UNCONDITIONAL BASES OF INVARIANT SUBSPACES OF A CONTRACTION WITH FINITE

DEFECTS.

S. TREIL

Abstract. The main result of the paper is that a system of invariant subspaces of a (completely non-unitary) Hilbert space contraction T with finite defects (rank(I−T^{∗}T) < ∞, rank(I−T T^{∗}) < ∞) is an unconditional basis (Riesz basis) if and only if it is uniformly minimal.

Results of such type are quite well known: for a system of eigenspaces of a contraction with defects 1–1 it is simply the famous Carleson interpolation theorem. For general invariant subspaces of operators with defects 1–1 such theorem was proved by V. I. Vasyunin. Then partial results for the case of finite defects were obtained by the author.

The present paper solves the problem completely.

Part of the paper was written in when the author was visiting Mathematical Science Research Institute, Berkeley, CA .

Contents

Notation 3 1. Introduction 4 1.1. A little more geometry. 5 Acknowledgment 8 2. The main result 8 3. Idea of the proof of the main result. 10 4. S- and S^{∗}-invariant subspaces of H^{2}(E) ⊕ L^{2}(E_{∗}) 11 5. Embedding theorems 12 6. Some auxiliary results. 13 7. Reduction to a scalar embedding theorem 17 8. Geometry of vector reproducing kernels. 20 9. Carleson measures and bases of reproducing kernels. 22 10. Carleson contours 23 11. The main construction 26 11.1. Approximation by interpolating Blaschke products 26 11.2. Choice of ε. 29 11.3. Reduction to “almost scalar” case. 29 12. The main Lemma 30 Idea of the proof of the main lemma. 31 References 35

Notation

ℂ  — complex plane
𝔻  — open unit disc, 𝔻 =^{def} {ξ ∈ ℂ : |ξ| < 1};
𝕋  — unit circle, 𝕋 =^{def} ∂𝔻;

m, | . | — the normalized (m(𝕋) = 1) Lebesgue measure on

𝕋;

f̂(k)

— kth Fourier coefficient of the function f defined on the unit circle 𝕋; f̂(k) =^{def} ∫_{𝕋} f(z)z^{−k}dm(z);

P_{E}  — orthogonal projection onto a closed subspace E;
L{… }  — linear span of the set …;
span{… } — closed linear span of a set …;
{H^{2}, H_{−}^{2}, \atop H^{∞}}  — Hardy classes,

H^{2} =^{def} {f ∈ L^{2} : f̂(k) = 0 for k < 0} , H_{−}^{2} =^{def} L^{2} ⊖ H^{2}

H^{∞} =^{def} {f ∈ L^{∞} : f̂(k) = 0 for k < 0} .

Hardy classes can be identified with the spaces of functions analytic in the unit disc 𝔻, see [1].

L^{∞}(E→E_{∗})

— the set of all bounded measurable functions on 𝕋 with values in the space of operators from E to E_{∗};

{L^{2}(E), \atop H^{2}(E)} — vector Lebesgue and Hardy classes consisting of functions with values in a (separable Hilbert) space E;

H^{∞}(E→E_{∗}) — the operator Hardy class H^{∞};

ℙ_{+}, ℙ_{−} — orthogonal projections onto H^{2} and H_{−}^{2} respec-

tively;

k_{λ}  — normalized (∥k_{λ}∥_{2} = 1) reproducing kernel in H^{2},

k_{λ}(z) := \frac{(1 − |λ|^{2})^{1/2}}{1 − \overline{λ}z} .

1. Introduction

The main result of the paper is that a system of invariant subspaces of a (completely non-unitary) Hilbert space contraction T with finite rank defect operators (rank(I − T^{∗}T) < ∞, rank(I − T T^{∗}) < ∞) is an unconditional basis if and only if it is uniformly minimal. Let us recall some definitions and a background, that explains why this problem is interesting and important.

We restrict ourselves to the case of a separable Hilbert space, although most of the definitions make sense perfectly in a Banach space as well.

