locally in the sense that k gk 2 is an in nite set. us 2 21 W D if tj 3; 5 occurs, we try to get back to as soon as possible as j increases. 2 ¹ º ® 2 ¯ e argument can then be repeated by induction. Note that if tj 3; 5 , we can simplify j as follows: 2 ¹ º .1 2/2 4 2 cos.'/2 1 1 2 2 cos.'/ j C C C ; D 4.1 2 2 cos.'//.1 2/ D 4 1 2 C C C where we used (19). 1 Let us consider the case tj 3 rst and assume gj 2 . More precisely, we D 1 D start our sequence with g1 g2 ::: until tj 3; 5 occurs the rst time D D D 2 2 ¹ º and consider the case where tj 3 occurs rst. en by de nition D 1 1 2 2 cos.'/ gj 1 C C C D 2 1 2 C and 1 1 2 2 cos.'/ gj 1.1 gj / C C j : C D 4 1 2 D C Observe that j and j 1 are not independent. Indeed, j 1 depends on h2j 1, C C C h2j 2, h2j 3 and h2j 4 whereas j depends on h2j 1, h2j , h2j 1 and h2j 2. C C C C C us if we x j , there are only 4 possible combinations for j 1. In particular, C if tj 3, then tj 1 has to be contained in 9; 10; 11; 12 . So there are four cases: D C ¹ º .1 2/2 4 2 cos.'/2 j 1 C .tj 1 9/; C D 4.1 2 2 cos.'//.1 2/ C D C C .1 2/2 4 2 cos.'/2 j 1 C .tj 1 10/; C D 4.1 2/2 C D C .1 2/2 4 2 cos.'/2 j 1 C .tj 1 11/; C D 4.1 2/2 C D C .1 2/2 4 2 cos.'/2 j 1 C .tj 1 12/: C D 4.1 2 2 cos.'//.1 2/ C D C C C On the spectrum and numerical range of tridiagonal random operators 249 1 In the rst case we have gj 2 2 : C D 1 1 1 2 2 cos.'/ gj 2.1 gj 1/ C C C C D 2 4 1 2 C 1 1 2 2 cos.'/ C D 4 1 2 C 1 1 4 C 4 .1 2/2 C j 1; C where we used (18) in line 2 and (17) and (18) in line 3. In the second case we 1 1 2 2 cos.'/ 1 1 have gj 2 2 C 1 2 2 if tj 2 5; 6 and gj 2 2 if not: C D C C 2 ¹ º C D 1 2 2 cos.'/ 1 1 1 2 2 cos.'/ gj 2.1 gj 1/ C C C C C 1 2 2 4 1 2 C C 1 .1 2 2 cos.'// 2 C D 4 .1 2/2 C 1 .1 4/2 C 4 .1 2/4 C j 1; C where we used (18) in line 2 and (17) in line 3. In the third case we have 1 1 2 2 cos.'/ gj 2 2 C 1C 2 : C D C 1 2 2 cos.'/ 1 1 1 2 2 cos.'/ gj 2.1 gj 1/ C C C C C C D 1 2 2 4 1 2 C C 1 1 2 2 cos.'/ 1 2 2 cos.'/ C C C D 4 1 2 1 2 C C 1 .1 2/2 4 2 cos.'/2 C D 4 .1 2/2 C j 1: D C 1 In the fourth case we have gj 2 2 : C D 1 1 1 2 2 cos.'/ gj 2.1 gj 1/ C C C C D 2 4 1 2 C 1 1 2 2 cos.'/ C D 4 1 2 C 1 .1 2/2 4 2 cos.'/2 C D 4 .1 2 2 cos.'