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Inequalities for the eigenvalues of 9 non-selfadjoint Jacobi operators 0 0 2 n Marcel Hansmann, Guy Katriel a ∗ ∗† J 3 1 ] P Abstract S . h WeproveLieb-Thirring-typeboundsoneigenvaluesofnon-selfadjoint t a Jacobioperators,whicharenearlyasstrongasthoseprovenpreviously m for the case of selfadjoint operators by Hundertmark and Simon. We [ use a method based on determinants of operators and on complex 1 function theory, extending and sharpening earlier work of Borichev, v Golinskii and Kupin. 5 2 7 1 1 Introduction and Results . 1 0 9 This paper is concerned with the study of the set of discrete eigenvalues 0 : of complex Jacobi operators J : l2(Z) l2(Z), represented by a two-sided v → infinite tridiagonal matrix i X r .. .. .. a . . . a b c  −1 0 0  J = a b c , 0 1 1  a b c   1 2 2   ... ... ...      ∗Institute of Mathematics, Technical University of Clausthal, 38678 Clausthal- Zellerfeld, Germany. †Partially supported by the Humboldt Foundation (Germany). 1 where ak k∈Z, bk k∈Z and ck k∈Z are bounded complex sequences. More { } { } { } precisely, for u l2(Z), J is defined via ∈ (Ju)(k) = a u(k 1)+b u(k)+c u(k +1). k−1 k k − We are interested in operators J that are compact perturbations of the free Jacobi operator J , which is defined as the special case with a = c 1 and 0 k k ≡ b 0, i.e. k ≡ (J u)(k) = u(k 1)+u(k +1). 0 − So, in the following, we will always assume that J J is compact, or equiv- 0 − alently that lim a = lim c = 1, and lim b = 0. k k k |k|→∞ |k|→∞ |k|→∞ As is well-known, the spectrum σ(J ) of J is equal to [ 2,2] and the com- 0 0 − pactness of J J implies that σ(J) = [ 2,2] ˙ σ (J), where [ 2,2] is the 0 d − − ∪ − essential spectrum of J and the discrete spectrum σ (J) C [ 2,2]consists d ⊂ \ − of a countable set of isolated eigenvalues of finite algebraic multiplicity, with possible accumulation points in [ 2,2]. − We define the sequence d = dk k∈Z as follows { } d = max a 1 , a 1 , b , c 1 , c 1 . (1) k k−1 k k k−1 k | − | | − | | | | − | | − | (cid:16) (cid:17) Note that the compactness of J J is equivalent to d converging to 0. 0 k − { } The main results of this paper provide information on σ (J), assuming the d stronger condition that d lp(Z), the space of p-summable sequences (we ∈ will see below that d lp(Z) implies that J J is an element of the Schatten 0 ∈ − class S , see Section 2 for relevant definitions). p Theorem 1. Let τ (0,1). If d lp(Z), where p > 1, then ∈ ∈ dist(λ,[ 2,2])p+τ − C(p,τ) d p . (2) λ2 4 1 ≤ k klp 2 λ∈Xσd(J) | − | Furthermore, if d l1(Z), then ∈ dist(λ,[ 2,2])1+τ − C(τ) d . (3) λ2 4 1+τ ≤ k kl1 2 4 λ∈Xσd(J) | − | 2 Remark 1. In the summation above, and elsewhere in this article, each eigenvalue is counted according to its algebraic multiplicity. Furthermore, the constants used in this paper are generic, i.e. the value of a constant may change from line to line. However, we will always indicate the parameters that a constant depends on. Theaboveestimates canberegardedas‘near-generalisations’ ofthefollowing Lieb-Thirring inequalities proven by Hundertmark and Simon [7] for selfadjoint Jacobi operators (i.e. for the case that a ,b ,c R and a = c for all k k k k k ∈ k). Theorem ([7], Theorem 2). Let J be selfadjoint and suppose that d lp(Z), ∈ where p 1. Then ≥ |λ+2|p−21 + |λ−2|p−21 ≤ C(p)kdkplp. (4) λ∈σd(XJ),λ<−2 λ∈σdX(J),λ>2 To see the relation between the above result and Theorem 1, we note that in the selfadjoint case the eigenvalues of J are in R [ 2,2], and we have \ − dist(λ,[ 2,2]) = λ 2 for λ > 2 and dist(λ,[ 2,2]) = λ+2 if λ < 2, so − | − | − | | − that (4) can be rewritten in the form