EQUIVALENCE UP TO A RANK ONE PERTURBATION

Alexei G. Poltoratski

Abstract. We prove that any two unitary operators with simple singular spectrum which contains the whole circle are unitarily equivalent up to a rank one operator.

Introduction

This note is devoted to the spectral analysis of rank one perturbations of unitary and selfadjoint operators. We study the following question: given two cyclic (i. e. having simple spectrum) operators A and B, when is A equivalent to B up to a rank one perturbation? More precisely, when does there exist a unitary operator U such that rank(UAU^{∗}−B) = 1? As usual, we are looking for an answer in terms of the spectra of A and B.

An analogous question for compact perturbations is answered by the Weyl – von Neumann Theorem [K]. It says that A is equivalent to B + K for some compact K iff the essential spectra σ_{ess}(A) and σ_{ess}(B) coincide.

A necessary and sufficient condition for A and B to be equivalent up to a trace class operator, which is found by Carey and Pincus [CP], is more delicate and involves additional spectral invariants. In addition to the essential spectra, the isolated eigenvalues of A and B must now obey certain rules.

In this paper we make the last step down the ladder and study the case when A and B are equivalent up to a rank one perturbation. A general necessary and sufficient condition in these settings seems out of reach: it is impossible to formulate in any reasonable terms. However, it is still possible to achieve a good assessment of the situation by fully describing the most important particular case.

It is quite well understood how isolated eigenvalues of an operator behave under rank one perturbations. On the other hand, by the Weyl – von Neumann Theorem, if A is equivalent to B up to a rank one perturbation, then σ_{ess}(A) = σ_{ess}(B). In particular, rank one perturbations do not affect absolutely continuous spectrum. Hence, it seems reasonable to restrict ourselves to the case when A and B have singular spectrum and

σ(A) = σ_{ess}(A) = σ(B) = σ_{ess}(B)

(where σ(A) and σ(B) denote the spectra of A and B respectively). Under this restriction we are able to give a complete answer to our question.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 30E20.

Key words and phrases. Linear operators, Perturbation Theory, Krein spectral shift.

Denote 𝕋 = {|z| = 1}. We will say that operators U and V are completely non equivalent if there are no non trivial closed invariant subspaces H_{1} and H_{2} of U and V respectively such that the restriction of U on H_{1} is unitarily equivalent to the restriction of V on H_{2}. Our main result is

Theorem 1. Let U and V be completely non equivalent singular unitary cyclic operators such that σ(U) = σ(V) = 𝕋. Then U and V are equivalent up to a rank one perturbation. I. e. there exist a unitary operator U and a rank one operator V such that

UUU^{∗} = V + V.

We will prove this result in Section 2. One can easily adapt the proof to replace unitary U and V with self-adjoint ones, and 𝕋 with ℝ. An analogous result for the case of pure point operators follows from a function theoretic construction suggested by Aleksandrov [A] (see also [P]). However, the general case requires a different approach. Our main tool in Section 2 is the spectral shift function, which was originally introduced by Lifshits for finite rank perturbations and later studied by Krein in the trace class situation (see [BY] for the history of this notion and further references).

The following question naturally arises: under which conditions on a closed set K ⊂ 𝕋, can we replace 𝕋 in the statement of Theorem 1 with K? Such necessary and sufficient conditions were formulated in [P] for the discrete self-adjoint case. They now can be proved to be valid for the general self-adjoint (unitary) case. We discuss this matter in the remark at the end of Section 2.

Note, that if two unitary (self-adjoint) operators A and B satisfy

σ(A) = σ_{ess}(A) = σ(B) = σ_{ess}(B) = K

then, by the Weyl–von Neumann Theorem, they are equivalent up to a compact operator. Theorems 1 and 8 (below) refine this statement showing that, when K is sufficiently ”good”, the operators are much closer related.

To conclude the introduction, we would like to point out some function theoretical consequences of Theorem 1. Let Kμ and P μ denote the Cauchy and Poisson integrals of measure μ on 𝕋 in the unit disk 𝔻:

Kμ(z) = ∫_{𝕋} \frac{1 + ξ̄z}{1 − ξ̄z} dμ(ξ)

and

P μ = ℜKμ(z) = ∫_{𝕋} \frac{1 − |z|^{2}}{|ξ − z|^{2}} dμ(ξ).

The results of [P] imply that Theorem 1 is equivalent to the following statement. For any two positive singular measures μ and ν on 𝕋 such that μ ⊥ ν and suppμ = suppν = 𝕋, there exist f ∈ L^{1}(μ), ||f|| = 1, f > 0 μ-a.e. and g ∈ L^{1}(ν), ||g|| = 1, g > 0 ν-a.e. such that

Kμ = \frac{1 − θ}{1 + θ} = (Kν)^{−1} .

From this statement one can obtain the following Corollary.

Corollary 2. For any two singular measures μ and ν on 𝕋 such that μ ⊥ ν and suppμ = suppν = 𝕋, there exists an inner function θ such that θ(ξ) = 1 μ-a. e. and θ(ξ) = −1 ν-a. e. Moreover,

{θ = 1} = {dμ/dm = ∞}

and

{θ = −1} = {dν/dm = ∞},

where m is the normalized Lebesgue measure on 𝕋.