//.1 2/ C C C j 1: D C 250 R. Hagger 1 So either gj 2 2 (and we included one special case that we need afterwards) or C 1 gj 2 gj 1. us either we are where we started with, namely 2 ,orweareinthe C D C third case, where j 1 is of type (11). But in this case we have h2j 1 h2j 3 and C C D C h2j 2 h2j 4 and thus we have again the same four cases for j 2 and so on. C D C C So either we end up with an in nite sequencewith gk gj 1 for all k>j (which D C is impossible by pseudo-ergodicity, but would still be just ne) or we eventually 1 go out with gk for some k j 2. us we are done by induction if we can 2 C control the case tj 5 as well. D e case tj 5 is very similar to the case tj 3, but we have to think back- D D wards this time, which is a little bit more complicated. If we have a look at the generators (i.e. h2j 1, h2j , h2j 1 and h2j 2) of the cases tj 3 and tj 5, it is C C D D intuitively clear, why this has to be the same but backwards. So assume tj 5. 2 D g 1 1 2 cos.'/ g 1 en j 2 C 1 2 and j 1 2 by de nition and thus D C C D 1 1 1 2 2 cos.'/ 1 1 2 2 cos.'/ gj 1.1 gj / C C C j : C D 2 4 1 2 D 4 1 2 D C C As already mentioned, we have to look backwards here, i.e. we want to control gj 1. Now there are ve cases. e rst case is j 1, which is trivial of course. D 1 e second case is where tj 1 2. In this case we have gj 1 2 : D D 1 1 2 2 cos.'/ gj .1 gj 1/ C D 4 1 2 C 1 1 4 C 4 .1 2/2 C j 1; where we used (18) in line 1 and (17) and (18) in line 2. e third case is where 1 1 2 2 cos.'/ tj 1 6. In this case we have gj 1 2 C 1 2 : D D C 1 1 2 2 cos.'/ 1 1 2 2 cos.'/ gj .1 gj 1/ C 1 C D 2 1 2 2 1 2 C C 1 1 2 2 cos.'/1 2 2 cos.'/ C C C D 4 1 2 1 2 C C 1 .1 2/2 4 2 cos.'/2 C D 4 .1 2/2 C j 1: D On the spectrum and numerical range of tridiagonal random operators 251 1 1 2 2 cos.'/ 1 e fourth case is where tj 1 10. We either have gj 1 2 C 1C 2 2 1 D D C if tj 2 3; 11 or gj 1 2 if not: 2 ¹ º D 1 1 2 2 cos.'/ 1 1 2 2 cos.'/ gj .1 gj 1/ C 1 C C 2 1 2 2 1 2 C C 1 .1 2 2 cos.'// 2 C D 4 .1 2/2 C 1 .1 4/2 C 4 .1 /4 C j 1; where we used (18) in line 2 and (17) and in line 3. Note that this case matches perfectly with the second case above. e fth case is where tj 1 14. In this 1 D case we have gj 1 2 : D 1 1 2 2 cos.'/ gj .1 gj 1/ C D 4 1 2 C 1 .1 2/2 4 2 cos.'/2 C D 4 .1 2 2 cos.'//.1 2/ C C C j 1: D 1 Again we conclude that either gj 1 2 (note that the inequality is in the other direction this time, which is good!) or gj 1 gj . us either we started where 1 D we ended, namely 2 (or even better, we started with something that is greater than 1 1 or equal to 2 and the sequence reduced to 2 , compare with the mentioned special case above), or we are in the third case, where tj 1 6. But in this case we have D h2j 1 h2j 3 and h2j 2 h2j 4 and thus we again have the same four cases for D D j 2 and so on. us we either end up at g1, which is ne or we eventually have 1 gk for some k j 1. In either case we are done by induction. 