dist(λ,[−2,2])p−21 ≤ C(p)kdkplp. (5) λ∈Xσd(J) One could try to generalise (5) to non-selfadjoint Jacobi operators, but we are not able to do this, and in fact we conjecture that (5) is not true in the non-selfadjoint case, due to a different behaviour of sequences of eigenvalues when converging to 2 or to ( 2,2), respectively. To get an analogue of (4) ± − which is valid in the non-selfadjoint case, we note that since dist(λ,[ 2,2])p 1 λ 2 p−21, λ > 2 − | − | λ2 4 21 ≤ 2 ( λ+2 p−21, λ < 2, | − | | | − inequality (4) implies that in the selfadjoint case dist(λ,[ 2,2])p − C(p) d p . (6) λ2 4 1 ≤ k klp 2 λ∈Xσd(J) | − | 3 Clearly, forp > 1, (6)isthesameas(2)whenτ = 0. Ontheotherhand, The- orem 1 requires τ > 0, so that (2) (when applied to selfadjoint operators) is slightlyweaker than(6), whichiswhy wesay thatitisa‘near-generalisation’. The same observation applies to inequality (3) in case that p = 1. An interesting open question is whether (2) and (3) are valid for τ = 0 in the non-selfadjoint case. We note that the methods used by Hundertmark and Simon in [7] depend in anessential way ontheselfadjointness oftheoperators, andsoarecompletely different from those used here. Our approach develops and sharpens ideas of Borichev, Golinskii and Kupin [1]. Using determinants of Schatten class operators, one defines a function g(λ) whose zeros coincide with the eigenvalues of J, and then these zeros are studied by applying complex function theory. Other applications of this approach can be found in [2, 3]. In [1], Theorem 2.3, the authors used this approach to show that for p > 1 and τ > 0, dist(λ,[ 2,2])p+1+τ − C(τ, J J ,p) J J p , (7) λ2 4 ≤ k − 0k k − 0kSp λ∈Xσd(J) | − | where . S denotesthep-thSchattennorm. WenotethattheSchattennorm k k p kJ −J0kSp is equivalent to kdklp, as is shown in Lemma 8 below. Inequality (7) was originally derived for Jacobi operators on l2(N) but it carries over to the whole-line case. The authors of [1]also derived a morerefined estimate in case p = 1, similar to (3), but here their proof seems to use special properties ofthe half-lineoperatorandis thus notdirectly transferableto the whole-line setting. We remark that inequality (7) is an easy consequence of Theorem 1 since dist(λ,[ 2,2])p+1+τ dist(λ,[ 2,2])p+τ − − , λ2 4 ≤ λ2 4 1 2 | − | | − | as a direct calculation shows. Theorem 1 improves upon (7) in another re- spectsincetheconstantsontheright-handsideof(2)and(3)areindependent ofJ. Togetridofthisdependence, wedevelop, inTheorem4below, avariant of the complex function result used in [1]. The proof of Theorem 1 further depends on a more subtle estimate on the function g(λ) (mentioned above) than the one used in [1], exploiting the structure of the Jacobi operators in an essential way. 4 In relation to Theorem 1, it is interesting to discuss another approach to gen- eralising inequality (4) to non-selfadjoint operators, developed by Golinskii and Kupin [6]. For θ [0, π) we define the following sectors in the complex ∈ 2 plane: Ω± = λ : 2 Re(λ) < tan(θ) Imλ . θ { ∓ | |} Theorem ([6], Theorem 1.5). Let θ [0, π). Then for p 3 ∈ 2 ≥ 2 |λ−2|p−21 + |λ+2|p−21 ≤ C(p,θ)kdkplp, (8) λ∈σdX(J)∩Ω+θ λ∈σdX(J)∩Ω−θ where C(p,θ) = C(p)(1+2tan(θ))p. Clearly, this theorem, when restricted to the selfadjoint case, gives (4). This is not a coincidence, but due to the fact that its proof is obtained by a reduction to the case of selfadjoint operators and employing (4). A drawback of this result is that the sum is not over all eigenvalues since it excludes a diamond-shaped region around the interval [ 2,2] (thus avoiding sequences − of eigenvalues converging to some point in ( 