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A BLASCHKE-TYPE CONDITION AND ITS APPLICATION TO COMPLEX JACOBI MATRICES 8 A. BORICHEV, L. GOLINSKII, S. KUPIN 0 0 Abstract. WeobtainaBlaschke-typenecessaryconditiononze- 2 ros of analytic functions on the unit disk with different types of n exponentialgrowthat the boundary. These conditions areused to a J prove Lieb-Thirring-type inequalities for the eigenvalues of com- 3 plex Jacobi matrices. ] h p Introduction - h t In the first part of the paper, we obtain some information on the dis- a tribution of the zeros of analytic functions from special growth classes. m We use this information to get interesting counterparts of famous Lieb- [ Thirring inequalities [8, 9] for complex Jacobi matrices. 2 ThetraditionalapproachtoLieb-ThirringboundsforcomplexJacobi v 7 matrices consists in deducing them from the bounds for corresponding 0 self-adjoint objects, see Frank-Laptev-Lieb-Seiringer [3] and Golinskii- 4 0 Kupin [6]. However, this method is quite limited. At best, it allows 2. us to get an information on a part of the point spectrum σp(J) of a 1 Jacobi matrix J, situated in a very special diamond-shaped region, see 7 [6, Theorem 1.5]. The information on the whole σ (J) is missing. 0 p : The main idea of our paper is to use functional-theoretic tools in the v i problem described above. Let J = J( ak , bk , ck ) be a complex X { } { } { } Jacobi matrix (2.1) such that J J lies in the Schatten-von Neumann 0 r − a class , p 1, and J = J( 1 , 0 , 1 ) (see Section 2 for termi- p 0 S ≥ { } { } { } nology). For an integer p, the regularized perturbation determinant u (λ) = det (J λ)(J λ)−1 is well-defined, and is an analytic func- p p 0 − − tion on Cˆ σ(J ) = Cˆ [ 2,2]. Its zero set coincides with σ (J) up to 0 p \ \ − multiplicities. We can obtain some information on the distribution of the zeros of the function u from its growth estimates in a neighbor- p hood of the boundary of Cˆ [ 2,2]. As usual, the domain Cˆ [ 2,2] is mapped conformally to the\u−nit disk D, so we are mainly int\er−ested in properties of corresponding analytic functions on D. Let (D) be the set of analytic functions on the unit disk D, f (D), Af = 0, and Z = z denote the set of zeros of f. ∈ f j A 6 { } Date: December 1, 2007. 1991 Mathematics Subject Classification. Primary: 30C15;Secondary: 47B36. Key words and phrases. Blaschke-type estimates, Lieb-Thirring inequalities for Jacobi matrices. 