2 So we are done with the case ' Œ' ; . is means that there is only the 2 2 case ' Œ0; ' left. All the other angles will follow by symmetry. Let us now 2 consider the table for the angles ' Œ0; ' . Remember that we have 2 N.'/ 2 cos.'/ .1 2/2 cos.'/2 .1 2/2 sin.'/2 D C C C 2 cos.'/ p.1 2/2 4 2 cos.'/2 D C C p here and let us drop the ' in N.'/ for the sake of readability. 252 R. Hagger Table 2 tj .h2j 1;h2j ;h2j 1;h2j 2/ j .'/ C C 1 1 .; ; ; / 4 .1 2/2 42 cos.'/2 2 .; ; ; / 4.N 2C cos.'//N .1 2/2 42 cos.'/2 3 .; ; ;/ 4.NC 2 cos.'//N .1 2/2 42 cos.'/2 4 .; ; ; / 4.N 2C cos.'//.N 2 cos.'// C .1 2/2 42 cos.'/2 5 .; ; ; / 4.NC 2 cos.'//N .1 2/2 42 cos.'/2 6 .; ; ; / C 4N 2 .1 2/2 42 cos.'/2 7 .; ; ;/ C 4N 2 .1 2/2 42 cos.'/2 8 .; ; ; / 4.N 2C cos.'//N C .1 2/2 42 cos.'/2 9 . ; ; ; / 4.N 2C cos.'//N .1 2/2 42 cos.'/2 10 . ; ; ; / C 4N 2 .1 2/2 42 cos.'/2 11 . ; ; ;/ C 4N 2 .1 2/2 42 cos.'/2 12 . ; ; ; / 4.NC 2 cos.'//N C .1 2/2 42 cos.'/2 13 . ; ; ; / 4.N 2C cos.'//.N 2 cos.'// C .1 2/2 42 cos.'/2 14 . ; ; ; / 4.NC 2 cos.'//N C .1 2/2 42 cos.'/2 15 . ; ; ;/ 4.N 2C cos.'//N C 2 2 2 2 16 . ; ; ; / .1 / 4 cos.'/ 4.N 2C cos.'//2 C We will nd the following equalities and inequalities useful: N 1 2; (20) C cos.'/ ; (21) 1 2 C On the spectrum and numerical range of tridiagonal random operators 253 N 2 cos.'/ .1 2/2 4 2 cos.'/2 D C p 4 2 2 4 .1 / (22) s C .1 2/2 C 1 4 C D 1 2 C 1 4 4 2 N 2 cos.'/ C C 1 2 C 1 2 C C (23) 1 4 2 4 C C D 1 2 C .1 2/2 4 2 cos.'/2 .N 2 cos.'//2 (24) C D 4 4 .1 2/2 4 2 cos.'/2 .1 2/2 C C .1 2/2 C (25) .1 4 2 4/.1 4/ C C C : D .1 2/2 C 1 N Using these, it isnot di cult to seethat j .'/ 2 for all ' Œ0; ' and j 12 2 (i.e. for all possible values of j .'/ in Table 2 and j .'/ for all ' Œ0; ' 4 2 and j N with tj 3; 5 . If 2 … ¹ º .1 2/2 4 2 cos.'/2 .N 2 cos.'//N; C 1 N then even j .'/ 4 for all ' Œ0; ' and j (i.e. also if tj 3; 5 ). In this 21 2 2 ¹ º case we can just choose gj for all j N and we are done. It thus remains to D 2 2 consider the case where .1 2/2 4 2 cos.'/2 > .N 2 cos.'//N: C e argument is now exactly the same as in the proof of Proposition 26. Proposition 27. Let .0; 1, U 1 1 , U0 0 , U1 and let 2 D ¹ º D ¹ º D ¹˙ º A ‰E.U 1; U0; U1/. Let ' Œ0; ', j j .'/ and tj for all j N be 2 2 WD 2 de ned as above. Further assume that .1 2/2 4 2 cos.'/2 > .N 2 cos.'//N: C en the sequence .gj /j N, de ned by the following prescription, satis es 2 gj Œ0; 1 and j gj 1.1 gj / for all j N: 2 C 2 254 R. Hagger 1 .1 2/2 42 cos.'