With some additional effort one can deduce from the last statement the following

Corollary 3. For any singular measure μ on 𝕋 and any ϕ ∈ L^{∞}(μ) there exists ψ ∈ H^{∞} such that ψ(ξ) = ϕ(xi) for μ-a. e. ξ and ||ψ||_{H^{∞}} = ||ϕ||_{L^{∞}(μ)}.

The Krein-Lifshits spectral shift for a unitary pair

We first define the Krein-Lifshits spectral shift for the rank one perturbation problem of unitary operators. This definition is very similar to the one given in [S] or [P] for the self-adjoint case.

Let U_{1} be a unitary cyclic operator, v, ||v|| = 1 its cyclic vector. Then we can consider the family of unitary rank one perturbations of U_{1}:

(1)  U_{α} = U_{1} + (α − 1)(·, U_{1}^{−1}v)v

α ∈ 𝕋.

Denote by μ_{α} the spectral measure of v for U_{α}

μ_{α}(B) = (v, E_{B}(U_{α})v)

for any Borel B ∈ 𝕋, where E_{B}(U_{α}) is the spectral projection of U_{α}. Note, that since ||v|| = 1, all μ_{α}’s are probability measures.

Since

Kμ_{α} = ((U_{α} + z)(U_{α} − z)^{−1}v, v)

for z ∈ 𝔻, after simple computations one can obtain

(2)  Kμ_{α} = \frac{(α − 1) + (α + 1)Kμ_{1}}{(α + 1) + (α − 1)Kμ_{1}}

(e. g. [Ar]).

If α ∈ 𝕋 then (1 + α)/(1 − α) is imaginary, i. e. is equal to ic for some real c. Define the spectral shift function u ∈ L^{∞}(𝕋) for the perturbation problem (U_{1} ↦ U_{α}) as

(3)  u = π/2 + arg(Kμ_{1} − ic)

where arg stands for the principal branch of the argument taking values in [0; 2π). All analytic functions in this paper are naturally defined a.e. on 𝕋 by their nontangential boundary values.

Next, we discuss some elementary properties of the spectral shift. Since such properties are well known (see for instance [MP], [S] or [P]), we do not supply all the proofs here.

First important property (which follows from (2)) is that (3) can be extended to

(4)  u = \frac{π}{2} + arg(Kμ_{1} − ic) = \frac{π}{2} − arg(Kμ_{α} + ic).

Formula (4) shows connections between the spectral shift u and spectral measures μ_{1} and μ_{α}. Since in this paper we will mostly operate with spectral measures, we will often call u satisfying (4) the spectral shift of the pair of measures (μ_{1}; μ_{α}) and u satisfying (3) (for some real c) a spectral shift of μ_{1}.

For a pair of probability measures μ_{1} and μ_{α} there exists at most one real c and function u satisfying (4). If a pair of measures (μ_{1}; μ_{α}) possesses a spectral shift, i. e. there exist u (and c) to satisfy (4), then there exist unitary operators U_{1} and U_{α} satisfying (1) such, that μ_{1} and μ_{α} are their spectral measures.

Since μ_{1} is a singular positive measure (U_{1} is a singular operator), function u takes only two values – 0 and π – on 𝕋. For each non-constant non-negative L^{∞}(𝕋) function u, ||u||_{∞} ≤ π there exist unique c ∈ ℝ and a pair of probability measures (ν; γ) for which u is the spectral shift. If u takes only values 0 and π, then ν and γ are singular.

Suppose we have a sequence of functions u_{k} ≥ 0, ||u_{k}||_{∞} ≤ π which converge in measure to function u_{0} as k → ∞. For each u_{k}, k ≥ 0 there exist pairs of measures (μ_{k}; ν_{k}) and constants c_{k} satisfying (4). Then μ_{k} → μ_{0}, ν_{k} → ν_{0} in ∗-weak topology and c_{k} → c_{0} as k → ∞.

We will say that two measures μ and ν are equivalent (μ ∼ ν) if there exists a positive μ-a. e. function f ∈ L^{1}(μ) such that ν = fμ. We will denote by μ|_{I} the restriction of μ on the set I.

If E ⊂ ℝ we will write that

p. v. ∫_{E} dμ(x) < ∞ (> −∞)

if

lim inf_{ϵ→0} ∫_{E\(−ϵ;ϵ)} dμ(x) < ∞ (lim sup > −∞).

Let u_{1} and u_{2} be the phase shifts of the pairs of measures (μ_{1}, ν_{1}) and (μ_{2}, ν_{2}) respectively. Then, as follows from the definition,

(5)  exp(iK(u_{1} − \frac{π}{2} ) + d_{1}) = Kμ_{1} − ic_{1} = (Kν_{1} + ic_{1})^{−1}.

for some real c_{1} and d_{1} and

(6)  exp(iK(u_{2} − \frac{π}{2} ) + d_{2}) = Kμ_{2} − ic_{2} = (Kν_{2} + ic_{2})^{−1}.

for some real c_{2} and d_{2}.

