CYCLICITY IN RANK-ONE PERTURBATION PROBLEMS

EVGENY ABAKUMOV, CONSTANZE LIAW, AND ALEXEI POLTORATSKI

Abstract. The property of cyclicity of a linear operator, or equivalently the property of simplicity of its spectrum, is an important spectral characteristic that appears in many problems of functional analysis and applications to mathematical physics. In this paper we study cyclicity in the context of rank-one perturbation problems for self-adjoint and unitary operators. We show that for a fixed non-zero vector the property of being a cyclic vector is not rare, in the sense that for any family of rank-one perturbations of self-adjoint or unitary operators acting on the space, that vector will be cyclic for every operator from the family, with a possible exception of a small set with respect to the parameter. We discuss applications of our results to Anderson-type Hamiltonians.

1. Introduction

Consider a self-adjoint operator T on a separable Hilbert space H. A vector φ ∈ H is called cyclic for an operator T, if

H = clos span{(T − λI)^{−1}φ : λ ∈ ℂ\ℝ}.

An operator T is called cyclic, if there exists a cyclic vector. For a bounded operator T, an equivalent definition is that

H = clos span{T^{n}φ : n ∈ ℕ ∪ {0}},

i.e., the span of the orbit of φ under T is dense in the Hilbert space. Cyclicity of an operator is equivalent to the property that the operator has simple spectrum. The property of simplicity of the spectrum often appears in problems originating from physics.

In this note we study cyclicity in the context of rank-one perturbation problems for selfadjoint and unitary operators. If A is a self-adjoint operator and φ is its cyclic vector one can consider the family of rank-one perturbations

A_{α} = A + α( · , φ)φ, for α ∈ ℝ. (1.1)

Similar families can be defined for unitary operators, see Sections 2.2 and 2.3 for the definitions. We show that the property of cyclicity for a fixed non-zero vector in the Hilbert space is not a rare event, in the sense that for any family of cyclic rank-one perturbations a fixed vector is cyclic for all operators in the family with an exception of some small sets of parameters.

2010 Mathematics Subject Classification. 47A16, 47B80, 81Q15. Key words and phrases. Cyclic vector, rank-one perturbation, random Hamiltonian. The first author was partially supported by the ANR project DYNOP and the ESF program ”Harmonic

and Complex Analysis and its Applications”.

The second author is partially supported by the NSF grant DMS-1101477. The third author is partially supported by the NSF grant DMS-1101278.

While similar statements, showing that any fixed non-zero vector is cyclic for almost every rank-one perturbation of any self-adjoint operator, exist in the literature (see [8, 9]), we attempt to make these statements more precise. In the rank-one setting we improve the existing results from ”almost every” to ”all but countably many” or even ”all but possibly one” in some cases.

In particular, in Section 3 we prove that an arbitrary non-zero vector φ in a Hilbert space is cyclic for all but countably many operators (A_{α})_{ac} for any family of self-adjoint (or unitary) rank-one perturbations A_{α}. An arbitrary non-zero vector is also cyclic for almost all operators (A_{α})_{s} for any such family. Here (A_{α})_{ac} and (A_{α})_{s} denote the absolutely continuous and singular parts of the operators A_{α} respectively, see section 2.1 for the definitions. This result strengthens a theorem by Jakšić and Last [8, 9].

In Section 4, we show that if a vector belongs to a certain natural class of vectors, associated with the family A_{α}, then we have cyclicity for all except possibly one operator A_{α}.

In Subsection 3.4 we discuss applications of our results to Anderson-type Hamiltonians and deduce and improve some of the results from [8, 9] and [21].

The theory of rank-one perturbations of self-adjoint and unitary operators, and its applications to Anderson-type models became an active area of research over the last 20 years. The interest to this part of perturbation theory is caused, to a large degree, by connections to the famous problem of Anderson localization.

In 1958 P. W. Anderson (see [1]) suggested that sufficiently large impurities in a semiconductor could lead to spatial localization of electrons, called Anderson localization. Although most physicists consider the problem solved, many mathematical questions with striking physical relevance remain open. The field has grown into a rich mathematical theory (see e.g. [6, 7, 11] for different Anderson models and [5, 13] for refined notions of Anderson localization).

While the property of localization for a random Anderson-type operator has many different definitions, one of the “weaker” definitions of localization is equivalent to the property that the spectrum of the operator is almost-surely singular. It is well-known that, if an Andersontype Hamiltonian is almost-surely singular, then it is almost-surely cyclic. Equivalently, if such an operator is not cyclic with positive probability, then it is delocalized.

The study of spectral behavior under rank-one perturbations proves to be one of the main tools in spectral analysis of Anderson-type models, in particular in problems concerning cyclicity, see for instance [8, 9, 21]. This connection served as one of the motivating factors for the current paper.

1.1. Notation/Beware. We consider three main classes of families of perturbations: rankone self-adjoint, rank-one unitary perturbations and Anderson-type Hamiltonians. Throughout we use the notations A_{α}, U_{γ} and H_{ω} to denote the corresponding families of perturbations, respectively. While all three classes are somewhat closely related, the two types of rank-one perturbations ((2.1) and (2.6) below) are almost interchangeable via the Cayley transform: The Aleksandrov-Clark theory and all its basic results discussed in Section 2.3 below can be equivalently re-stated in the case of the real line (upper half-plane). For example, the Cauchy transform in 𝔻, see the second equation of (2.7), is replaced with its analogue in ℂ_{+}, see equation (2.2) for the definition. Similarly, results on rank-one perturbations of selfadjoint operators can be re-formulated for the families of unitary rank-one perturbations, see for instance [19]. It is a well known feature of complex function theory that some of the proofs of the half-plane statements look more natural in the settings of the unit disk and vice versa. Similarly, in this paper we utilize both self-adjoint and unitary settings in our statements and proofs.