/2 if t1 5, choose g1 1 2 C.N 2 cos.'//N ; D D if there is some k N such that t1 ::: tk 6 and tk 1 5, choose 1 .1 2/2242 cos.'/2 D D D C D g1 1 2 C.N 2 cos.'//N ; D 1 if neither is true, choose g1 ; D 2 1 .1 2/2 42 cos.'/2 if tj 2; 6; 10; 14 and tj 1 5, choose gj 1 1 2 C.N 2 cos.'//N ; 2 ¹ º C D C D if tj 2; 6; 10; 14 , there is some k>j such that tj 1 ::: tk 6 and C 2 ¹ º 1 .1 2/2 42 cos.'/2 D D D tk 1 5, choose gj 1 1 2 C.N 2 cos.'//N ; C D C D 1 .1 2/2 42 cos.'/2 if tj 3, choose gj 1 2 C.N 2 cos.'//N ; D C D if tj 11, there is some k j such that tk ::: tj 11 and tk 1 3, D 1 .1 2/2 42 cos.'/2 D D D D choose gj 1 2 C.N 2 cos.'//N ; C D 1 if none of the above is true, choose gj 1 2 . C D Proof. e proof is exactly the same as the proof of Proposition 26. We only have to change the numbers. at gj Œ0; 1 holds for all j N follows from (20), 2 2 (22), and (25). So it remains to prove j gj 1.1 gj /. Above we observed that 1 C j unless tj 3; 5 . us if the cases tj 3 and tj 5 do not occur, then 4 2 ¹ º D D j gj 1.1 gj / is obviously satis ed. So we are left with the cases tj 3 and C D tj 5 again. D 1 Let us considerthe case tj 3 rst and assume that gj . en by de nition D D 2 1 .1 2/2 4 2 cos.'/2 gj 1 C C D 2 .N 2 cos.'//N and 1 .1 2/2 4 2 cos.'/2 gj 1.1 gj / C j : C D 4 .N 2 cos.'//N D Now there are four possible cases for j 1: C .1 2/2 4 2 cos.'/2 j 1 C .tj 1 9/; C D 4.N 2 cos.'//N C D .1 2/2 4 2 cos.'/2 j 1 C .tj 1 10/; C D 4N 2 C D .1 2/2 4 2 cos.'/2 j 1 C .tj 1 11/; C D 4N 2 C D .1 2/2 4 2 cos.'/2 j 1 C .tj 1 12/: C D 4.N 2 cos.'//N C D C On the spectrum and numerical range of tridiagonal random operators 255 1 In the rst case we have gj 2 2 : C D 1 1 .1 2/2 4 2 cos.'/2 gj 2.1 gj 1/ C C C D 2 4 .N 2 cos.'//N 1 2.N 2 cos.'//N .1 2/2 4 2 cos.'/2 C C D 4 .N 2 cos.'//N 1 2.1 4/ .1 2/2 4 2 cos.'/2 C C C 4 .N 2 cos.'//N 1 .1 2/2 4 2 cos.'/2 C D 4 .N 2 cos.'//N j 1; D C where we used (20) and (22) in line 2. In the second case we have 1 .1 2/2 4 2 cos.'/2 1 gj 2 1 C C D 2 .N 2 cos.'//N 2 1 if tj 2 5; 6 and gj 2 2 if not: C 2 ¹ º C D gj 2.1 gj 1/ C C 1 .1 2/2 4 2 cos.'/2 1 .1 2/2 4 2 cos.'/2 1 C 1 C 2 .N 2 cos.'//N 2 .N 2 cos.'//N 1 2N.N 2 cos.'// .1 2/2 4 2 cos.'/2 . C C /2 D 4 .N 2 cos.'//N 1 N.N 2 cos.'// 1 4 .1 2/2 4 2 cos.'/2 . C C C C /2 4 .N 2 cos.'//N 1 N.N 2 cos.'// 2N cos.'/ 4 2 cos.'/2 . C /2 4 .N 2 cos.'//N 1 .N 2 cos.'//2 D 4 N 2 1 . .1 2/2 4 2 cos.'/2/2 C2 D 4 p N j 1; D C where we used (20) and (22) in line 2 and (20) and (21) in line 3. 256 R. Hagger 1 .1 2/2 42 cos.'/2 In the third case we have gj 2 2 C.N 2 cos.'