If I is an open subset of 𝕋 and u_{1} = u_{2} on I then, μ_{1}|_{I} ∼ μ_{2}|_{I} and ν_{1}|_{I} ∼ ν_{2}|_{I}. More generally we have the following lemma.

Lemma 4 [P]. Let spectral shift functions u_{1}, u_{2} satisfy (5) and (6) for some singular measures μ_{1,2}, ν_{1,2} and real constants c_{1,2}, d_{1,2}. Let K ⊂ 𝕋 be a measurable set. Put c = exp(d_{1} − d_{2}). Then

i)

p. v. ∫_{[−π;π]} (u_{1}(e^{i(t+x)}) − u_{2}(e^{i(t+x)}))\frac{dt}{t} < ∞

for μ_{1}-a. e. point e^{ix} ∈ K iff the restriction of μ_{1}^{s} on K is absolutely continuous with respect to μ_{2};

p. v. ∫_{[−π;π]} (u_{1}(e^{i(t+x)}) − u_{2}(e^{i(t+x)}))\frac{dt}{t} > −∞

for ν_{1}-a. e. point e^{ix} ∈ K iff the restriction of ν_{1}^{s} on K is absolutely continuous with respect to ν_{2};

ii) if μ_{2}^{s}-a. e. x ∈ K is a Lebesgue point of u_{1} − u_{2} and

p. v. ∫_{[−π;π]} (u_{1}(e^{i(t+x)}) − u_{2}(e^{i(t+x)}))\frac{dt}{t} = f(x) < ∞

for μ_{2}^{s}-a. e. e^{ix} ∈ K then the restriction of μ_{1}^{s} on K is equal to the restriction of ce^{f}μ_{2}^{s} on K; if ν_{2}^{s}-a. e. x ∈ K is a Lebesgue point of u_{1} − u_{2} and

p. v. ∫_{[−π;π]} (u_{1}(e^{i(t+x)}) − u_{2}(e^{i(t+x)}))\frac{dt}{t} = f(x) > −∞

for ν_{2}^{s}-a. e. e^{ix} ∈ K then the restriction of ν_{1}^{s} on K is equal to the restriction of ce^{−f}ν_{2}^{s} on K.

When one of the measures μ_{1,2} is a point mass, Lemma 4 gives (see [P]):

Lemma 5 [MP]. Let u be the spectral shift of a pair of measures (μ; ν). Measure μ has a point mass at e^{ix} iff

(7)  ∫_{x−1}^{x+1}(πχ_{\{e^{it}:t∈(x;x+1)\}}(e^{iy}) − u(e^{iy}))\frac{dy}{y − x} < ∞.

Measure ν has a point mass at x iff

(8)  ∫_{x−1}^{x+1}(πχ_{\{e^{it}:t∈(x−1;x)\}}(e^{iy}) − u(y))\frac{dy}{x − y} < ∞.

We will also need the following important example.

Example 6. Let E ⊂ 𝕋 be a closed set, |E| = 0. Denote E = 𝕋 \ ∪I_{n} where I_{n} are disjoint open arcs. Suppose μ > 0 be a measure such that supp μ = E and

\frac{dμ}{dm} = ∞

at every point of E. Let u be a spectral shift of μ. Then there exist measure ν and real constants c and d such that:

(9)  exp(iK(u − \frac{π}{2} ) + d) = Kμ − ic = (Kν + ic)^{−1}.

We claim that ν is a pure point measure with at most one point mass at each I_{n}.

Indeed, since the derivative of μ with respect to the Lebesgue measure is infinite on E, ℜKμ tends to ∞ non tangentially at every point of E. This together with (9) implies that |Kν| tends to −ic non tangentially at every point of E. Since ν is a singular measure, |Kν| tends to ∞ ν-a. e. Therefore ν(E) = 0. Since, by (9), ν is concentrated at those points where Kμ − ic tends to 0 and Kμ is analytic on 𝕋 outside of E, ν is pure point. Since μ is positive, ℑKμ is monotonic on every I_{n}, and therefore ν has at most one point mass at each of them.

Our main tool in the proof of Theorem 1 is the following lemma.

Lemma 7. Let μ, ν be singular measures on 𝕋, I ⊂ 𝕋 be an open arc, E ⊂ I be a closed set, |E| = 0. Suppose that I ⊂ supp μ and I ⊂ supp ν.

Then for any ϵ > 0 there exist closed subsets F and G of I, and measures μ′ and ν′ satisfying the following conditions

1)|F | = |G| = 0, μ(G) = ν(F) = 0 and E ⊂ F,

2)μ′ ∼ μ|_{F} and ν′ ∼ ν|_{G},

3)the pair (μ′, ν′) possesses a phase shift u,

4)u = π on 𝕋 \ I and

∫_{I}(π − u(ξ))dm(ξ) < ϵ.

Remark. In the statement of Lemma 6, 4) can be replaced with:

4′)u = 0 on 𝕋 \ I and

∫_{I} u(ξ)dm(ξ) < ϵ.

If x, y ∈ 𝕋, x ≠ y, we will denote by (x; y) the open arc going from x to y counterclockwise.

