Lieb-Thirring Inequalities for Complex Finite Gap Jacobi Matrices

We establish Lieb-Thirring power bounds on discrete eigenvalues of Jacobi operators for Schatten class complex perturbations of periodic and more generally finite gap Jacobi matrices.


Introduction
In this paper we consider bounded non-selfadjoint Jacobi operators on ℓ 2 (Z) represented by tridiagonal matrices with bounded complex parameters {a n , b n , c n } n∈Z . Our goal is to obtain Lieb-Thirring inequalities for complex perturbations of periodic and, more generally, almost periodic Jacobi operators with absolutely continuous finite gap spectrum. We are motivated by the recent progress on Lieb-Thirring inequalities for Jacobi matrices [1,9,10,11,13], and in particular by the finite gap results [14,5,6,8,4]. See also [12]. Before explaining our new results, let us briefly go through what is already known. The spectral theory for perturbations of the free Jacobi matrix, J 0 , (i.e., the case of a n = c n ≡ 1 and b n ≡ 0) is well-understood, see [18]. Let E = σ(J 0 ) = [−2, 2] and suppose J is a selfadjoint Jacobi matrix (i.e., a n = c n > 0) such that δJ = J −J 0 is a compact operator, that is, J is a compact selfadjoint perturbation of J 0 . Hundertmark and Simon [13] proved the following Lieb-Thirring inequalities, with some explicit constants L p, E independent of J. Here, σ d (J) is the discrete spectrum of J. It was also shown in [13] that the inequality is false for p < 1.
More recently, (1.2) was extended to selfadjoint perturbations of periodic and almost periodic Jacobi matrices with absolutely continuous finite gap spectrum [14,5,6,4]. When E is a finite gap set (i.e., a finite union of disjoint, compact intervals), the role of J 0 as a natural background operator is taken over by the so-called isospectral torus, denoted T E . See, e.g., [2,3,19] for a deeper discussion of this object. For J ′ ∈ T E and a compact selfadjoint perturbation J = J ′ + δJ, Frank and Simon [6] proved (1.2) for p = 1 while the case of p > 1 is established in [4]. The constant L p, E is now independent of J and J ′ and only depends on p and the underlying set E.
It is also worth mentioning a general result of Kato [15]. Specialized to the case of selfadjoint perturbations of Jacobi matrices from T E , it states that where · p denotes the Schatten norm. In contrast to the Lieb-Thirring bounds, the power on the eigenvalues in (1.3) is the same as on the perturbation. Kato's inequality is optimal for perturbations with large sup norm. On the other hand, the Lieb-Thirring bound with p = 1 is optimal for perturbations with small sup norm (cf. [13]). A fact that seemingly went unnoticed is that one can combine (1.2) and (1.3) into one ultimate inequality which is optimal for both large and small perturbations (at least when p = 1). This inequality takes the form where the constant C p, E is independent of J and J ′ . In recent years, several results have also been established for non-selfadjoint perturbations of selfadjoint Jacobi matrices [1,9,10,8,11]. For compact non-selfadjoint perturbations J = J 0 + δJ of the free Jacobi matrix J 0 , a near generalization (with an extra ε) of the Lieb-Thirring bound (1.2) was obtained by Hansmann and Katriel [10] using the complex analytic approach developed in [1]. Their non-selfadjoint version of the Lieb-Thirring inequalities takes the following form: For every 0 < ε < 1, where the eigenvalues are repeated according to their algebraic multiplicity and the constant L p, ε is independent of J. Whether or not this inequality continues to hold for ε = 0 is an open problem.
For perturbations of Jacobi matrices from the isospectral tori associated with finite gap sets E, an eigenvalue power bound was obtained in [8]. Shortly thereafter, this was superseded by a generalization of Kato's inequality due to Hansmann [11]. In the special case of a compact non-selfadjoint perturbation where the eigenvalues are repeated according to their algebraic multiplicity and K p is a universal constant that depends only on p.
The purpose of the present article is to obtain a near generalization of the Lieb-Thirring bound (1.4) for compact non-selfadjoint perturbations J = J ′ + δJ of Jacobi matrices J ′ from the isospectral tori T E associated with finite gap sets E. If ∂E denotes the set of endpoints of the intervals that form E, our main result can be formulated as follows: For every p ≥ 1 and any ε > 0, where the eigenvalues are repeated according to their algebraic multiplicity and the constant L ε, p, E is independent of J ′ and J. We point out that (1.7) is new even for perturbations of the free Jacobi matrix J 0 since, unlike (1.5), it is nearly optimal not only for small but also for large perturbations. As with (1.5), it is an open problem whether or not (1.7) remains true for ε = 0.