1 2 A. BORICHEV,L. GOLINSKII,S. KUPIN Theorem 0.1. Given a finite set E = ζ , E T, let f j j=1,...,N (D), f(0) = 1, and { } ⊂ ∈ A | | D (0.1) f(z) exp , | | ≤ dist(z,E)q (cid:16) (cid:17) with q 0. Then for any ε > 0, ≥ (0.2) (1 z )dist(z,E)(q−1+ε)+ C(ε,q,E)D. −| | ≤ zX∈Zf Here, x = max x,0 . + { } Clearly, (0.2) is a Blaschke-type condition. The classical Blascke condition is valid for functions from Hardy spaces Hp(D),0 < p , ≤ ∞ or, more generally, from the Nevanlinna class , see Garnett [4, Ch. N 2], and follows from the Poisson–Jensen formula: (0.3) (1 z ) sup log f(rζ) dm log f(0) , −| | ≤ | | − | | r T zX∈Zf Z where dm is the normalized Lebesgue measure on T. With small modi- fications one can write its analogs for the Bergman and the Korenblum spaces, see Hedenmalm-Korenblum-Zhu [7]. The specifics of our particular problem lead us to consider weights with a finite number of “exponential singularities” at the boundary in addition to the radial growth of the weight. Note that for q < 1 the function f (0.1) is in , and (0.2) follows N directly from (0.3). So in the proof of (0.2) we can assume q 1. ≥ Theorem 0.2. Let f (D), f(0) = 1, and ∈ A | | D 1 (0.4) f(z) exp , | | ≤ (1 z )pdist(z,E)q (cid:16) −| | (cid:17) where p,q 0. Then for any ε > 0, ≥ (0.5) (1 z )p+1+εdist(z,E)(q−1+ε)+ C(ε,p,q,E)D . 1 −| | ≤ zX∈Zf One can vary the degrees of the singularities ζ in (0.1) and (0.4). j More precisely, Theorem 0.3. Let −1 −1 N N h (z) = z ζ qj , h (z) = (1 z )p z ζ qj , 1 j 2 j | − | −| | | − | ! ! j=1 j=1 Y Y with q , q 0. Furthermore, let f (D), f (0) = 1, j j=1,...,N j j j { } ≥ ∈ A | | j = 1,2, satisfy f (z) exp(K h (z)). j j j | | ≤ A BLASCHKE-TYPE CONDITION AND ITS APPLICATION 3 Then N (0.6) (1 z ) z w (qj−1+ε)+ C(ε, q ,E)K , j j 1 −| | | − | ≤ { } zX∈Zf1 Yj=1 N (0.7) (1 z )p+1+ε z w (qj−1+ε)+ C(ε,p, q ,E)K . j j 2 −| | | − | ≤ { } zX∈Zf2 Yj=1 For the sake of simplicity, we prove relations (0.2), (0.5). The proofs of (0.6), (0.7) are similar. The starting idea of the work is close in spirit to an interesting paper by Demuth-Katriel [1]. To obtain counterparts of Lieb-Thirring bounds, the authors look at the difference of two semigroups generated by two continuous Schrödinger operators and they apply the classical Poisson-Jensen formula. They work with nuclear and Hilbert-Schmidt perturbations and the potential has to be from the Kato class. Our methods seem to be more straightforward. The computations are sim- pler and they are valid for -perturbations, p 1. In particular, we p S ≥ do not require the self-adjointness of the perturbed operator. As usual, D = z : z < r , D = D ,T = z : z = r ,T = T , r 1 r 1 { | | } { | | } and B(z ,δ) = z : z z < δ . C is a constant changing from one 0 0 { | − | } relation to another one. Weproceed as follows. In Section 2 we prove Theorem 0.1 andderive Theorem 0.2 from it. In Section 3 we discuss applications to complex Jacobi matrices. 