Table Of ContentInequalities 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 eigen-
value 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 selfad-
joint 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 in-
teresting 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 eigen-
values 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: in-
equality (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 H¨older’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