Let H be a separable Hilbert space, and let E_{n} be closed subspaces of H. Here and below, unless the converse is stated, by a subspace we always mean a closed linear subspace. Suppose that the subspaces E_{n} are linearly independent, that means that among all finite linear combinations ∑ f_{n}, f_{n} ∈ E_{n}, only trivial ones (all f_{n} = 0) equal 0. Assume also that the system of subspaces E_{n} is complete, i.e. H = span{E_{n} : n = 1, 2…. }, where span{… } stands for the closed linear span of the set {… }. If the collection {E_{n} : n = 1, 2, … } is finite, then it is a standard linear algebra fact that the system {E_{n} : n = 1, 2, … } is a basis for H, that means any vector f ∈ H admits a unique representation

f = ∑ f_{n} ,  f_{n} ∈ E_{n} .  (1.1)

If the system {E_{n} : n = 1, 2, … } is infinite, the things become more complicated. It is natural to define a basis as a system such that any vector f admits a representation (1.1) where the series converges in the norm of H. And that is how bases in infinite-dimensional spaces are defined. Clearly, to be a basis the system has to be linearly independent and complete. Unfortunately (or maybe fortunately) for infinite systems it is not sufficient, and we shall show that below.

There is a simple description of infinite bases due to Banach. To formulate it we need some notation. Given a complete linearly independent system of subspaces E_{n}, one can define (on a dense set L{E_{n} : n = 1, 2, … } consisting of all finite linear combinations of f_{n} ∈ E_{n}) projections P_{n}

P_{n}(∑ f_{k}) = ∑_{k=1}^{n} f_{k} ,  f_{k} ∈ E_{k}.  (1.2)

Theorem 1.1 (Banach Basis Theorem). A complete linearly independent system of subspaces E_{n}, n = 1, 2, … is a basis if and only if all projections P_{n} defined above are uniformly bounded, ∥P_{n}∥ ≤ C < ∞.

In the definition of a basis the ordering of subspaces E_{n} is essential, because the convergence of a series usually depends on the order. If we require that in the decomposition (1.1) the series converges unconditionally, i.e. independently of order, we obtain what is called an unconditional basis.

Unconditional bases admit a description similar to that given by the Banach Basis Theorem above. For a finite subset σ of ℕ, one can define on L{E_{n} : n = 1, 2, … } the projections P_{σ}

P_{σ}(∑ f_{k}) = ∑_{k∈σ} f_{k} , f_{k} ∈ E_{k}.

Theorem 1.2. A complete linearly independent system of subspaces E_{n}, n = 1, 2, … is an unconditional basis if and only if all projections P_{σ} defined above are uniformly bounded, sup_{σ} ∥P_{σ}∥ ≤ C < ∞, where supremum is taken over all finite subsets σ of ℕ.

The above two theorems hold in any Banach space. In a Hilbert space we have the wonderful notion of orthogonal basis. It is natural to consider bases that are orthogonal “up to isomorphism”, i.e. that can be transformed into an orthogonal basis by a bounded invertible operator. Such bases are called Riesz bases. An equivalent definition is that a Riesz basis is a basis that is orthogonal in an equivalent Hilbert norm.

It turns out that for a Hilbert space a system of subspaces is an unconditional basis if and only if it is a Riesz basis, [1, 4]. So in the sequel we will use both terms. We need a couple more of important definitions. With each Riesz basis one can associate a bounded invertible operator J that maps this basis into an orthogonal one. Such an operator (which is clearly not unique) is called an orthogonalizer of the system of subspaces.

It is convenient to “normalize” an orthogonalizer by assuming that its restriction on each subspace E_{n} is an isometry. Such “normalized” orthogonalizer is now unique up to a unitary multiplier on the left, and in what follows by orthogonalizer we always mean a normalized orthogonalizer.

If J is a (normalized) orthogonalizer for a Riesz basis {E_{n}}_{n=1}^{∞}, then the quantity C({E_{n}}_{n=1}^{∞}) = ∥J ∥ · ∥J^{−1}∥ could serve as a measure of the non-orthogonality of {E_{n}}_{n=1}^{∞}.

1.1. A little more geometry. We need a couple more of notions that generalize linear independence to the case of an infinite system of subspaces. We say that a system {E_{n}}_{n=1}^{∞} is minimal if all projections P^{n} defined on L{E_{n} : n = 1, 2, … }

P^{n}(∑ f_{k}) = f_{n} , f_{k} ∈ E_{k}

are bounded, and uniformly minimal if all P^{n} are uniformly bounded: sup_{n} ∥P^{n}∥ =: 1/δ < ∞. The constant δ is called the constant of uniform minimality of the system {E_{n}}_{n=1}^{∞}.