//N : C D 1 .1 2/2 4 2 cos.'/2 1 .1 2/2 4 2 cos.'/2 gj 2.1 gj 1/ C 1 C C C D 2 .N 2 cos.'//N 2 .N 2 cos.'//N 1 .1 2/2 4 2 cos.'/2 N 2 cos.'/ C 4 .N 2 cos.'//N N 1 .1 2/2 4 2 cos.'/2 C D 4 N 2 j 1 D C 1 like in the second case. In the fourth case we have gj 2 2 : C D 1 1 .1 2/2 4 2 cos.'/2 gj 2.1 gj 1/ C C C D 2 4 .N 2 cos.'//N 1 1 1 4 2 4 C C 2 4 .1 2/2 C 1 1 4 C D 4 .1 2/2 C 1 .1 2/2 4 2 cos.'/2 C 4 .N 2 cos.'//N C j 1; D C where we used (20), (22) and (25) in line 1 and (20), (23) and (25) in line 3. 1 So either gj 2 2 or gj 2 gj 1. As in the proof of Proposition 26 we con- C C D C 1 clude that we eventually go out with gk for some k j 2. us we are 2 C done by induction if we can control the case tj 5 as well. D 1 .1 2/2 42 cos.'/2 1 So assume tj 5. en gj 1 2 C.N 2 cos.'//N and gj 1 2 by D D C D de nition and thus 1 .1 2/2 4 2 cos.'/2 gj 1.1 gj / C C D 4 .N 2 cos.'//N j : D Again there are ve cases here. e rst case is j 1, which is again trivial. D On the spectrum and numerical range of tridiagonal random operators 257 1 e second case is where tj 1 2. In this case we have gj 1 2 : D D 1 1 .1 2/2 4 2 cos.'/2 gj .1 gj 1/ C D 2 4 .N 2 cos.'//N 1 2.N 2 cos.'//N .1 2/2 4 2 cos.'/2 C C D 4 .N 2 cos.'//N 1 2.1 4/ .1 2/2 4 2 cos.'/2 C C C 4 .N 2 cos.'//N 1 .1 2/2 4 2 cos.'/2 C D 4 .N 2 cos.'//N j 1; D C where we used (20) and (22) in line 2. e third case is where tj 1 6. In this 1 .1 2/2 42 cos.'/2 D case we have gj 1 1 2 C.N 2 cos.'//N : D 1 .1 2/2 4 2 cos.'/2 1 .1 2/2 4 2 cos.'/2 gj .1 gj 1/ 1 C C D 2 .N 2 cos.'//N 2 .N 2 cos.'//N 1 .2N.N 2 cos.'// .1 2/2 4 2 cos.'/2 C C D 4 .N 2 cos.'//N .1 2/2 4 2 cos.'/2 C .N 2 cos.'//N 1 .N.N 2 cos.'// 1 4 .1 2/2 4 2 cos.'/2 C C C C 4 .N 2 cos.'//N .1 2/2 4 2 cos.'/2 C .N 2 cos.'//N 1 .N.N 2 cos.'// 2N cos.'/ 4 2 cos.'/2 C 4 .N 2 cos.'//N .1 2/2 4 2 cos.'/2 C .N 2 cos.'//N 1 N 2 cos.'/ .1 2/2 4 2 cos.'/2 C D 4 N .N 2 cos.'//N 1 .1 2/2 4 2 cos.'/2 C D 4 N 2 j 1; D where we used (20) and (22) in line 2 and (20) and (21) in line 3. e fourth case 1 .1 2/2 42 cos.'/2 1 is where tj 1 10. In this case we either have gj 1 2 C.N 2 cos.'//N 2 D D 258 R. Hagger 1 if tj 2 3; 11 or gj 1 2 if not: 2 ¹ º D 1 .1 2/2 4 2 cos.'/2 gj .1 gj 1/ 1 C 2 .N 2 cos.'//N 1 .1 2/2 4 2 cos.'/ 2 1 C 2 .N 2 cos.'//N 1 .N 2 cos.'//2 D 4 N 2 1 . .1 2/2 4 2 cos.'/2/2 C2 D 4 p N j 1: D C 1 e fth case is where tj 1 14. In this case we have gj 1 2 : D D 1 1 .1 2/2 4 2 cos.'/2 gj .1 gj 1/ C D 2 4 .N 2 cos.'//N 1 1 1 4 2 4 C C 2 4 .1 2/2 C 1 1 4 C D 4 .1 2/2 C 1 .1 2/2 4 2 cos.'/2 C 4 .N 2 cos.'