1. Proofs of Theorems 0.1 and 0.2 Given a circular arc Γ = [eit1,eit2], denote by ω(z;t ,t ) its harmonic 1 2 measurewithrespecttotheunitdiskD. Anexplicitformulaisavailable (see Garnett [4, Ch. 1, Exercise 3]) 1 t t 1 2 ω(z;t ,t ) = α(z) − , 1 2 π − 2 (cid:18) (cid:19) where Γ is seen from a point z under the angle α. We use the notation ω for the symmetric arc γ t(γ) γ Γ = ζ T : 1 ζ γ = [e−it(γ),eit(γ)], sin = . { ∈ | − | ≤ } 2 2 Here γ = γ(E) < 1/500N is a small parameter which will depend on E. By the Mean Value Theorem γ t(γ) γ (1.1) ω (0) = . γ π ≤ π ≤ 2 Next, 1 t(γ) 1 (1.2) ω (z) γ ≥ 2 − π ≥ 4 4 A. BORICHEV,L. GOLINSKII,S. KUPIN for 1 z γ. An outer function | − | ≤ ^ e, 1 ζ γ, (1.3) g (z) = eωγ(z)+iωγ(z), g (ζ) = | − | ≤ γ | γ | 1, 1 ζ > γ, (cid:26) | − | where ζ T, will play a key role in what follows. Clearly, ω = log g , γ γ ∈ | | and (1.4) 1 g (z) e, γ ≤ | | ≤ for z D. ∈ We need a bound for the Blaschke product z λ b (z) := − λ ¯ 1 λz − in the case when the parameter λ and variable z are “well-separated”. Lemma 1.1. Let M = 200N. Then for 1 z = γ, 1 λ Mγ one | − | | − | ≥ has 1 1 1 (1.5) log log . b (z) ≤ 4Nγ λ λ | | | | Proof. Obviously, we asume λ = 0. Consider the domain 6 1 λ U := D B(λ,r), r = −| | , \ 4 and two harmonic functions on U 1 (1+ε)2 z 2 V (z) := log , W (z) := −| | , λ b (z) λ (1+ε)w z 2 λ | | | − | where parameters ε = ε(λ) > 0 and w = w(λ) T are chosen later on. Clearly, V (ζ) = 0 < W (ζ) for ζ T, and w∈e want to bound these λ λ ∈ functions on ∂B(λ,r). This is easy for V : λ 1 4 ¯ = 1 λz , b (z) 1 λ | − | λ | | −| | and for z = λ+reit we have 1 λ 1 λ¯z = 1 λ 2 −| | λ¯eit − −| | − 4 ¯ so 1 λz < 3(1 λ ) and | − | −| | V (z) < log30 < 5, λ where z ∂B(λ,r). ∈ The problem is more delicate for the lower bound on W . Let λ = λ λ eiθ and w = eiϕ. Then, for z ∂B(λ,r) | | ∈ (1.6) 1 (1+ε)2 z 2 > 1 z 2 = 1 λ 2 r2 2r λ cos(θ t) > (1 λ ). −| | −| | −| | − − | | − 3 −| | A BLASCHKE-TYPE CONDITION AND ITS APPLICATION 5 Next, we want to have a bound O((1 λ )2) for the denominator of − | | W λ (1.7) (1+ε)w z 2 = w z 2 +ε2 +2εRe(1 w¯z). | − | | − | − For the first term, w z 2 = 1 2Rew¯z + λ 2 +r2 +2r λ cos(θ t) = S +S , 1 2 | − | − | | | | − S = (1 λ )2+r2 2r(1 λ )cos(θ t), 1 −| | − −| | − S = 2 λ +2rcos(θ t) 2Rew¯z. 2 | | − − For S one already has S < 2(1 λ )2. To get 1 1 −| | S = 2 λ (1 cos(θ ϕ))+2r(cos(θ t) cos(t ϕ)) = O (1 λ )2 , 2 | | − − − − − −| | we choose w in the following way. If θ t(Mγ/2), we put(cid:0)ϕ = θ, so(cid:1) | | ≥ w = λ/ λ and S = 0. If θ < t(Mγ/2), we put 2 | | | | t(Mγ/2), 0 θ < t(Mγ/2), ϕ = ≤ t(Mγ/2), t(Mγ/2) < θ < 0. (cid:26) − − Then, by (1.1), ϕ θ < t(Mγ/2) πMγ/4. On the other hand, by | − | ≤ the hypothesis of lemma Mγ 1 λ = 1 λ + λ (1 eiθ) ≤ | − | −| | | | − πMγ (cid:12) (cid:12) 1 λ + θ 1 λ + , (cid:12) (cid:12) ≤ −| | | | ≤ −| | 4 so 1 λ Mγ(1 π/4) and ϕ θ π (1 λ ). Hence −| | ≥ − | − | ≤ 4−π −| | θ ϕ θ ϕ θ +ϕ 2t S 4sin2 − +4r sin − sin − 2 ≤ 2 2 2 (cid:12) (cid:12) 2 (cid:12) (cid:12) π (cid:12) π (cid:12) (1 λ(cid:12))2 + (1 λ )2(cid:12)< 26(1 λ )2 ≤ 4 π −| | 5(4 π) −| | −| | (cid:18) − (cid:19) − and w z 2 < 28(1 λ )2. | − | −| | Note also that in both cases above we have ϕ t(Mγ/2), that is | | ≥ Mγ (1.8) 1 w . | − | ≥ 2 To fix the second term in (1.7) we take 0 < ε < (1 λ )2, so −| | ε2 +2εRe(1 w¯z) 5ε < 5(1 λ )2, − ≤ −| | and eventually (1+ε)w z 2 < 31(1 λ )2. | − | −| | It follows now from the above inequality and (1.6) that W admits λ the lower bound 1/3(1 λ ) 1 1 W (z) > −| | > , λ 31(1 λ )2 1001 λ −| | −| | 6 A. BORICHEV,L. GOLINSKII,S. KUPIN where z ∂B(λ,r). So, ∈ 1 1 log W (z) > λ λ 100 | | for these z. By the Maximum Principle V < 500W on U, and by λ λ letting ε 0 we obtain → 1 1 1 z 2 log < 500log −| | , b (z) λ w z 2 λ | | | | | − | where z U. ∈ Note that by the assumption of the lemma 1 λ = l Mγ, so | − | ≥ 1 λ 1 λ = l, and if z ∂B(λ,r), then −| | ≤ | − | ∈ 1 λ 3 1 λ reit l −| | Mγ > 2γ, | − − | ≥ − 4 ≥ 4 which means that the arc 1 z = γ, z < 1 lies inside U. For such {| − | | | } z by (1.8) w z 1 w 1 z M/2 1 γ, so | − | ≥ | − |−| − | ≥ − 1 z 2 2(cid:0) (cid:1) −| | < . w z 2 M/2 1 2γ | − | − 2 The proof is complete. (cid:0) (cid:1) An invariant form of (1.5) is 1 1 1 log log , b (z) ≤ 4Nγ λ λ | | | | for any ζ T with the properties ζ z = γ, ζ λ Mγ. ∈ | − | | − | ≥ We decompose the unit disk into a union of disjoint sets D = Ω Ω , Ω = z D : 2−n−1 < z ζ 2−n , n,k n,k k { ∈ | − | ≤ } ! n,k [ [ where k = 1,2,...,N, n = L,L + 1,.... Here L = L(E) is a large parameter, so that (1.9) dist(Ω ,ζ ) 2−L, L,k s ≥ where s = k, k = 1,...,N. The latter obviously yields the same 6 inequality for dist(Ω ,ζ ) for all n L. n,k s ≥ Let us fix a pair (n,k), and define numbers 2−L−1/M, s = k, γ = 6 s 2−n−1/M, s = k, (cid:26) where s = 1,...,N, and M is from Lemma 1.1. Now (1.9) reads (1.10) dist(Ω ,ζ ) Mγ . n,k s s ≥ Define three sets of arcs Γ := ∂B(ζ ,γ ) D, Γ˜ := ζ T : ζ ζ γ , s s s s s s { ∈ | − | ≤ } \ A BLASCHKE-TYPE CONDITION AND ITS APPLICATION 7 the arcs Γ˜ and Γ˜ are separated by Γˆ T, s = 1,...,N. Set s s+1 s ∆ D in a way that ⊂ n,k ⊂ N N ∂∆ = Γ Γˆ . n,k s s ! ! s=1 s=1 [ [ [ Let z m be a finite number of zeros of f (counting