Proof. Denote F_{0} = E. WLOG

(10)  \frac{d(μ|_{F_{0}})}{dm} = ∞

everywhere on F_{0} = E (if this is not true, we can always choose an equivalent measure with this property). Put μ_{0} = μ|_{F_{0}}.

Step 1.

Chose c ∈ ℝ so that

(11)  {ℑKμ_{0} > c} ⊂ I

and

(12)  |{ℑKμ_{0} > c}| < ϵ/2.

There exist a spectral shift function u_{0} and a measure ν_{0} satisfying

(13)  exp(iK(u_{0} − \frac{π}{2} ) + d_{0}) = Kμ_{0} − ic_{0} = (Kν_{0} + ic_{0})^{−1}

for some real c_{0} and d_{0}. Suppose F_{0} = I \ ∪I_{n}^{0} where I_{n}^{0} = (x_{n}^{0}; y_{n}^{0}) are disjoint open arcs. Condition (10), in the same way as in Example 6, implies that ν_{0} = ∑ α_{k}^{0}δ_{z_{k}^{0}} where z_{k}^{0} ∈ I_{n_{k}} for some sequence n_{k}.

Now we will replace point masses z_{k}^{0} with ”pieces” of measure ν.

For each k let V_{k}^{0} ⊂ I_{n_{k}}^{0} be a neighborhood of z_{k}^{0} such that

(14)  | ∪ V_{k}^{0}| < ϵ/4

and

(15)  ∫_{\{x:e^{ix}∈∪V_{k}^{0}\}} \frac{1}{|x − y|}dx < 1/2

for any y ∈ F_{0}.

Choose closed set H_{1}, |H_{1}| = 0 so that ν(H_{1} ∩ V_{k}^{0}) > 0 for any k. Consider measure γ_{1} ∼ ν|_{H_{1}} chosen so that dγ_{1}/dm = ∞ at any point of H_{1}.

Let v_{1} be a phase shift function of measure γ_{1}. In each neighborhood V_{k}^{0} choose points a_{k}^{0} and b_{k}^{0} so that z_{k}^{0} ∈ (a_{k}^{0}; b_{k}^{0}) and

(16)  ∫_{\{arga_{k}^{0}<x<arga_{k}^{0}+1)} \frac{v_{1}(e^{ix})dx}{x − arga_{k}^{0}} = ∞,

(17)  ∫_{\{argb_{k}^{0}−1<x<argb_{k}^{0})} \frac{(π − v_{1}(e^{ix}))dx}{argb_{k}^{0} − x} = ∞

and ν has no point masses at a_{k}^{0}, b_{k}^{0}. Note that it is always possible to choose a_{k}^{0} and b_{k}^{0} satisfying (16) and (17) because γ_{1} has non zero mass on V_{k}^{0}.

Put G_{1} = H_{1} ∩∪(a_{k}^{0}; b_{k}^{0}). Define function u_{1} to be equal to −v_{1} on each interval (a_{k}^{0}; b_{k}^{0}) and to u_{0} outside of ∪(a_{k}^{0}; b_{k}^{0}). Then u_{1} is the phase shift of a pair of measures (μ_{1}; ν_{1}) such that μ_{1} = f_{1}μ|_{F_{0}} + ∑ α_{k}^{1}δ − z_{k}^{1} and ν_{1} = g_{1}ν|_{G_{1}} for some positive functions f_{1} ∈ L^{1}(μ|_{F_{0}}), g_{1} ∈ L^{1}(ν|_{G_{1}}) and for some sequence of points {z_{k}^{1}} ⊂ ∪V_{k}^{0} \ G_{1}.

Indeed, on F_{0} = E we have μ_{0} ∼ μ_{1} by (15) and Lemma 4. On ℝ \ (F_{0} ∪ ⋃[a_{k}^{0}; b_{k}^{0}]) measures μ_{1} and ν_{1} are not supported because u is locally constant there. Also, μ_{1} and ν_{1} do not have point masses at points a_{k}^{0}, b_{k}^{0} because by (16) and (17) conditions (7) and (8) are not satisfied there. Since u_{1} = −v_{1} on each (a_{k}^{0}; b_{k}^{0}), ν_{1} ∼ ν_{0} on ∪(a_{k}^{0}; b_{k}^{0}). Since dν_{0}/dm = ∞ on G_{1}, we have that μ|_{∪(a_{k}^{0};b_{k}^{0})} is a discrete measure with point masses in ∪(a_{k}^{0}; b_{k}^{0}) \ G_{1}.

Step n, n is even.

After step n − 1 we obtained a shift function u_{n−1} of a pair of measures (μ_{n−1}, ν_{n−1}) such that μ_{n−1} = f_{n−1}μ|_{F_{n−2}} + ∑ α_{k}^{n−1}δ_{z_{k}^{n−1}} and ν_{n−1} = g_{n−1}ν|_{G_{n−1}}, where |F_{n−2}| = 0, F_{n−2} ⊃ E, |G_{n−1}| = 0 and {z_{k}^{n−1}} ⊂ ∪V_{k}^{n−1} \ G_{n−1}. Note that the pair (μ_{n−1}, ν_{n−1}) ”almost” satisfies the conditions of the Lemma, except for the discrete part ∑ α_{k}^{n−1}δ_{z_{k}^{n−1}} of μ_{n−1}. We must, therefore, replace those point masses with ”pieces” of μ.