2. Preliminaries

2.1. Cyclicity for normal operators. Recall that an operator in a separable Hilbert space is called normal if T^{∗}T = T T^{∗}. By the spectral theorem the operator T is unitarily equivalent to M_{z}, multiplication by the independent variable z, in a direct sum of Hilbert spaces

H = ⊕ ∫ H(z) dμ(z)

where μ is a scalar positive measure on ℂ. The measure μ is called a spectral measure of T.

If T is a unitary or self-adjoint operator, its spectral measure μ is supported on the unit circle or on the real line, respectively. Via Radon decomposition, μ can be decomposed into a singular and an absolutely continuous parts μ = μ_{s} + μ_{ac}. The singular component μ_{s} can be further split into singular continuous and pure point parts. For unitary or self-adjoint T we denote by T_{ac} the restriction of T to its absolutely continuous part, i.e. T_{ac} is unitarily equivalent to

M_{t}|_{⊕ ∫ H(t)dμac(t)}.

Similarly, define the singular, singular continuous and the pure point parts of T, denoted by T_{s}, T_{sc} and T_{pp}, respectively.

In terms of the spectral representation described above, the property of cyclicity, as defined in the introduction, is equivalent to the property that all Hilbert spaces H(t) are one-dimensional and the space

⊕ ∫ H(t)dμ(t)

can be identified with L^{2}(μ). Cyclic vectors for T correspond to functions with full support in L^{2}(μ), i.e. those functions that are non-zero almost everywhere with respect to μ.

2.2. Self-adjoint rank-one perturbations. Let T be a normal operator on a Hilbert space H and let φ ∈ H be a non-zero vector. An alternative definition of the spectral measure of T can be given as follows. Notice that there exists a unique measure μ on ℂ such that

((T − λI)^{−1}φ, φ)_{H} = ∫ \frac{dμ(t)}{t − λ},

for all λ outside of the spectrum of T. If μ is such a measure we say that μ is the spectral measure of T with respect to the vector φ. Note that such a measure is unique, once T and φ are fixed. The operator T is bounded if and only if μ is compactly supported.

Let A be a self-adjoint operator and let φ be its cyclic vector. Consider the family of rank-one perturbations

A_{α} = A + α( · , φ)φ, for α ∈ ℝ. (2.1)

It is not difficult to show that then φ will be a cyclic vector for A_{α} for all α ∈ ℝ. Denote by μ_{α} the spectral measure of A_{α} with respect to φ. In these notations μ = μ_{0}.

In virtue of the spectral theorem, one can always assume that H = L^{2}(μ), A = M_{t} and φ = 1 ∈ L^{2}(μ). Denote by V_{α} the operator of spectral representation for A_{α}, i.e. the unitary operator V_{α} : L^{2}(μ) → L^{2}(μ_{α}) such that V_{α}A_{α} = M_{t}V_{α} and V_{α}φ = 1. An explicit formula for V_{α} was recently derived in [15].

For unbounded A (i.e. not compactly supported μ), we always assume that the spectral measure μ corresponding to φ satisfies

∫_{ℝ} \frac{dμ(t)}{1 + |t|} < ∞.

Using the standard terminology, this means that we consider the class of singular form bounded perturbations and assume that φ ∈ H_{−1}(A) ⊃ H, i.e. that Aφ ∈ H. Notice that if φ ∉ H_{−1}(A) then the formal expression (2.1) does not possess a unique self-adjoint extension, see for instance [14].

The Aronszajn–Donoghue theory analyzes the spectrum of the perturbed operator under rank-one perturbations. We will use the following well-known statement:

Theorem 2.1 ([22]). For non-equal coupling constants α ≠ β, the singular parts (μ_{α})_{s} and (μ_{β})_{s} are mutually singular.

Of fundamental importance to many spectral problems is the Cauchy transform. If τ is the spectral measure of a self-adjoint operator A corresponding to the vector φ then the Cauchy transform of τ,

K_{τ} (z) = \frac{1}{π} ∫_{ℝ} \frac{dτ(t)}{t − z} , z ∈ ℂ \ ℝ, (2.2)

is equal to the corresponding resolvent function of A:

K_{τ} (z) = ((A − zI)^{−1}φ, φ) = ∫ \frac{dμ(t)}{t − z}.

This connection allows one to apply complex analysis in spectral problems.

We use the notation

(K_{τ})_{+}(x) = lim_{y→0} K_{τ} (x + iy)  and  (K_{τ})_{−}(x) = lim_{y→0} K_{τ} (x − iy)

for x ∈ ℝ. By a theorem of Privalov,

(K_{τ})_{−}(x) − (K_{τ})_{+}(x) = 2πi \frac{dτ}{dx}(x) (2.3)

for Lebesgue-a.e. x ∈ ℝ.

2.3. Aleksandrov–Clark theory and unitary rank-one perturbations. By H^{p} we will denote the standard Hardy spaces in the unit disk. Recall that a function θ ∈ H^{∞} is called inner, if |θ(z)| = 1 for almost every |z| = 1. The (scalar valued) model space K_{θ} is defined as K_{θ} = H^{2} ⊖ θH^{2}. Such spaces play an important role in complex function theory and functional analysis, see for instance [17].

The model operator on a space K_{θ} is defined as S_{θ} = P_{θ}S, where P_{θ} denotes the orthogonal projection onto K_{θ}, while S is the shift operator given by Sf(z) = zf(z) for f ∈ H^{2}. The adjoint to the shift operator is the so-called backward shift operator defined as

S^{∗}f = \frac{f(z) − f(0)}{z} .

Let θ be an inner function. To simplify the formulas we will assume that θ(0) = 0. In [3] Clark showed that the family of rank-one perturbations

(2.4)  Ũ_{γ} = S_{θ} + γ( · , S^{∗}θ)1 for γ ∈ 𝕋

consists of unitary operators on K_{θ}, and - vice versa - that all unitary rank-one perturbations of the model operator S_{θ} = P_{θ}S|_{K_{θ}} are given by (2.4).