2,2)). However, we shall show − that by a suitable integration, the inequalities (8) can be used to derive an inequality where the sum is over all the eigenvalues: Theorem 2. Let τ (0,1). If d lp(Z), where p 3, then ∈ ∈ ≥ 2 dist(λ,[ 2,2])p+τ − C(p,τ) d p . (9) λ2 4 1+τ ≤ k klp 2 λ∈Xσd(J) | − | We emphasise that the proof of Theorem 2 does not involve any complex analysis and is thus completely different from the proof of Theorem 1. Let us note the similarities, and the differences, between these two results: inequality (9) is in fact somewhat stronger than (2), because of the τ in the denominator of (9). However, while Theorem 2 requires the condition p 3, ≥ 2 Theorem 1 is valid for p 1, just like the corresponding inequality (4) in the ≥ selfadjoint case. We can thus conclude that the approach based on complex analysis provides very satisfying results for non-selfadjoint Jacobi operators, which are almost as strong as those obtained in the selfadjoint case by specialised methods relying on the selfadjointness of the operators. 5 Let us give a short overview of the contents of this paper: in Section 2, we gather information about Schatten classes and infinite determinants. In Section 3 we present some complex analysis results that are used in the proof of Theorem 1, which is provided in Section 4. In the final Section 5 we are concerned with the proof of Theorem 2. 2 Preliminaries For a Hilbert space let C( ) and B( ) denote the classes of closed and H H H of bounded linear operators on , respectively. We denote the ideal of all H compact operators on by S and the ideal of all Schatten class operators ∞ H by S ,p > 0, i.e. a compact operator C S if p p ∈ ∞ C p = µ (C)p < , S n k k p ∞ n=1 X where µ (C) denotes the n-th singular value of C. n Schatten class operators obey the following Hölder’s inequality: let p,p ,p 1 2 be positive numbers with 1 = 1 + 1 . If C S (i = 1,2), then the operator p p1 p2 i ∈ pi C = C C S and 1 2 p ∈ kCkSp ≤ kC1kSp1kC2kSp2, (10) (see e.g. Simon [11], Theorem 2.8). Let C S , where n N. Then one candefine the (regularized) determinant n ∈ ∈ n−1 λj det (I C) = (1 λ)exp , n − − j " !# λ∈σ(C) j=1 Y X having the following properties (see e.g. Dunford and Schwartz [4], Gohberg and Kre˘ın [5], or Simon [11]): 1. I C is invertible if and only if det (I C) = 0. n − − 6 2. det (I) = 1. n 3. For A,B B( ) with AB,BA S : n ∈ H ∈ det (I AB) = det (I BA). (11) n n − − 6 4. If C(λ) S depends holomorphically on λ Ω, where Ω C is open, n ∈ ∈ ⊂ then det (I C(λ)) is holomorphic on Ω. n − 5. If C S for some p > 0, then C S , where p ⌈p⌉ ∈ ∈ p = min n N : n p , ⌈ ⌉ { ∈ ≥ } and the following inequality holds, det (I C) exp Γ C p , (12) ⌈p⌉ p S | − | ≤ k k p (cid:16) (cid:17) where Γ is some positive constant, see [4, page 1106]. We remark that p Γ = 1 for p 1, Γ = 1 and Γ e(2+logp) in general, see Simon [10]. p p ≤ 2 2 p ≤ ForA,B B( ) with B A S , the p -regularized perturbation determi- p ∈ H − ∈ ⌈ ⌉ nantofAbyB Aisawell defined holomorphicfunctiononρ(A) = C σ(A), − \ given by d(λ) = det (I (λ A)−1(B A)). ⌈p⌉ − − − Furthermore, λ ρ(A) is an eigenvalue of B of algebraic multiplicity k if 0 0 ∈ and only if λ is a zero of d( ) of the same multiplicity. 0 · 3 Complex analysis Let D = z C : z < 1 . The following result is due to Borichev, Golinskii { ∈ | | } and Kupin [1]. Theorem 3. Let h : D C be holomorphic with h(0) = 1, and suppose that → N 1 1 log h(z) K , | | ≤ (1 z )α z ξj βj −| | j=1 | − | Y where ξ = 1 (1 j N), and the exponents α,β are nonnegative. Let j j | | ≤ ≤ τ > 0. Then the zeros of h satisfy the inequality N (1 z )α+1+τ z ξ (βj−1+τ)+ C(α, β , ξ ,τ)K, j j j −| | | − | ≤ { } { } z∈D,h(z)=0 j=1 X Y where x = max(x,0), and each zero of h is counted according to its multi- + plicity. 