We will do it in the same way as in step 1. First we choose neighborhoods V_{k}^{n} of points z_{k}^{n} such that

(18)  ∪V_{k}^{n} ⊂ ∪V_{k}^{n−1},

(19)  | ∪ V_{k}^{n}| < ϵ/2^{n+1}

and

(20)  ∫_{\{x:e^{ix}∈∪V_{k}^{n}\}} \frac{dx}{|x − y|} < 1/2^{n}

at each point y ∈ F_{n−2} ∪ G_{n−1}. After that inside ∪V_{k}^{n} we choose a closed set H_{n} such that μ(V_{k}^{n} ∩ H_{n}) > 0 for any k. Also let H_{n} satisfy

(21)  μ(∪_{k}V_{k}^{n} \ H_{n}) < 1/n.

We choose measure γ_{n} ∼ μ|_{H_{n}} so that dγ_{n}/dm = ∞ at each point of H_{n}. Denote by v_{n} a shift function of γ_{n}. Inside each neighborhood V_{k}^{n} choose points a_{k}^{n} and b_{k}^{n} so that z_{k}^{0} ∈ (a_{k}^{0}; b_{k}^{0}) and

(22)  ∫_{\{arga_{k}^{n}<x<arga_{k}^{n}+1\}} \frac{v_{n}dx}{x − arga_{k}^{n}} = ∞,

(23)  ∫_{\{argb_{k}^{n}−1<x<argb_{k}^{n}\}} \frac{(π − v_{n})dx}{argb_{k}^{n} − x} = ∞

and μ has no point masses at a_{k}^{n}, b_{k}^{n}.

Put F_{n} = F_{n−2} ∪ [∪(a_{k}^{n}; b_{k}^{n}) ∩ H_{n}]. Define the spectral shift function u_{n} to be equal to v_{n} on ∪(a_{k}^{n}; b_{k}^{n}) and to u_{n−1} elsewhere. Then in the same way as in step 1 we can show that u_{n} is the shift function of the pair (μ_{n}, ν_{n}), where μ_{n} = f_{n}μ|_{F_{n}} for some positive function f_{n} ∈ L^{1}(μ|_{F_{n}}), and ν_{n} = g_{n}ν|_{G_{n−1}} + ∑ α_{k}^{n}δ_{z_{k}^{n}} for some positive function g_{n} ∈ L^{1}(ν|_{G_{n−1}}), positive constants α_{k}^{n} and points z_{k}^{n} from ∪V_{k}^{n} \ F_{n}. Closed sets F_{n} and G_{n} satisfy |G_{n}| = |F_{n}| = 0, F_{n} ⊃ E.

Step n, n is odd.

Our construction is similar to the one we used for the even n. Essentially, we only have to replace μ with ν and ν with μ.

I. e. after step n−1 we obtained a shift function u_{n−1} of a pair of measures (μ_{n−1}, ν_{n−1}) such that μ_{n−1} = f_{n−1}μ|_{F_{n−1}} and ν_{n−1} = g_{n−1}ν|_{G_{n−2}} + ∑ α_{k}^{n−1}δ_{z_{k}^{n−1}}, where |F_{n−1}| = 0, F_{n−1} ⊃ E, |G_{n−2}| = 0, and {z_{k}^{n−1}} ⊂ ∪V_{k}^{n−1} \ F_{n−1}. First we choose neighborhoods V_{k}^{n} of points z_{k}^{n−1} satisfying (17),(18) and (19) for each y ∈ F_{n−1} ∪G_{n−2}. After that inside ∪V_{k}^{n} we choose a closed set H_{n} satisfying (21) and such that ν(V_{k}^{n} ∩ H) > 0 for any k. We choose measure γ_{n} ∼ ν|_{H_{n}} so that dγ_{n}/dm = ∞ at each point of H_{n}. Denote by v_{n} a shift function of γ_{n}. Inside each neighborhood V_{k}^{n} choose points a_{k}^{n} < z_{k}^{n} and b_{k}^{n} > z_{k}^{n} so that

(24)  ∫_{\{arga_{k}^{n}<x<arga_{k}^{n}+1\}} \frac{π − v_{n}dx}{x − arga_{k}^{n}} = ∞,

(25)  ∫_{\{argb_{k}^{n}−1<x<argb_{k}^{n}\}} \frac{v_{n}dx}{argb_{k}^{n} − x} = ∞

and ν has no point masses at a_{k}^{n}, b_{k}^{n}. Put G_{n} = G_{n−2} ∪ ∪(a_{k}^{n}; b_{k}^{n}). Define the spectral shift function u_{n} to be equal to v_{n} on ∪(a_{k}^{n}; b_{k}^{n}) and to u_{n−1} elsewhere. Then in the same way as before we can show that u_{n} is the shift function of the pair (μ_{n}, ν_{n}), where ν_{n} = g_{n}ν|_{G_{n}} for some closed set G_{n}, |G_{n}| = 0 and positive g_{n} ∈ L^{1}(ν|_{G_{n}}), and μ_{n} = f_{n}μ|_{F_{n−1}} + ∑ α_{k}^{n}δ_{z_{k}^{n}} for some set F_{n−1}, |F_{n−1}| = 0, , F_{n−1} ⊃ E, positive function f_{n} ∈ L^{1}(μ|_{F_{n−1}}), positive constants α_{k}^{n} and points z_{k}^{n} from ∪V_{k}^{n} \ G_{n}.