It is well-known that the the vector 1 ∈ K_{θ} is cyclic for all operators Ũ_{γ} in the above family. By σ_{γ} denote the spectral measures of Ũ_{γ} with respect to the function 1, i.e. such measures that the identity

((Ũ_{γ} + zI)(Ũ_{γ} − zI)^{−1}1, 1) = ∫_{𝕋} \frac{ξ + z}{ξ − z} dσ_{γ}(ξ)

holds true for all γ ∈ 𝕋. In fact, every inner function θ with θ(0) = 0 corresponds in a one-to-one fashion with a family of Clark measures {σ_{γ}}_{γ∈𝕋}. Further, for inner functions θ, the measures σ_{γ} are purely singular for all γ ∈ 𝕋; and vice versa.

One of the main results of the Clark theory says that the spectral measures of Ũ_{γ} are defined by the identity

(2.5)  \frac{θ + γ}{θ − γ} = ∫_{𝕋} \frac{ξ + z}{ξ − z} dσ_{γ}(ξ).

The Clark operator is the unitary operator Φ_{γ} : K_{θ} → L^{2}(σ_{γ}) such that Φ_{γ}Ũ_{γ} = M_{z}Φ_{γ}, where M_{z} is the operator that acts as multiplication by the independent variable in L^{2}(σ_{γ}). In other words, the Clark operator is the spectral representation of Ũ_{γ}.

Notice that the spectral representation of V_{α} : L^{2}(σ) → L^{2}(σ_{α}) from the previous subsection on self-adjoint rank-one perturbations corresponds to the composition operator Φ_{γ}Φ_{1}^{∗} : L^{2}(σ_{1}) → L^{2}(σ_{γ}) in the case of unitary rank-one perturbations.

The situation becomes more complicated without the assumption that the spectral measure is purely singular: In the case of non-trivial absolutely continuous spectrum the model space consists of pairs of functions analytic inside and outside of the unit disk, see [16].

However, many of the formulas of the Aleksandrov-Clark theory remain valid in the case where the spectral measures are not purely singular. If μ is a positive finite measure on the unit circle, denote by U_{1} the operator of multiplication by z in L^{2}(μ). Let θ be a bounded holomorphic function in the unit disk 𝔻 that satisfies (2.5) for γ = 1 and σ_{1} = μ. If μ is not singular, θ is not inner but still belongs to the unit ball of H^{∞}, i.e. |θ| ≤ 1 in 𝔻. Nonetheless, one can still consider a family of measures σ_{γ} defined by (2.5). This family will consist of spectral measures of unitary rank-one perturbations of U_{1} corresponding to the vector 1 ∈ L^{2}(μ), with U_{γ} defined as

(2.6)  U_{γ} = U_{1} + (γ − 1)(·, U_{1}^{∗}1)1, γ ∈ 𝕋,

see [20].

We will also use the following two theorems from the Aleksandrov-Clark theory. Let m denote the Lebesgue measure on 𝕋.

Theorem 2.2 (Aleksandrov’s spectral averaging, see e.g. [20]). For f ∈ L^{1}(𝕋, dm) we have

∫ fdm = ∫ (∫ fdσ_{γ}) dm(γ).

It is well known that the adjoint Φ_{γ}^{∗} : L^{2}(σ_{γ}) → K_{θ} of the Clark operator can be represented using the normalized Cauchy transform

Φ_{γ}^{∗}h = \frac{𝒦_{hσ_{γ}}}{𝒦_{σ_{γ}}} ,

where 𝒦 stands for the Cauchy transform in 𝔻:

(2.7)  𝒦_{σ_{γ}}(z) = ∫_{𝕋} \frac{dσ_{γ}(ξ)}{1 − ξ̄z} ,  and  𝒦_{hσ_{γ}}(z) = ∫_{𝕋} \frac{h(ξ)dσ_{γ}(ξ)}{1 − ξ̄z} .

Theorem 2.3 ([18]). For any f ∈ L^{1}(σ_{γ}),

lim_{r→1} \frac{𝒦_{fσ_{γ}}(rz)}{𝒦_{σ_{γ}}(rz)} = f(z), for (σ_{γ})_{s}-a.e. z ∈ 𝕋.

Theorems 2.2 and 2.3, hold true even if the family of spectral measures possesses non-trivial absolutely continuous parts, although the normalized Cauchy transform of an arbitrary function in L^{2}(σ_{γ}) cannot be interpreted as a vector from K_{θ}.

Via the standard agreement, every function from a Hardy space H^{p} is identified with its boundary values on the circle 𝕋. It follows from Theorem 2.3 that the boundary values of any f ∈ K_{θ} exist σ_{γ}-a.e for any γ ∈ 𝕋 and the Clark operator Φ_{γ} : K_{θ} → L^{2}(σ_{γ}) simply sends f into its boundary values.

For a function f ∈ K_{θ}, denote by f̃ the function θf̄ on 𝕋. Note that f ∈ K_{θ} and f(0) = 0 imply that θf̄ ∈ K_{θ}. A function f ∈ K_{θ} is called a Hermitian element, if f̃ = f. Notice that Hermitian functions satisfy f(0) = 0.

The following simple statement plays an important role in Section 4.

Theorem 2.4 ([18]). Let f ∈ K_{θ}. Then f is a Hermitian element if and only if

(2.8)  arg(Φ_{γ}f) = \frac{arg γ}{2} (mod π), σ_{γ}-a.e.,

and ∫ fdσ_{γ} = 0 for some γ ∈ 𝕋. If f is a Hermitian element then f satisfies (2.8) and ∫ fdσ_{γ} = 0 for any γ ∈ 𝕋.

2.4. Spaces of Paley–Wiener functions. For a > 0 the class of Paley–Wiener functions on ℝ is given by

PW_{a} = {f̂: f ∈ L^{2}(−a, a)},

where f̂(z) = ∫ e^{−izt}f(t)dt denotes the classical Fourier transform of f. Alternatively, the Paley–Wiener space can be characterized as the space of entire functions of exponential type at most a whose boundary values on the real line are square summable with respect to Lebesgue measure.