//N C j 1; D C where we used (20), (22), and (25) in line 1 and (20), (23), and (25) in line 3. As in the proof of Proposition 26 we conclude that we either end up at g1, which is 1 ne or we eventually have gk for some k j 1. In either case we are done 2 by induction. Using the sequences obtained in Proposition 26 and Proposition 27, we can now apply Lemma 19 to prove eorem 23: Proof of eorem 23. Let A ‰E.U 1; U0; U1/. e inclusion 2 N.A2/ conv.N.B2/ N.B2/ N.B2// 1 [ 2 [ 3 is clear by eorem 11 and the fact that op.B2/ op.B/2 (see Proposition 1). D2 To prove the other inclusion, we have to show r'.A / N.'/ for all ' Œ0; 2/, 2 where N.'/ is given by Proposition 25. Using the transformations ' ' 7! On the spectrum and numerical range of tridiagonal random operators 259 and ' ' , it is clear that is su ces to consider ' Œ0; . Indeed, N.'/ is 7! C 2 2 invariant under these transformations and in the Tables 1 and 2 only the roles of and are interchanged. To apply Lemma 19 to E.'/, we have to assure C 1 i' i' Ej;j 1.'/ e .C /j;j 1 e .C /j;j 1 > 0 C D 2j C C C C C j and Ej;j .'/ > 0 for all ' Œ0; . e latter can be achieved by shifting and the 2 2 former can only fail if 1 and ' 0. But in this case we trivially have D D r0.E.'// E.'/ 4 N.'/ k k D by the Wiener estimate (e.g. [17, p. 25]). Moreover, we clearly have N.'/ > sup Ej;j .'/ j N 2 as Ej;j 2 cos.'/; 0; 2 cos.'/ for all j N and ' Œ0; (cf. Proposi- 2 ¹ º 2 2 2 tion 25). We can thus apply Lemma 19, using the sequences from Proposition 26 and Proposition 27,3 to obtain 2 r'.A / r0.E.'// N.'/ for all ' Œ0; D 2 2 and hence all ' Œ0; 2/. 2 e inclusion for more general operators A M.U 1; U0; U1/ now follows 2 from eorem 11 and Proposition 1 again. In Figure 1 we can see that N.A2/ is indeed a tighter upper bound to the spectrum than N.A/. Moreover, it shows that sp.A/ is not equal to N.A/ and thus p not convex. is con rms and improves the numerical results obtained in [3]. A rigorous proof of this observation can be found in Section 3.2. 3.2. A proof that pN.A2/ N.A/. In this section we provide formulas for N.A/, N.A/2 and N.A2/ in terms of graphs of explicit functions. ese follow from elementary computations using eorem 16, eorem 23 and Proposition 24. ese formulas then allow us to show that N.A2/ is indeed a proper subset of N.A/. p 3 Including the trivial case where .1 2/2 4 2 cos.'/2 .N 2 cos.'