Conclusion of proof.

After step n of our construction we obtain the spectral shift function u_{n} of a pair of measures (μ_{n}, ν_{n}). Let u_{n} and (μ_{n}; ν_{n}) satisfy

exp(iK(u_{n} − \frac{π}{2} ) + d_{n}) = Kμ_{n} − ic_{n} = (Kν_{n} + ic_{n})^{−1}

for some real c_{n}, d_{n}. Note that {u_{n} ≠ u_{n−1}} ⊂ ∪V_{k}^{n} and therefore by (19) the sequence u_{n} converges in measure to a function u. Let u be the spectral shift of a pair of measures (μ′, ν′), i. e.

exp(iK(u − \frac{π}{2} ) + d′) = Kμ′ − ic′ = (Kν′ + ic′)^{−1}

for some real c′, d′. We claim that (μ′, ν′) satisfies the conditions of the lemma with F = Clos ∪ F_{2n} and G = Clos ∪ G_{2n+1}.

Indeed, |F | = 0 because

F = Clos[∪F_{2n}] = ∪F_{2n} ∪ F′

where

(26)  F′ ⊂ [∩_{n} ∪_{k} V_{k}^{n}] \ ∪_{n}H_{n} ⊂ ∩_{n} ∪_{k} V_{k}^{n}.

Since | ∪_{k} V_{k}^{n}| tends to 0 as n → ∞ by (19), |F | = 0.

Also, (26) and (21) imply that μ(F′) = 0. Therefore

(27)  μ|_{F} = μ|_{∪F_{2n}}

and μ(G) = 0. Hence we have to show that μ′ ∼ μ|_{∪F_{2n}}.

Since u_{n} → u in measure, μ_{n} → μ′ in ∗-weak topology, c_{n} → c′ and d_{n} → d′. Consider the sequence {μ_{2n}}. Each measure μ_{2n} is equivalent to μ|_{F_{2n}} where F_{2n} ⊂ F_{2n+2}. Inequality (20) and Lemma 4 imply that

(28)  1 − \frac{1}{2^{n−1}} < \frac{dμ_{2n+2}}{dμ_{2n}} (y) < 1 + \frac{1}{2^{n−1}}

at any point y of F_{2n} for sufficiently big n (note that exp(d_{n−1} − d_{n}) → 1). Since

||μ_{2n+2}|| = ||μ_{2n}|| = ||μ|_{F_{2n}}|| = 1,

(28) implies that for large n

||μ_{2n+2} − μ_{2n}|| < \frac{1}{2^{n−2}} .

Therefore μ_{2n} → μ′ in norm. Since μ_{2n} ∼ μ|_{F_{2n}}, μ′ ∼ μ|_{∪F_{2n}} = μ|_{F}.

Similarly, |G| = 0, ν(F) = 0 and ν′ ∼ ν|_{G}. ▲

Proof of Theorem 1. Since U and V are completely non equivalent, spectral measures of U and V are mutually singular. Thus, for any pair of singular probability measures (μ; ν) on 𝕋 such that μ ⊥ ν we have to show that there exists an equivalent pair (μ_{0}; ν_{0}) possessing a phase shift u_{0}. We will do it by constructing u_{0}. The main part of our construction will consist of recursive applications of Lemma 7.

Let M and N be disjoint subsets of 𝕋 such that |M| = |N| = 0, μ(M) = 1, μ(𝕋\M) = 0 and ν(N) = 1, ν(𝕋 \ N) = 0.

Step 1.

Let E_{1} be a closed subset of M such that

(29)  μ(M \ E) < 1.

By Lemma 7 there exist closed sets F_{1} and G_{1} such that |F_{1}| = |G_{1}| = 0 , μ(G_{1}) = ν(F_{1}) = 0, E_{1} ⊂ F_{1} and (f_{1}μ|_{F_{1}}; g_{1}ν|_{G_{1}}) possesses a phase shift, for some positive summable f_{1} and g_{1}.

Step n, n is even.

After step n − 1 we obtained closed sets F_{n−1} and G_{n−1} of zero measure such that μ(G_{n−1}) = ν(F_{n−1}) = 0 and (f_{n−1}μ|_{F_{n−1}}; g_{n−1}ν|_{G_{n−1}}) possesses a phase shift u_{n−1}, for some positive summable functions f_{n} and g_{n}. Denote 𝕋 \ (F_{n−1} ∪ G_{n−1}) = ∪_{k}I_{k}^{n} where I_{k}^{n} are disjoint open arks.

Choose closed subset E_{n} of N so that E_{n} ⊂ ∪_{k}I_{k}^{n} and

(30)  ν(N \ (G_{n−1} ∪ E_{n})) < 1/n.