Schatten Norm Estimates
In this section we establish the fundamental estimates that are needed to prove our main result, Theorem 3.3. Throughout, S p will denote the Schatten class and · p the corresponding Schatten norm for p ≥ 1. To clarify our application of complex interpolation, we occasionally use · ∞ to denote the operator norm.
Theorem 2.1. Suppose J ′ is a selfadjoint Jacobi matrix and D ≥ 0 is a diagonal matrix of Schatten class S p for some p ≥ 1. Denote by dρ n the spectral measure of (J ′ , δ n ), that is, the measure in the Herglotz representation of the nth diagonal entry of (J ′ − z) −1 , Proof. We consider first the case p = 1. Let {P (t)} t∈R denote the projectionvalued spectral family of the selfadjoint operator J ′ . Recall that for any measurable and bounded function f on σ(J ′ ), the bounded operator f (J ′ ) is given by the functional calculus, 3), substituting into (2.1), and recalling that the measure in the Herglotz representation is unique yield Applying (2.3) to f (t) = 1/|t − z| and using (2.4) then imply We also note that if, in addition, the function f (t) in (2.3) is nonnegative, then f (J ′ ) is a bounded, selfadjoint, and nonnegative operator.
Fix z ∈ C σ(J ′ ). In the following we assume without loss of generality that Im(z) ≥ 0. Define the nonnegative functions and note that (2.10) Using (2.9), the triangle inequality, and the fact that for nonnegative operators the trace norm coincides with the trace, we obtain the estimate Let D n,n denote the diagonal entries of D. Since D is nonnegative and diagonal, we have n∈Z D n,n = tr(D) = D 1 . (2.12) Hence, by linearity of the trace it follows from (2.11), (2.10), and (2.5) that This is exactly the case p = 1 in (2.2).
To obtain (2.2) for p > 1, we use complex interpolation. Define the map ζ → T (ζ) = D ζp/2 |J ′ − z| −1 D ζp/2 (2.14) from the strip 0 ≤ Re(ζ) ≤ 1 into the space of bounded operators. Then for any u, v ∈ ℓ 2 (Z), the scalar function is continuous on the strip 0 ≤ Re(ζ) ≤ 1, analytic in its interior, and bounded. In addition, since D iy ≤ 1 and it follows that Taking x = 1/p, raising both sides to the power p, and noting that T (1/p) = D 1/2 (J ′ − z) −1 D 1/2 and D p 1 = D p p finally yields (2.2). In what follows, E ⊂ R will denote a finite gap set, that is, and ∂E will be the set of endpoints of E, that is, For a probability measure dρ supported on E, we define the associated mfunction by Reflectionless measures appear prominently in spectral theory of finite and infinite gap Jacobi matrices (see, e.g., [2,16,18,19]). In particular, the isospectral torus associated to E is the set of all Jacobi matrices J ′ that are reflectionless on E (i.e., the spectral measure of (J ′ , δ n ) is reflectionless for every n ∈ Z) and for which σ(J ′ ) = E. It is well known (see for example [19]) that dρ is a reflectionless probability measure on E if and only if m(z) is of the form We now provide an estimate for the variant of the m-function for dρ n that appear in Theorem 2.1.
Theorem 2.2. Let E ⊂ R be a finite gap set and suppose dρ is a reflectionless probability measure on E. Then for every p > 1, where the constant K p, E is independent of dρ. In addition, for every ε > 0, where the constant K ε, E is independent of dρ.
Proof. Denote the bands of E as in (2.20). Since dρ is reflectionless on a finite gap set, it follows from the Stieltjes inversion formula and (2.24) that dρ is absolutely continuous with density given by for some γ j ∈ [β j , α j+1 ], j = 1, . . . , N − 1. Fix 1 ≤ k ≤ N and rearrange the terms in (2.27) as follows (2.28) Since α j < β j ≤ γ j ≤ α j+1 < β j+1 for j = 1, . . . , N − 1, the two products in (2.28) are at most 1 for every t ∈ [α k , β k ] and thus Applying this estimate for the individual bands of E implies that By [10, Lemma 11], each integral in the sum can be estimated bŷ (2.31) Since the function z → dist(z, ∂E)(1 + |z|)/|z − α k ||z − β k | is continuous on C {α k , β k } and bounded near α k , β k , and ∞, it is bounded on C E, and thereforê In order to obtain (2.26), note that since E is a bounded set we have the trivial bound This inequality yields the estimatê and (2.26) hence follows from (2.25).  is an analytic function on D such that |h(0)| = 1 and for some K, α, β, γ ≥ 0,

2)
where each zero is repeated according to its multiplicity.
In [8], an analogous result on the distribution of zeros of analytic functions on Ω = C E was obtained via a reduction to the unit disk case. For our purposes we will need the following extension of [8,Theorem 0.1] where an additional decay assumption at infinity is imposed in exchange for a stronger conclusion. The extension follows from the reduction to the unit disk case developed in [8] combined with the above version (Theorem 3.1) of the unit disk result. We omit the proof as it is a straightforward modification of the one presented in [8].
Theorem 3.2. Let E ⊂ R be a finite gap set and suppose f (z) is an analytic function on Ω = C E such that |f (∞)| = 1 and for some K, p, q, r ≥ 0, Then for every ε > 0, there exists a constant C p,q,r,ε independent of f (z) such that the zeros of f (z) satisfy
We are now ready to present our finite gap version of the Lieb-Thirring inequalities for non-selfadjoint perturbations of Jacobi matrices from the isospectral torus T E . Theorem 3.3. Let E ⊂ R be a finite gap set and suppose J, J ′ are two-sided Jacobi matrices such that J ′ ∈ T E and J = J ′ + δJ is a compact perturbation of J ′ . Then for every p ≥ 1 and any ε > 0, where the eigenvalues are repeated according to their algebraic multiplicity and the constant L ε, p, E is independent of J and J ′ .