The Paley–Wiener space PW_{a} is closely related to the model space K_{θ} for the inner function

θ_{a}(z) = θ(z) = e^{−2a \frac{1+z}{1−z}}

in the unit disk. To establish the connection, consider the conformal map

ψ(z) = \frac{z − i}{z + i}

from ℂ_{+} to 𝔻. Denote ϑ(z) = ϑ_{a}(z) = e^{2iaz}. Note that

ϑ_{a}(z) = θ_{a}(ψ(z)).

By K_{ϑ}^{ℝ} denote the space obtained from K_{θ} by composing all functions from K_{θ} with ψ, i.e.

K_{ϑ}^{ℝ} = {f(ψ)|f ∈ K_{θ}}.

Then the space

e^{−iaz}K_{ϑ}^{ℝ} = {e^{−iaz}f(ψ)|f ∈ K_{ϑ}}

is equal to the space of entire functions of exponential type at most a and with boundary values on ℝ that are square summable with respect to the measure (1 + x^{2})^{−1}dx.

Hence we have

PW_{a} ⊂ e^{−iaz}K_{ϑ_{a}}^{ℝ} for 0 < a.

Further, one can prove that the codimension is equal to 1 and

e^{−iaz}K_{ϑ_{a}}^{ℝ} ⊖ PW_{a}

consists of constant functions.

3. Arbitrary non-zero vectors yield cyclic vectors for almost all parameters

Let A be a self-adjoint (possibly unbounded) operator on a separable Hilbert space H and let φ be a cyclic vector for A. Define the family of self-adjoint rank-one perturbations of A, A_{α} as in Subsection 2.2. Recall that (A_{α})_{ac} and (A_{α})_{s} denote the absolutely continuous part and the singular part of the operator A_{α}, respectively.

Further notice that (A_{α})_{s} ⊕ (A_{α})_{ac}, since (μ_{α})_{s} ⊥ (μ_{α})_{ac}.

Theorem 3.1. Let A_{α} be a family of self-adjoint rank-one perturbations in a Hilbert space H given by (2.1). Let 0 ≠ f ∈ H. Then

1) The function f is a cyclic vector for (A_{α})_{ac} for all but a countable number of α ∈ ℝ.

2) The function f is a cyclic vector for (A_{α})_{s} for Lebesgue a.e. α ∈ ℝ.

In the following subsection we prove an equivalent reformulation of this theorem in terms of its spectral representation.

3.1. Proof of Theorem 3.1. As we mentioned before, in view of the spectral theorem, instead of dealing with a general family of self-adjoint rank-one perturbations given by equation (2.1) one can consider the self-adjoint rank-one perturbations A_{α} = M_{t} + α( · , 1)_{L^{2}(μ)}1 on

L^{2}(μ). Let the spectral operator V_{α} : L^{2}(μ) → L^{2}(μ_{α}) be as defined in Subsection 2.2. In these settings Theorem 3.1 can be stated as follows.

Theorem 3.2. Let 0 ≠ f ∈ L^{2}(μ). Then

1) The function f_{α} = V_{α}f ∈ L^{2}(μ_{α}) is not equal to zero (μ_{α})_{ac}-a.e. for all but a countable number of α ∈ ℝ.

2) The function f_{α} = V_{α}f ∈ L^{2}(μ_{α}) is not equal to zero (μ_{α})_{s}-a.e. for Lebesgue a.e. α ∈ ℝ.

Remarks. (a) Obviously one cannot expect to obtain the above conclusion of cyclicity for all α ∈ ℝ. It is always possible to start with f that is zero on a set of positive μ_{0}-measure that is, therefore, not cyclic for α = 0. In fact, under the conditions of Theorem 3.2, we cannot replace ”countable” with ”finite.”

(b) In the case of purely singular spectral measures for some natural classes of f the conclusion can be strengthened to ”all but one” α, see Section 4 below.

Our next example says that, in general, if f is not Hermitian, then f can be non-cyclic for uncountably many corresponding rank-one perturbations. In particular, in the conclusion of Theorem 3.2 the distinction between the singular and the absolutely continuous is necessary. As was mentioned above, throughout the rest of the paper we will switch between selfadjoint and unitary settings as a matter of convenience. An analogous discussion can always be carried out in the other case.

Example 1. Consider the setting of rank-one unitary perturbation described in Subsection 2.3. We will construct a bounded holomorphic function θ such that for the family of spectral measures (σ_{γ})_{γ∈𝕋} defined by (2.5) and the corresponding family of unitary operators U_{γ} there exists a non-zero function f ∈ L^{2}(σ_{1}) such that f_{γ} = Φ_{γ}Φ_{1}^{∗}f is non-cyclic in L^{2}(σ_{γ}) for an uncountable set of γ’s. Note that in this example θ is not inner and the corresponding operators have nontrivial absolutely continuous parts.

Let C be a Cantor (closed uncountable) subset of the unit circle 𝕋. Let w be the continuous function on 𝕋 defined by w(ξ) = dist^{2}(ξ, C). Denote by (σ_{γ})_{γ∈𝕋} the system of probability measures which is the family of Clark measures for some inner function θ, and such that the measure σ_{1} coincides, up to a multiplicative constant, with the measure w dm, where m is the Lebesgue measure on 𝕋. Note that, by definition of σ_{1}, we have

∫ \frac{1}{|x − y|^{2}} dσ_{1}(y) < ∞

for any x ∈ C. It is well known (see, e.g. [4]) that the last condition implies that each point of C is a point mass for one of the measures σ_{γ}. Since C is uncountable, we conclude that uncountably many σ_{γ}’s have atoms on the set C.