multiplicity) { j}j=1 in Ω , and n,k z z j b := b (z) = − , j zj 1 z¯ z j − for j = 1,...,m. Consider the functions 4D g (z) = gρs(zζ¯), ρ = , s γs s s γq s 1 1 g (z) = gρj,s(zζ¯), ρ = log , j,s γs s j,s Nγ z s j | | with s = 1,...,N, j = 1,...,m. Recall that g are defined in (1.3). s For z Γ we have s ∈ ¯ log b (z)g (z) = log b (z) +ρ log g (zζ ) . | j j,s | | j | j,s | γs s | By (1.10) we can apply Lemma 1.1, which along with (1.2) gives 1 1 1 log b (z)g (z) log z + log = 0, j j,s j | | ≥ 4Nγ | | 4Nγ z s s j | | so (1.11) b (z)g (z) 1, j j,s | | ≥ for z Γ , j = 1,...,m. s Not∈e also, that, for z D (see (1.4)) ∈ b (z)g (z) e. j j,s | | ≤ Next, by (1.2) D D ¯ (1.12) log g (z) = ρ log g (zζ ) = , | s | s | γs s | ≥ γq z ζ q s s | − | where z Γ . s ∈ Lemma 1.2. A function N f(z) (1.13) F(z) := , gˆ (z) := g (z) m b (z)gˆ (z) N g (z) j j,l j=1 j j l=1 l l=1 Y is analytic on D, and satisfies Q Q (1.14) log F(z) 0, | | ≤ for z N Γ , ∈ l=1 l D (1.15) S log F(ζ) , | | ≤ dist(ζ,E)q 8 A. BORICHEV,L. GOLINSKII,S. KUPIN for ζ N Γˆ . Furthermore, ∈ l=1 l m N S 1 1 (1.16) log F(0) log 2D γ1−q. | | ≥ 2 z − l j j=1 | | l=1 X X Proof. For z Γ , s = 1,...,N, we have s ∈ D D log f(z) = , | | ≤ dist(z,E)q z ζ q s | − | and b = gˆ = g = 1 for the rest of the boundary of ∆ . So j j s n,k | | | | | | bounds (1.14) and (1.15) follow immediately from definition (1.13). To prove (1.16), we write m N log F(0) = log b (0) +log gˆ (0) log g (0) . j j l | | − | | | | − | | Xj=1(cid:16) (cid:17) Xl=1 Apply the bound for the harmonic measure (see (1.1)) 4D γ log g (0) l = 2Dγ1−q, | l | ≤ γq 2 l l N N 1 1 1 γ 1 1 l log gˆ (0) = ρ log g (0) log = log , | j | j,l | γl | ≤ N γ z 2 2 z l j j l=1 l=1 | | | | X X so 1 1 1 1 log b (0) +log gˆ (0) log z + log = log . j j j | | | | ≤ | | 2 z −2 z j j | | | | 2 The proof of the lemma is complete. Proof of Theorem 0.1. DefineanouterfunctionF∗ inDbyitsboundary values 1, ζ N Γ˜ , F∗(ζ) = D ∈ l=1 l | |  exp dist(ζ,E)q , ζ ∈ SNl=1Γˆl. As F∗ 1 on T, then F∗ (cid:16) 1 in the w(cid:17)hole diSsk, so by Lemma 1.2 | | ≥ | | ≥ F F∗ on ∂∆ , and hence by the Maximum Modulus Principle n,k | | ≤ | | F F∗ in ∆ . In particular, n,k | | ≤ | | log F(0) log F∗(0) . | | ≤ | | The upper bound for the RHS follows from N log F∗(0) = log F∗(ζ) dm = D dist(ζ,E)−qdm. | | | | ZT l=1 ZΓˆl X It is easy to estimate a typical integral in the above sum (note that by the definition γ γ for all l = 1,...,N) k l ≤ 2πq−1 1 q−1, q > 1, dist(ζ,E)−qdm q−1 γk ZΓˆl ≤ ( log γ1k(cid:16), (cid:17) q = 1. A BLASCHKE-TYPE CONDITION AND ITS APPLICATION 9 Hence for q > 1 (for q = 1 the argument is the same) q−1 1 log F∗(0) C(q)ND , | | ≤ γ (cid:18) k(cid:19) so by (1.16) m m q−1 1 1 (1.17) (1 z ) log < C(q,N)D . j −| | ≤ z γ j=1 j=1 | j| (cid:18) k(cid:19) X X By the definition of γ and Ω we have k n,k 2Mγ = 2−n z ζ = dist(z ,E), k j k j ≥ | − | so (1.17) implies m m (1 z )dist(z ,E)q−1+ε (1 z )(2Mγ )q−1+ε j j j k −| | ≤ −| | j=1 j=1 X X CD(2Mγ )ε = CD2−nε. k ≤ Summation over n L, and then over k = 1,...,N gives ≥ (1 z )dist(z ,E)q−1+ε CD, j j −| | ≤ zXj∈Ω where Ω := Ω . n,k n,k The same reasoning with γ = 1/2LM, s = 1,...,N applies to the S s domain Ω := z D : z ζ > 2−L, k = 1,...,N . The proof is 0 k { ∈ | − | } 2 complete. Proof of Theorem 0.2. Denote τ := 1 2−n, n = 0,1... and put n − f (z) := f(τ z). Then n n D 1 f (z) exp . | n | ≤ (1 τ z )pdist(τ z,E)q n n (cid:16) −| | (cid:17) An elementary inequality 1 z z ζ 1+ z (1.18) −| | | − | | | < 2, 1 τ z ≤ τz ζ ≤ 1+τ z − | | | − | | | which holds for z D, ζ T and 0 τ < 1, gives ∈ ∈ ≤ 1 dist(τ z,E) > dist(z,E) n 2 so D 2 f (z) exp , | n | ≤ dist(z,E)q (cid:16) (cid:17) where D = 2np+qD . If Z = z , Z = z then 2 1 f { j} fn { j,n} z j z = : z < τ . j,n j n τ | | n 10 A. BORICHEV,L. GOLINSKII,S. KUPIN By Theorem 0.1 z z r (1.19) 1 | j| dist j,E C(ε,q,E)2np+qD , 1 − τ τ ≤ n n zj:|Xzj|<τn(cid:16) (cid:17) (cid:16) (cid:17) r = (q 1+ε) . We readily continue as + − z z r j j LHS of (1.19) 1 | | dist ,E ≥ − τ τ n n τn−2≤X|zj|<τn−1(cid:16) (cid:17) (cid:16) (cid:17) 1 (1 z )dist(z ,E)r, j j ≥ 4r+1 −| | τn−2≤X|zj|<τn−1 and, consequently, 2−n(p+ε) (1 z )dist(z ,E)r C2−nεD . j j 1 −| | ≤ τn−2≤X|zj|<τn−1 Since 1 z 1 τ then j n−2 −| | ≤ − 1 2−n(p+ε) (1 z )p+ε, ≥ 4p+ε −| j| and finally (1 z )p+ε+1dist(z ,E)r C2−nεD . j j 1 −| | ≤ τn−2≤X|zj|<τn−1 It remains only to sum up over n from 2 to . 2 ∞ 2. Applications to complex Jacobi matrices We are interested in complex-valued Jacobi matrices of the form b c 0 ... 1 1 a b c ... 1 2 2 (2.1) J = J( ak , bk , ck ) = 0 a b ... { } { } { } 2 3  ... ... ... ...     where a ,b ,c C. We assume J to be a compact perturbation of the k k k ∈ freeJacobimatrixJ = J( 1 , 0 , 1 ),or,equivalently, lim a = 0 k→+∞ k { } { } { } lim c = 1, lim b = 0. It is well-known that inthis situation k→+∞ k k→+∞ k σ (J) = [ 2,2]. The point spectrum of J is denoted by σ (J); the ess p − eigenvalues λ σ (J) have finite algebraic (and geometric) multiplic- p ∈ ity, and the set of their limit points lies on the interval [-2,2] (see, e.g., [5, Lemma I.5.2]). The structure of σ (J) and, especially, its behavior near [ 2,2], is p − an important part of the spectral analysis of complex Jacobi matrices. We quote a theorem from Golinskii-Kupin [6] to give a flavor of the known results.