Note that such a choice of E_{n} is possible because ν(F_{n−1}) = 0. Since E_{n} is closed, there exist open arcs J_{k}^{n} such that ClosJ_{k}^{n} ⊂ I_{k}^{n} (k = 1, 2, ...), E_{n} ⊂ ∪J_{k}^{n} ⊂ ∪I_{k}^{n} and

(31)  dist(∪_{k}J_{k}^{n}, F_{n−1} ∪ G_{n−1}) = δ_{n} > 0.

By Lemma 7, for each k there exist closed sets F_{k}^{n}, G_{k}^{n} ⊂ J_{k}^{n} such that G_{k}^{n} ⊃ E_{n} ∩ J_{k}^{n} and (f_{k}^{n}μ|_{F_{k}^{n}}; g_{k}^{n}ν|_{G_{k}^{n}}) possesses a spectral shift u_{k}^{n}. We choose u_{k}^{n} to satisfy condition 4) from the statement of Lemma 7 for I = J_{k}^{n} and

(32)  ϵ = ϵ_{k}^{n} = δ_{n}/2^{n+k}

if u_{n−1} = π on I_{k}^{n} and choose u_{k}^{n} to satisfy condition 4′) if u_{n−1} = 0 on I_{k}^{n} (note that u_{n−1} is constant on each I_{k}^{n}).

Define u_{n} to be equal to u_{k}^{n} on each I_{k}^{n}. We claim that then u_{n} is a phase shift of a pair of measures (f_{n}μ|_{F_{n}}; g_{n}ν|_{G_{n}}) where f_{n} and g_{n} are positive summable functions, F_{n} = F_{n−1} ∪ Clos ∪_{k} F_{k}^{n} and G_{n} = G_{n−1} ∪ Clos ∪_{k} G_{k}^{n}. Indeed, let u_{n} be the phase shift of a pair (μ_{n}; ν_{n}). Recall that on each J_{k}^{n}, where u_{n−1} was equal to π, functions u_{k}^{n} satisfy condition 4) from Lemma 7, and on those J_{k}^{n}, where u_{n−1} was equal to 0, functions u_{k}^{n} satisfy condition 4′), with ϵ satisfying (32). Therefore

(33)  |{u_{n} ≠ u_{n−1}}| < δ_{n}/2^{n} < 1/2^{n−1}.

Also by (32) and (33)

(34)  ∫_{[−π;π]} |u_{n}(e^{i(t+x)}) − u_{n−1}(e^{i(t+x)})|\frac{dt}{|t|} < ∑_{k} ϵ_{n}^{k}/δ_{n} < 1/2^{n}

for any point e^{ix} ∈ F_{n−1} ∪ G_{n−1}. Hence, by Lemma 4, μ_{n}|_{F_{n−1}} ∼ μ|_{F_{n−1}} and ν_{n}|_{F_{n−1}} ∼ ν|_{F_{n−1}}. Also, on each J_{k}^{n}, since u_{n} is equal to u_{k}^{n}, we have that μ_{n}|_{I_{k}^{n}} ∼ μ|_{F_{k}^{n}} and ν_{n}|_{I_{k}^{n}} ∼ ν|_{G_{k}^{n}}. Therefore μ_{n} = f_{n}μ|_{F_{n}} and ν_{n} = g_{n}ν|_{G_{n}} for some positive summable f_{n} and g_{n}. By construction G_{n} ⊃ E_{n}. Also, since μ(G_{n−1}) = ν(F_{n−1}) = 0, μ(G_{k}^{n}) = ν(F_{k}^{n}) = 0 (by condition 1) of Lemma 7), Clos ∪ F_{k}^{n} \ ∪F_{k}^{n} ⊂ (F_{n−1} ∪ G_{n−1}) and Clos ∪ G_{k}^{n} \ ∪G_{k}^{n} ⊂ (F_{n−1} ∪ G_{n−1}), we have μ(G_{n}) = ν(F_{n}) = 0.

Step n, n is odd.

The construction here is essentially the same as in the previous case.

After step n − 1 we obtained closed sets F_{n−1} and G_{n−1} of zero measure such that μ(G_{n−1}) = ν(F_{n−1}) = 0 and (f_{n−1}μ|_{F_{n−1}}; g_{n−1}ν|_{G_{n−1}}) possesses a phase shift u_{n−1}, for some positive summable functions f_{n} and g_{n}. Let us again denote 𝕋\(F_{n−1}∪G_{n−1}) = ∪_{k}I_{k}^{n}.

Choose closed subset E_{n} of M so that E_{n} ⊂ ∪_{k}I_{k}^{n} and

(35)  μ(M \ (F_{n−1} ∪ E_{n})) < 1/n.

After that in the same way as above we construct the phase shift function u_{n} of a pair (μ_{n}; ν_{n}) = (f_{n}μ|_{F_{n}}; g_{n}ν|_{G_{n}}). Function u_{n} satisfies (33) and (34). The sets F_{n} and G_{n} are zero-measure closed sets such that F_{n} ⊃ E_{n} and μ(G_{n}) = ν(F_{n}) = 0.

Conclusion of proof.

This part is similar to the corresponding part in the proof of Lemma 7.