Let now F be an outer function with modulus equal to w almost everywhere on 𝕋. Consider f = F/w. Then f is a unimodular function on 𝕋, and we have fσ_{1} = fw = F, hence 𝒦_{fσ_{1}} = 0 on the set C. So we have

\frac{𝒦_{fσ_{1}}}{𝒦_{σ_{1}}} = 0 on C.

In virtue of the following Lemma 3.3, it follows that Φ_{γ}Φ_{1}^{∗}f = 0 on the (uncountable) set of those γ ∈ 𝕋 for which σ_{γ} has a point mass on C. Hence f is not cyclic for uncountably many operators U_{γ}.

Lemma 3.3 (Aronszajn–Krein-type formula). Under the hypotheses of Theorem 3.2 we have

K_{f_{α}μ_{α}} = \frac{K_{f_{β}μ_{β}}}{1 + (α − β)K_{μ_{β}}}  and  \frac{K_{f_{α}μ_{α}}}{K_{μ_{α}}} = \frac{K_{f_{0}μ_{0}}}{K_{μ_{0}}} . (3.1)

Proof of Lemma 3.3. First consider the case where 1 ∈ H = L^{2}(μ). Let z ∈ ℂ\ℝ. Combining the second resolvent equation and the fact that A_{α} = A_{β} + (α − β)( · , 1)1 we obtain

(A_{β} − zI)^{−1}1 − (A_{α} − zI)^{−1}1 = (α − β)((A_{α} − zI)^{−1} · , 1)(A_{β} − zI)^{−1}1.

Application to a vector f ∈ H and pairing with 1 yields

((A_{β} − zI)^{−1}f, 1) − ((A_{α} − zI)^{−1}f, 1) = (α − β)((A_{α} − zI)^{−1}f, 1)((A_{β} − zI)^{−1}1, 1).

Recall that V_{α}A_{α} = M_{t}V_{α}, V_{α}1 = 1 and V_{α}f = f_{α}. With this we obtain

K_{f_{β}μ_{β}} − K_{f_{α}μ_{α}} = (α − β)K_{f_{α}μ_{α}}K_{μ_{β}},

or equivalently,

K_{f_{α}μ_{α}} = \frac{K_{f_{β}μ_{β}}}{1 + (α − β)K_{μ_{β}}} = \frac{K_{f_{β}μ_{β}}}{K_{μ_{β}}} \frac{K_{μ_{β}}}{1 + (α − β)K_{μ_{β}}} = \frac{K_{f_{β}μ_{β}}}{K_{μ_{β}}} K_{μ_{α}}. (3.2)

In the last equality we used the well-known Aronszajn–Krein formula, which can be obtained from the first equality of (3.2) by using f = φ (or equivalently f_{α} = 1).

To obtain the second formula of (3.1), we divide both sides by K_{μ_{α}}.

If 1 ∈ H_{−1}(A)\H the resolvent formula is slightly more complicated

(A_{α} − λI)^{−1}f = (A_{β} − λI)^{−1}f − \frac{(α − β) ((A_{β} − λI)^{−1}f, 1)}{1 + (α − β) ((A_{β} − λI)^{−1}1, 1)}(A_{β} − λI)^{−1}1

for f ∈ H_{−1}(A), see e.g. [12]. When paired with the vector 1 this yields

K_{f_{α}μ_{α}} = (1 − \frac{(α − β)K_{μ_{β}}}{1 + (α − β)K_{μ_{β}}} ) K_{f_{β}μ_{β}} = \frac{K_{f_{β}μ_{β}}}{1 + (α − β)K_{μ_{β}}} .

The remainder of the proof for 1 ∈ H_{−1}(A)\H now follows similarly to the case of regular perturbations 1 ∈ H. □

Proof of part 1) of Theorem 3.2. Define the set

Σ_{α} = {x ∈ supp(μ_{α})_{ac} : f_{α}(x) = 0}.

The goal is to show that (μ_{α})_{ac}(Σ_{α}) = 0 for all but a countable number of parameters α.

Assume that f_{α} is not cyclic for uncountably many α ∈ ℝ, i.e. assume that for some S ⊂ ℝ, S uncountable, we have (μ_{α})_{ac}(Σ_{α}) > 0 for all α ∈ S. Then |Σ_{α}| > 0 for all α ∈ S. Since S is uncountable

(3.3)  |Σ_{α} ∩ Σ_{β}| > 0  for some α, β ∈ S with α ≠ β.

Let us fix α and β satisfying (3.3) and investigate the jump behavior in the first equation of (3.1) below. By Fatou’s jump theorem, see equation (2.3), we have

(K_{f_{α}μ_{α}})_{−}(x) − (K_{f_{α}μ_{α}})_{+}(x) = 2πi \frac{d(f_{α}μ_{α})}{dx}(x) = 0 Lebesgue-a.e. x ∈ (Σ_{α} ∩ Σ_{β}),

because f_{α} = 0 on Σ_{α}.

Similarly K_{f_{β}μ_{β}} has no jump Lebesgue almost everywhere on Σ_{α} ∩ Σ_{β}. On the other hand K_{μ_{β}} has a non-zero jump Lebesgue almost everywhere on (Σ_{α} ∩ Σ_{β}) ⊂ supp(μ_{α})_{ac}.

Hence, while the right hand side in the first equation of (3.1) has a jump, the left hand side does not jump Lebesgue-a.e. on Σ_{α}∩Σ_{β} where we have (3.3), and we arrive at a contradiction. Therefore the assumption that f_{α} is not cyclic for uncountably many α ∈ ℝ cannot be maintained. □

Proof of part 2) of Theorem 3.2. Denote by S_{α} the essential support of the measure (μ_{α})_{s} defined as the set of points where the Radon derivative of (μ_{α})_{s} is infinite. It follows from the the Aronszajn–Donoghue theorem, Theorem 2.1, that the sets S_{α} are disjoint. Define the set

Ω_{α} = {x ∈ S_{α} : f_{α}(x) = 0}.