7 For the holomorphic function h that we will consider below, there will be some additional information on the speed of convergence of log h(z) 0 | | → as z 0. The following modification of the above theorem takes this into → account. Theorem 4. Let h : D C be holomorphic with h(0) = 1, and suppose that → z γ N 1 log h(z) K | | , (13) | | ≤ (1 z )α z ξj βj −| | j=1 | − | Y where ξ = 1 (1 j N), and the exponents α,β and γ are nonnegative. j j | | ≤ ≤ Let 0 < τ < 1. Then the zeros of h satisfy the inequality (1 z )α+1+τ N −| | z ξ (βj−1+τ)+ C(α, β ,γ, ξ ,τ)K, (14) j j j z (γ−1+τ)+ | − | ≤ { } { } z∈D,h(z)=0 | | j=1 X Y where each zero is counted according to its multiplicity. We note that the results of the previous theorem differ from the results obtained in Theorem 3 both in the hypothesis, which requires log h(z) to | | vanish at 0 at a specified rate, and requires τ < 1, and in its conclusion, which includes the 1 term. |z| Proof of Theorem 4. In view of Theorem 3, we only need to consider the case γ > 1 τ. Set D = z C : z < r . Using the boundedness of 1 on − r { ∈ | | } |z| D D , we obtain from Theorem 3 that 1 \ 2 (1 z )α+1+τ N −| | z ξ (βj−1+τ)+ C(α, β ,γ, ξ ,τ)K. z γ−1+τ | − j| ≤ { j} { j} z∈D\DX1,h(z)=0 | | jY=1 2 Hence, the proof is completed by showing that 1 C(α, β ,γ, ξ ,τ)K. (15) z γ−1+τ ≤ { j} { j} z∈D1X,h(z)=0 | | 2 8 Let N (D ) denote the number of zeros of h in D (multiplicities taken into h r r account). Then we can rewrite the sum in (15) as follows: 1 1 |z| = (γ 1+τ) dt tγ−2+τ z γ−1+τ − z∈D1X,h(z)=0 | | z∈D1X,h(z)=0Z0 2 2 ∞ = (γ 1+τ) dt tγ−2+τ 1 −   Z0 |z|<min(12X,t−1),h(z)=0 2  ∞  = (γ 1+τ) dt tγ−2+τN (D )+ dt tγ−2+τN (D ) . (16) h 1 h t−1 − 2 (cid:20)Z0 Z2 (cid:21) To estimate the last two integrals the following lemma is used. Lemma 5. Assume (13). Then for r (0, 1] we have ∈ 2 N (D ) C(α, β , ξ )Krγ. h r j j ≤ { } { } Proof of Lemma 5. Let 0 < r < s < 1. From Jensen’s identity (see e.g. Rudin [9], Theorem 15.18) and assumption (13) we obtain 1 s 1 s N (D ) = log log h r log(s) r ≤ log(s) z r z∈DXr,h(z)=0 (cid:16) (cid:17) r z∈DXr,h(z)=0 (cid:18)| |(cid:19) 1 s 1 1 2π log = log h(seiθ) dθ ≤ log(s) z log(s)2π | | r z∈DXs,h(z)=0 (cid:18)| |(cid:19) r Z0 1 Ksγ 1 2π N 1 dθ. ≤ log(rs)(1−s)α2π Z0 j=1 |seiθ −ξj|βj Y Choosing s = 3r (i.e. s 3) concludes the proof of the lemma. 2 ≤ 4 Returning to (16), we can use Lemma 5 and the fact that γ > 1 τ to − conclude 2 dt tγ−2+τN (D ) C(α, β ,γ, ξ ,τ)K. h 1 j j 2 ≤ { } { } Z0 9 Similarly, using that τ < 1, Lemma 5 implies that ∞ ∞ dt tγ−2+τN (D ) C(α, β , ξ )K dt t−2+τ h t−1 j j ≤ { } { } Z2 Z2 C(α, β ,γ, ξ ,τ)K. j j ≤ { } { } This concludes the proof of Theorem 4. We now translate the result of Theorem 4 into a result about holomorphic functions on C [ 2,2], which is the one we will use below. \ − Corollary 6. Let g : C [ 2,2] C be holomorphic with lim g(λ) = 1. |λ|→∞ \ − → For α,β 0 suppose that ≥ K log g(λ) 0 . (17) | | ≤ dist(λ,[ 2,2])α λ2 4 β − | − | Let 0 < τ < 1 and set η = α+1+τ, 1 (18) η = (2β +α 1+τ) . 2 + − Then, dist(λ,[ 2,2])η1 − C(α,β,τ)K , (19) λ2 4 η1−η2 ≤ 0 λ∈C\[−X2,2],g(λ)=0 | − | 2 where each zero of g is counted according to its multiplicity Proof. We note that z z+z−1 maps D 0 conformally onto C [ 2,2]. 7→ \{ } \ − So we can define h(z) = g(z +z−1), z D 0 . ∈ \{ } Setting h(0) = 1, h is holomorphic on D and K log h(z) 0 , (20) | | ≤ dist(λ,[ 2,2])α λ2 4 β − | − | where λ = z + z−1. To express the right-hand side of the last equation in terms of z, the following Lemma is needed. Its proof is given below. 10