By (33) u_{n} tend in measure to a function u_{0}. Let u_{0} be the shift function of a pair (μ_{0}; ν_{0}). Then μ_{n} and ν_{n} weakly converge to μ_{0} and ν_{0} respectively. Since u_{n} → u_{0}, condition (34) and Lemma 4 imply that

(36)  1 − 1/2^{n+1} < c_{n} \frac{f_{n}}{f_{n+1}} < 1 + 1/2^{n+1}

μ_{n}-a. e. for some c_{n} → 1. Since all our measures have norm 1, (36) implies that μ_{n} → μ_{0} in norm. Since μ_{n} ∼ μ|_{F_{n}}, μ_{0} << μ. Since (by (36)) μ(F_{n}) > 1 − 1/2^{n−1} and

(37)  μ_{n} = \frac{f_{n}}{f_{n+1}} μ_{n+1}|_{F_{n}},

where functions f_{n} and f_{n+1} satisfy (36), μ << μ_{0}. Hence μ ∼ μ_{0}. Similarly ν ∼ ν_{0}. ▲

Remark. A closed set K can replace 𝕋 in the statement of Theorem 1 iff it is not very ”porous”. More precisely, the following generalization of Theorem 1 can be proved. We will state it in self-adjoint settings.

Definition. We will say that two disjoint sets of real numbers A and B are well-mixed if they satisfy the following conditions:

1) for any two points x, y ∈ A each of the sets (x; y) and ℝ \ [x; y] contains at least one point from B.

2) for any two points x, y ∈ B each of the sets (x; y) and ℝ \ [x; y] contains at least one point from A.

We will denote by σ_{p.p.}(A) the set of eigenvalues of operator A.

Theorem 8. Let K ⊂ 𝕋 be a closed set with no isolated points. Denote by I_{1}, I_{2}, ... the disjoint open intervals I_{k} = (x_{k}; y_{k}) such that K = ℝ \ ⋃ I_{n}. Denote ∂I = {x_{1}, y_{1}, x_{2}, y_{2}, ...}.

Then the following two conditions are equivalent:

i)Any two completely non equivalent cyclic self-adjoint operators A and B such that the sets

(σ_{p.p.}(A) \ σ_{p.p.}(B)) ∩ ∂I

and

(σ_{p.p.}(B) \ σ_{p.p.}(A)) ∩ ∂I

are well-mixed and σ(A) = σ(B) = K are unitarily equivalent up to a rank one perturbation.

ii) If y ∈ K \ {x_{1}, y_{1}, x_{2}, y_{2}, ...} then

∫_{(y−1;y+1)\K} \frac{dx}{|y − x|} < ∞;

if y = x_{k} or y = y_{k} for some k ∈ ℕ then

∫_{(y−1;y+1)\(K∪I_{k})} \frac{dx}{|y − x|} < ∞.

In particular if the spectrum of A and B is an interval [a; b] such that

{a, b} ⊄ (σ_{p.p.}(A) \ σ_{p.p.}(B)) and {a, b} ⊄ (σ_{p.p.}(B) \ σ_{p.p.}(A)) ,

then A is equivalent to B up to a rank one perturbation.

Theorem 8 was proved in [P] for pure point operators. One can easily combine the proof with the methods of the present paper to prove the general case. Here is another (simpler) way to generalize Theorem 1. We prohibit our operators to have eigenvalues at the endpoints of the complementary intervals.

Theorem 9. Let K ⊂ ℝ be a closed set. Denote by I_{1} = (x_{1}; y_{1}), I_{2} = (x_{2}; y_{2}), ... disjoint open intervals such that K = ℝ \ ⋃ I_{n}. Let A and B be two completely non equivalent self-adjoint cyclic operators. Suppose that σ(A) = σ(B) = K and σ_{p.p.}(A) ∩ {x_{1}, y_{1}, x_{2}, y_{2}, ...} = σ_{p.p.}(B) ∩ {x_{1}, y_{1}, x_{2}, y_{2}, ...} = ∅. Then A and B are equivalent up to a rank one perturbation.

This Theorem was also proved in [P] in the pure point case.

References

[A] A. B. Aleksandrov, Private communications.

[Ar] N. Aronszajn, On a problem of Weyl in the theory of singular Sturm- Liouville equations, Amer. J. Math. 79 (1957), 597-610.

[BY] M. Sh. Birman and D. R. Yafaev, The spectral shift function. The work of M. G. Krein and its further development., St. Petersburg Math. J. 4 (1993), 833-870.

[CP] R. W. Carey and J. D. Pincus, Unitary equivalence modulo the trace class for self-adjoint operators, Amer J. Math. 98 (1976), 481-514.

[K] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966.

[MP] M. Martin and M. Putinar, Lectures on Hyponormal operators, vol. 39, Operator Theory: Advances and Applications, 1989.

[P] A. Poltoratski, The Krein spectral shift and rank one perturbations of spectra, Preprint.

[S] B. Simon, Spectral analysis of rank one perturbations and applications, Proc. 1993 Vancouver Summer School in Mathematical Physics.

Dept. of Math., Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139

E-mail address: agp@math.mit.edu