The goal is to show that (μ_{α})_{s}(Ω_{α}) = 0 for Lebesgue a.e. α ∈ 𝕋.

Assume that f_{α} is not cyclic for a set of α’s with positive Lebesgue measure, i.e. assume that for some S ⊂ ℝ, |S| > 0 we have

μ_{α}(Ω_{α}) > 0 for all α ∈ S.

In virtue of Theorem 2.3, (or, more precisely, its analog for the real line) we have for the boundary values

\frac{K_{f_{α}μ_{α}}(x + iy)}{K_{μ_{α}}(x + iy)} ⟶^{y→0} 0 (3.4)

for (μ_{α})_{s}-a.e. x ∈ Ω_{α} and all α ∈ S. Therefore, there exists a set M ⊂ ℝ, |M| > 0 such that for all x ∈ M there exists an α such that (3.4) is satisfied.

Using Lemma 3.3 (below), the analytic function

\frac{K_{f_{α}μ_{α}}}{K_{μ_{α}}} = \frac{K_{f_{0}μ_{0}}}{K_{μ_{0}}}

has zero boundary values on M, a set of Lebesgue measure greater than zero.

Hence

\frac{K_{f_{0}μ_{0}}}{K_{μ_{0}}} ≡ 0

and we must have f_{0} ≡ 0. But this contradicts the hypothesis that f is a non-zero vector. □

Let us prove the lemma that was used in the above proofs. Recall the definition (2.2) of the Cauchy transform.

3.2. A Corollary of Lemma 3.3. Let us mention another consequence of Lemma 3.3, although we will not use this fact later in this paper. Consider a family of Aleksandrov– Clark measures {σ_{γ}}_{γ∈𝕋}.

Corollary 3.4. We have

\frac{𝒦_{f_{γ}σ_{γ}}}{𝒦_{f_{1}σ_{1}}} (z) = \frac{1 − θ(z)}{1 − γ̄θ(z)} m − a.e. z ∈ ℂ\𝕋.

In particular, the function

\frac{𝒦_{f_{γ}σ_{γ}}}{𝒦_{fσ_{1}}} (3.5)

is independent of the choice of f = f_{1} ∈ L^{2}(σ_{1}).

Remark. It was R. G. Douglas who observed the independence of the expression (3.5) from the choice of f ∈ L^{2}(σ_{1}).

Proof. In order to see the second statement notice that by Lemma 3.3, the expression (3.5) is independent from the choice of f ∈ L^{2}(σ_{1}), i.e.

\frac{𝒦_{f_{γ}σ_{γ}}}{𝒦_{fσ_{1}}} = \frac{𝒦_{g_{γ}σ_{γ}}}{𝒦_{gσ_{1}}} for all f, g ∈ L^{2}(σ_{1}).

We obtain the first statement by expanding

\frac{𝒦_{f_{γ}σ_{γ}}}{𝒦_{fσ_{1}}} = \frac{\frac{𝒦_{fγ σγ}}{𝒦_{σγ}} 𝒦_{σ_{γ}}}{\frac{𝒦_{fσ_{1}}}{𝒦_{σ_{1}}} 𝒦_{σ_{1}}}

and use Lemma 3.3 for g = 1 to cancel the fractions in the numerator and denominator. Further apply equation

𝒦_{σ_{γ}}(z) = \frac{1}{1 − γ̄θ(z)}

(confer of [4]) to the remaining 𝒦_{σ_{γ}} in the numerator as well as 𝒦_{σ_{1}} in the denominator. □

3.3. Anderson-type Hamiltonians. The following operator is a generalization of most Anderson models discussed in literature.

For n = 1, 2, ... consider the probability space Ω_{n} = (ℝ, B, μ_{n}), where B is the Borel sigmaalgebra on ℝ and μ_{n} is a Borel probability measure. Let Ω = ∏_{n=0}^{∞} Ω_{n} be a product space with the probability measure ℙ on Ω introduced as the product measure of the corresponding measures on Ω_{n} on the product sigma-algebra A. The elements of Ω are points in ℝ^{∞}, ω = (ω_{1}, ω_{2}, ...), ω_{n} ∈ Ω_{n}.

Let H be a separable Hilbert space. Consider a self-adjoint operator H on H and let φ_{1}, φ_{2}, … be a countable collection of unit vectors in H. For each ω ∈ Ω define an Anderson-type Hamiltonian on H as a self-adjoint operator formally given by

(3.6)  H_{ω} = H + V_{ω},  V_{ω} = ∑_{n} ω_{n}( · , φ_{n})φ_{n}.

Except for degenerate cases, the perturbation V_{ω} is almost-surely a non-compact operator. It is hence not possible to apply results from classical perturbation theory to study the spectra of H_{ω}, see e.g. [2] and [10].

In the case of an orthogonal sequence {φ_{n}}, this operator was studied in [8] and [9].

Probably the most important special case of an Anderson-type Hamiltonian is the discrete random Schrödinger operator on l^{2}(ℤ^{d})

Hf(x) = − △ f(x) = − ∑_{|n|=1}(f(x + n) − f(x)), φ_{n}(x) = δ_{n}(x) = \{ \begin{matrix}1 & x = n ∈ ℤ, \\ 0 & else. \end{matrix}

3.4. An application of Theorem 3.1 to Anderson-type Hamiltonians. Let H_{ω} be the Anderson-type Hamiltonian introduced in equation (3.6). Fix ω_{0} ∈ Ω. Assume φ ∈ H_{−1}(H_{ω_{0}}) is a cyclic vector for the self-adjoint operator H_{ω_{0}}. Consider operators H_{ω_{0}} +α( · , φ)φ, α ∈ ℝ.

Then (by Theorem 3.1) any non-zero f ∈ H is cyclic for H_{ω_{0}} + α( · , φ)φ for almost all α ∈ ℝ. In particular, for Lebesgue almost every α, the operators H_{ω_{0}} + α( · , φ)φ are cyclic.