//N . C 260 R. Hagger Figure 1. e boundary of N.A2/ (blue), the boundary of N.A/ (red) and a lower bound to sp.A/ consisting of spectra of periodic operators and the closed unit disk (black, see [2] p and [3]) in the case 1. D Proposition 28. Let .0; 1, U 1 1 , U0 0 , U1 and consider 2 D ¹ º D ¹ º D ¹˙ º an operator A ‰E.U 1; U0; U1/. en 2 N.A/ x iy C f.x/ y f.x/; .1 / x 1 ; D ¹ C 2 W C C º where f Œ .1 /; 1 R W C C ! is given by x 2 .1 /2 .1 / 1 for x .1 /; C ; 1 2 C 2.1 2/ 8 r C h C i ˆ 2 2 ˆ 2 .1 / p .1 / ˆ 2.1 / x for x C ; ; ˆ C C 2 2.1 2/ 2.1 2/ ˆ ˆp C C i ˆ 2 p 2 p 2 ˆ x .1 / .1 / f.x/ ˆ.1 / 1 for x ; ; D ˆ C 1 2 2.1 2/ 2.1 2/ <ˆ r C C .1p /2 .1p /2 ˆ 2.1 2/ x for x ; C ; ˆ C 2 2 2 ˆ 2.1 / 2.1 / ˆp h C C ˆ 2 ˆ x 2 p.1 / p ˆ.1 / 1 for x C ; 1 : ˆ 1 2 ˆ r 2 2.1 / C ˆ C h C i ˆ :ˆ p On the spectrum and numerical range of tridiagonal random operators 261 Proof. By eorem 16, the numerical range of A is given by the convex hull of the two ellipses ei# e i# # Œ0; 2/ and ei# e i# # Œ0; 2/ . ¹ C W 2 º ¹ W 2 º e assertion thus follows by an elementary computation. Proposition 29. Let .0; 1, U 1 1 , U0 0 , U1 and take an 2 D ¹ º D ¹ º D ¹˙ º operator A ‰E.U 1; U0; U1/. en 2 N.A/2 x iy C f.x/ y f.x/; .1 /2 x .1 /2 ; D¹ C 2 W C C º where f Œ .1 /2; .1 /2 R W C C ! is given by 2 2 x 2 2 .1 / 1 C 2 for x Œ .1 / ; 4/; 8 1 2 C r C ˆ x2 ˆ1 2 for x Œ 4;4; f.x/ ˆ 2 D ˆ C 4.1 / 2 <ˆ C 2 2 x 2 2 ˆ.1 / 1 for x .4; .1 / : ˆ 1 2 2 C ˆ r C ˆ Proof. Using Re:ˆ.z2/ .Re z/2 .Im z/2 and Im.z2/ 2 Re z Im z for z C, D D 2 this follows from Proposition 28 by another elementary computation. Proposition 30. Let .0; 1, U 1 1 , U0 0 , U1 and take an 2 D ¹ º D ¹ º D ¹˙ º operator A ‰E.U 1; U0; U1/. en 2 N.A2/ x iy C g.x/ y g.x/; .1 /2 x .1 /2 ; D C 2 W C C where ® ¯ g Œ .1 /2; .1 /2 R W C C ! is given by 2 .1 2/2 for x Œ .1 / ; 2 1C 4 /, 2 C C x 2 2 g.x/ .1 2/ 1 C D 1 2 I r C .1 2/2 for x Œ 2 1C 4 ; /, 2 C .1 2/2 g.x/ C x D p1 2 4 C p1 2 4 I C C C C 262 R. Hagger for x Œ ;, 2 g.x/ .1 2/2 x2 D C I .1 2/2 p for x Œ; 2 1C 4 /, 2 C C .1 2/2 C x p1 2 4 p1 2 4 I C C C C .1 2/2 2 for x Œ2 1C 4 ; .1 / /, 2 C C C x 2 2 .1 2/ 1 : 1 2 r C Proof. is follows from eorem 23 and Proposition 24 by yet another tedious but elementary computation. us N.A/2 is surrounded by (parts of) two parabolas and two ellipses whereas N.A2/ is surrounded by (parts of) a circle, two ellipses and four straight lines (see Figure 2 for the case 1 ). It is readily seen that the ellipses are the same, D 2 respectively. 1.5 1 0.5 0 -0.5 -1 -1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Figure 2. e two parabolas and the two ellipses (blue, dotted), the circle (red, dotted), N.A/2 (blue, solid) and N.A2/ (red, solid) in the case 1 . D 2 On the spectrum and numerical range of tridiagonal random operators 263 eorem 31. Let .0; 1, U 1 1 , U0 0 , U1 and take an 2 D ¹ º 2 D ¹ º D ¹˙ º 2 operator A ‰E.U 1; U0; U1/. en N.A / is a proper subset of N.A/ . 