In the case where φ ∈ clos span{φ_{n}}, φ = ∑ a_{n}φ_{n}, we say that φ corresponds to the (possibly non-unique) sequence a = (a_{1}, a_{2}, …). Further, the operators H_{ω_{0}} + α( · , φ)φ correspond to ω belonging to the one dimensional affine subspace

l(ω_{0}, a) = {ω_{0} + α(a_{1}, a_{2}, a_{3}, …)| α ∈ ℝ}.

Cyclicity of the operators for almost every ω in any one-dimensional affine subspace is a stronger statement than ℙ-almost-sure cyclicity that can be found in the literature for some particular cases of our model. In terms of almost-sure cyclicity we obtain the following result.

If l(ω_{0}, a) is a one-dimensional affine subspace of ℝ^{∞}, one can introduce Lebesgue measure on l as

m(S) = |{α| ω_{0} + α(a_{1}, a_{2}, a_{3}, …) ∈ S}|

for any Borel subset S of l.

Corollary 3.5. Suppose that H_{ω_{0}} is self-adjoint for some ω_{0} and that ∑ a_{n}φ_{n}, a = (a_{1}, a_{2}, …), is cyclic for H_{ω_{0}}. Consider a one-dimensional affine subspace of ℝ^{∞}, l = l(ω_{0}, a). Then any non-zero vector φ is cyclic for all (H_{ω})_{ac}, ω ∈ l, except possibly countably many ω, and cyclic for almost every (H_{ω})_{s}, ω ∈ l, with respect to Lebesgue measure on l.

In particular, suppose that the probability measure ℙ is a product of absolutely continuous measures, H_{ω} is self-adjoint ℙ-almost surely and some φ_{n} is cyclic for H_{ω}, ℙ-almost surely. Then any non-zero φ ∈ H is cyclic for H_{ω}, ℙ-almost surely.

It is well-known that if an Anderson-type Hamiltonian is singular almost-surely then it is cyclic almost-surely. The proof of almost-sure cyclicity of the singular part (H_{ω})_{s} and almostsure cyclicity of certain specific vectors can be found in [9] and for the discrete Schrödinger operator in [21]. The second part of Corollary 3.5 extends (from the singular part to the full Anderson-type Hamiltonian H_{ω}) these results showing that if one of the vectors φ_{n} is almost sure cyclic then any non-zero vector possesses that property.

Proof of Corollary 3.5. The first statement follows immediately from Theorem 3.1.

Let ω be such that φ_{n} is a cyclic vector for H_{ω}. For a = (0, …, 0, a_{n}, 0, …, 0) ∈ ℝ^{∞} every 0 ≠ φ ∈ H is cyclic for a.e. point in l(ω, a). Since the union of such subspaces covers ℙ-almost all points of ℝ^{∞}, we obtain the statement. □

4. General Hermitian elements and rank-one perturbations

Let U be a unitary operator. For a vector φ consider the space X defined as the closure of the set of real finite linear combinations of elements of the form

(U + U^{∗})^{n}φ  and  \frac{1}{i}(U − U^{∗})^{n}φ  for  n ∈ ℤ.

Then a vector f ∈ H is Hermitian with respect to U and the vector φ, if f ∈ X and f ⊥ φ.

An analogous definition can be given for self-adjoint operators. For a bounded self-adjoint operator A on a separable Hilbert space H and a vector φ, let (Re A)φ denote the closure of the space of linear combinations of A^{n}φ, n ∈ ℕ with real coefficients. We say that a vector f ∈ H is Hermitian with respect to the operator A and the vector φ, if f ∈ (Re A)φ and f ⊥ φ. For general (unbounded) operators (Re A)φ can be defined as the closed span of

((A − zI)^{−1} + (A − z̄I)^{−1}) φ, z ∈ ℂ_{+}.

Note that in the settings of Section 2.3, the space X defined above is the set of Hermitian functions from K_{θ} defined there, see the proof of Theorem 4.1 below.

Let U be a unitary operator on a separable Hilbert space H. Consider the family

U_{γ} = U + (γ − 1)( · , U^{−1}b)_{H}b (4.1)

of rank-one perturbations, γ ∈ 𝕋, b ∈ H with ∥b∥_{H} = 1. It is well known that U_{γ} is unitary for all γ ∈ 𝕋. Clearly we have U = U_{1}. Without loss of generality, assume that b is cyclic for U, i.e.

clos span{U^{k}b : k ∈ ℤ} = H.

Theorem 4.1. Consider the family U_{γ} of rank-one unitary perturbations given by (4.1). Assume that U_{γ} has purely singular spectrum for some (all) γ ∈ 𝕋. Let 0 ≠ f ∈ H be Hermitian with respect to U = U_{1} and b. Fix a constant c ∈ ℂ\{0}. Then the vector f − cb is cyclic for U_{γ} for all γ ∈ 𝕋\{e^{2i arg c}}.

4.1. Proof of Theorem 4.1. Via the spectral theorem, without loss of generality one can assume that U = U_{1} is an operator of multiplication by z in L^{2}(σ_{1}), σ_{1} is the spectral measure of U corresponding to b and b = 1 ∈ L^{2}(σ_{1}). Define the inner function θ so that σ_{1} is its Clark measure. Let K_{θ}, γ, σ_{γ} as well as Φ_{γ} and Φ_{γ}^{∗} be as defined in Subsection 2.3. Consider operators Ũ_{γ} defined in (2.4). Then U_{γ} = Ũ_{γ}.

Recall that f ∈ K_{θ} is called Hermitian if f = f̃, where f̃ = θf̄.

Using Theorem 2.4 one can show that this definition is equivalent to that of a Hermitian element with respect to operator U = Ũ_{1} and vector b = S^{∗}θ. The condition f(0) = 0 (for f ∈ K_{θ}) is translated into f ⊥ b.