2 Proof. Let f be as in Proposition 29 and B1, B2, B3 as in eorem 23. We will show that f is concave, which implies that N.A/2 is convex. It then 2 2 2 2 remains to show that N.A/ contains N.B1 /, N.B2 / and N.B3 / by eorem 23. 2 2 Using Corollary 13 and the parametrizations of @N.B1 / and @N.B3 / provided by Proposition 24, it is easily seen that 2 2 2 N.B / N.B1/ N.A/ 1 D and 2 2 2 N.B / N.B3/ N.A/ : 3 D 2 It will thus su ce to consider N.B2 /. Clearly, f is continuously dierentiable with x 2 1 2 x 2 2 1=2 2 C 1 C for x Œ .1 /2; 4/; 1 2 1 2 1 2 2 C 8 C x C C ˆ for x Œ 4;4; f 0.x/ ˆ 2 D ˆ 2.1 / 2 <ˆ C x 2 1 2 x 2 2 1=2 2 1 for x .4; .1 /2: ˆ 2 2 2 ˆ 1 1 1 2 C ˆ C C C ˆ Moreover,:f 0 is piecewise continuously dierentiable with 1 2 x 2 2 3=2 1 C for x Œ .1 /2; 4/; 1 2 1 2 2 C 8 C C ˆ 1 f .x/ ˆ for x . 4;4/; 00 ˆ 2 D ˆ 2.1 / 2 <ˆ C 1 2 x 2 2 3=2 1 for x .4; .1 /2: ˆ 2 2 ˆ 1 1 2 C ˆ C C us f .x/ <:ˆ 0 for x Œ .1 /2; .1 /2 4;4 , which implies that f 00 2 C C n ¹ º is concave. Let g Œ .1 2/; 1 2 R W C C ! be de ned by .1 2/2 x2 so that C N.B2/ px iy C g.x/ y g.x/; .1 2/ x 1 2 2 D C 2 W C C (see Proposition® 24). ¯ 264 R. Hagger Assume rst that 4 1 2. en C x2 f.x/ g.x/ 1 2 .1 2/2 x2 D () C 4.1 2/ D C C p x2 2 1 2 .1 2/2 x2 () C 4.1 2/ D C C x2 x4 0 () 2 C 16.1 2/2 D C x 0 () D for x Œ .1 2/; 1 2. us the graphs of f and g only intersect at 2 C C x 0. Since both f and g are continuous, it su ces to plug in some values D (e.g. .1 2/) to conclude f g and thus N.B2/ N.A/2. As we mentioned ˙ C 2 at the beginning of the proof, this implies N.A2/ N.A/2. Now let 4 < 1 2. For x Œ 4;4, this is the same as above. For C 2 x .4; 1 2 we have 2 C x 2 2 f.x/ g.x/ .1 2/ 1 .1 2/2 x2 D () 1 2 D C r C x 2 2 p .1 2/2 1 .1 2/2 x2: () 1 2 D C C 4 But this quadratic equation only has the solutions x 2 and x 1 , D D C which are not contained in .4; 1 2. us the graphs of f and g do not C intersect in .4; 1 2. Similarly, the graphs of f and g do not intersect in C Œ .1 2/; 4/. 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MR 0077805 Zbl 0067.08403 Received July 18, 2014; revised April 15, 2015 Raael Hagger, Institute of Mathematics, Hamburg University of Technology, Schwarzenbergstr. 95 E, 21073 Hamburg, Germany Institute for Analysis, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany e-mail: ra[email protected]