Recall that by Theorem 2.3, the non-tangential limit of f ∈ K_{θ} exist σ_{γ}-a.e. for all γ ∈ 𝕋. Let us denote this non-tangential limit by

f_{γ}(z) = lim_{ξ→z} f(ξ) σ_{γ}-a.e..

In fact, we can identify these boundary values with one function, say f (slightly abusing notation), on the circle which is defined σ_{γ}-a.e. for all γ. Indeed, we restricted ourselves to purely singular measures and by the Aronszajn–Donoghue theorem, Theorem 2.1, we have σ_{γ} ⊥ σ_{η} for γ ≠ η. Our statement now follows from Theorem 4.2 below.

Theorem 4.2. Let 0 ≠ f ∈ K_{θ} be a Hermitian function and fix any constant c ∈ ℂ\{0}. Then the level sets

{z ∈ 𝕋 : f(z) = c}

have zero σ_{γ}-measure for all γ ∈ 𝕋\{e^{2i arg c}}. In particular, the function f − c is cyclic for Ũ_{γ} for all γ ∈ 𝕋\{e^{2i arg c}}.

Proof of Theorem 4.2. Pick f and c according to the hypotheses of the theorem. By Theorem 2.4 we have

arg f = \frac{arg γ}{2} (mod π)

with respect to σ_{γ} almost everywhere.

If γ is such that f = c on a set S ⊂ 𝕋 with σ_{γ}(S) > 0, then we have

\frac{arg γ}{2} (mod π) = arg c

and therefore γ = e^{2i arg c}. □

Remark. In the statement of Theorem 4.2, and therefore Theorem 4.1, the constant c cannot be equal to 0. Indeed, consider K_{z^{n}} that consists of all polynomials of degree less than n. Let β_{1}, ...β_{n−2} be points on 𝕋 such that β_{k}^{n} = γ_{k} are different points. Let

p(z) = a_{n−1}z^{n−1} + ... + a_{1}z

be a polynomial with roots at 0, β_{1}, ...β_{n−2}. Then

p̃(z) = ā_{1}z^{n−1} + ... + ā_{n−1}z

has roots at the same points. Notice that p + p̃ is a Hermitian element of K_{z^{n}} whose zero set Z = {0, β_{1}, ...β_{n−2}} satisfies σ_{γ_{k}}(Z) = 1/n > 0 for k = 1, 2, ..., n − 2.

Let us mention the following examples that illustrate Theorem 4.2.

Level sets of self-reciprocal polynomials. A polynomial

p(z) = ∑_{m=0}^{n−1} a_{m}z^{m}

of degree less than n is a Hermitian element of K_{z^{n}}, if and only if

a_{0} = 0  and  a_{m} = \overline{a_{n−m}},  m = 1, …, n − 1. (4.2)

Polynomials that satisfy (4.2) are called self-reciprocal. Note that the Clark measure σ_{γ} of θ = z^{n} is concentrated on the set of n-th roots of γ.

Hence, if c ≠ 0 and p is a self-reciprocal polynomial, then by Theorem 4.2 all roots of the equation p = c on 𝕋 must be contained in a set of n-th roots of γ for γ ∈ 𝕋 given in the statement of the theorem.

Naturally, this simple fact can also be proved directly. If z is such that p(z) = c then

c = p(z) = z^{n}\overline{p(z)} = z^{n}c̄.

Hence z^{n} = c/c̄ where |c/c̄| = 1 and arg(c/c̄) = 2 arg c. The proof of Theorem 4.2 can be viewed as a generalization of this argument.

The non-zero level sets of Paley–Wiener functions. Recall the definition of Paley– Wiener functions from Subsection 2.4.

The following statement is an analog to Euler’s Formula e^{iθ} = cos θ +i sin θ for Paley–Wiener functions.

Proposition 4.3. Let f ∈ PW_{a}. Then e^{iaz}f = g_{1} + ig_{2} where g_{1}, g_{2} are entire functions such that g_{1}, g_{2} ∈ e^{iaz} PW_{a} and each level set

{x ∈ ℝ| g_{i}(x) = c}, i = 1, 2; c ≠ 0,

is contained in the arithmetic progression

\{2 arg c + \frac{2πn}{a}\}_{n∈ℤ} .

Proof. Recall that K_{ϑ_{a}}^{ℝ} was defined as the function space obtained by “mapping” the model space K_{θ}, where

θ_{a}(z) = θ(z) = e^{−2a \frac{1+z}{1−z}} , 0 < a ≤ 1

from 𝔻 to ℝ using the standard conformal map ψ : ℂ_{+} → 𝔻, see Section 2.4. Then

e^{iaz} PW_{a} ⊂ K_{ϑ_{a}}^{ℝ}.

Hence we have e^{iaz}f ∈ K_{ϑ}^{ℝ}.

Without loss of generality a = 1. For the inner function θ = e^{− \frac{1+z}{1−z}} in 𝔻, the Clark measure (σ_{γ})_{γ∈𝕋} is concentrated on the sequence

ψ({arg γ + 2πn}) ⊂ 𝕋.

Like in the remark following Theorem 4.2, we can decompose f into f = g_{1} + ig_{2} where the translations g̃_{1} and g̃_{2} of the functions g_{1} and g_{2} from the upper half plane to the disc are Hermitian in K_{θ}. By Theorem 4.2

{z ∈ 𝕋 : g̃_{i} = c} ⊂ ψ({arg γ + 2πn}), arg γ = 2 arg c.

Hence the level sets of the functions g_{i} are contained in arithmetic progressions given in the statement. □

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Laboratoire d’Analyse et de Mathématiques Appliquées, UMR CNRS 8050, Université Paris-Est, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2, France

E-mail address: evgueni.abakoumov@univ-mlv.fr

Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843, USA

E-mail address: conni@math.tamu.edu
URL: http://www.math.tamu.edu/~conni

Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843, USA

E-mail address: alexeip@math.tamu.edu
URL: http://www.math.tamu.edu/~alexei.poltoratski/