One can easily see that the constant of uniform minimality δ admits a very simple geometrical interpretation, namely

δ = inf_{n} inf_{f∈E_{n}, ∥f∥=1} dist{f, L{E_{k} : k ≠ n} }  (1.3)

In other words, δ is the greatest lower bound for the sine of the angle between E_{n} and L{E_{k} : k ≠ n} }.

If a system {E_{n}}_{n=1}^{∞} is minimal, one can define the so called biorthogonal or dual system {E′_{n}}_{n=1}^{∞},

E′_{n} = (P^{n})^{∗} H .

It is easy to see that a minimal system is uniformly minimal if and only if its dual is uniformly minimal.

It is also trivial that any minimal system is linearly independent, any uniformly minimal system is minimal, any basis is uniformly minimal, and any Riesz basis is a basis. In general there are great gaps between any two of the above properties. For example consider a system of one dimensional spaces E_{n} = L{z^{n}}, n ∈ ℕ in its closure in the weighted space L^{2}(w), where w is a non-negative L^{1} function (weight) on the unit circle 𝕋. Then the system is:

1. Linearly independent iff w ≢ 0;

2. Minimal iff log w ∈ L^{1};

3. Uniformly minimal iff 1/w ∈ L^{1};

4. Basis iff w satisfies Muckenhoupt (A_{2}) condition, i.e. iff

sup_{I} (\frac{1}{|I|} ∫_{I} w) · (\frac{1}{|I|} ∫_{I} w^{−1}) < ∞;

5. Unconditional (Riesz) Basis iff w ∈ L^{∞}, 1/w ∈ L^{∞}.

Let us explain that a little bit. The statement 1 is trivial. Let us prove 2. The condition log w ∈ L^{1} is equivalent to

dist_{L^{2}(w)}(1, L{z^{n} : n ≥ 1} > 0,

that means the projection P^{0}, P^{0}(∑_{0}^{∞} α_{n}z^{n}) = α_{0} · 1 is bounded.

Since for the projection P^{1}, P^{1}(∑_{0}^{∞} α_{n}z^{n}) = α_{1} · z we have P^{1}f = zP^{0}\overline{z}(I − P^{0})f, the projection P^{1} is bounded. Similarly all projections P^{n}, P^{n}(∑_{0}^{∞} α_{k}z^{k}) = α_{n}z^{n} are bounded.

To prove 3 one needs to notice that the system of positive exponents {z^{n}}_{0}^{∞} is uniformly minimal if and only if the system of all exponents {z^{n}}_{−∞}^{∞} is. But for the last system the norms of all projections P^{n}, P^{n} ∑ α_{k}z^{k} = α_{n}z^{n} coincide, and it is an easy exercise on Cauchy– Schwartz inequality to check that ∥P^{0}∥^{2} = ∫_{𝕋} w · ∫_{𝕋} w^{−1}.

By Theorem 1.1 the system {z^{n}}_{0}^{∞} is a basis if and only if sup_{n} ∥P_{n}∥ < ∞, where P_{n}(∑ α_{k}z^{k}) = ∑_{k=0}^{n} α_{k}z^{k}. The last is equivalent to the boundedness of the Riesz projection P_{+}, P_{+} ∑ α_{n}z^{n} = ∑_{0}^{∞} α_{n}z^{n} in the weighted space L^{2}(w). That is equivalent to the boundedness of the Hilbert transform T, T = −iP_{+} + i(I − P_{+}) and it is well known (see [6]) that T is bounded if and only if w satisfy the Muckenhoupt (A_{2}) condition.

To prove 5 notice that if {z^{n}}_{0}^{∞} is a Riesz basis, the norm

∥ ∑ α_{k}z^{k}∥_{L^{2}(w)} is equivalent to the norm (∑ |α_{k}|^{2})^{1/2} = ∥ ∑ α_{k}z^{k}∥_{L^{2}}. The last is possible if and only if w, 1/w ∈ L^{∞}.

As one can see here the gap between Riesz Bases and uniformly minimal systems is as big as the gap between L^{∞} and L^{1}

However sometimes the gap does not exist. For example, let us consider a system {k_{λ}}_{λ∈σ} of (normalized) reproducing kernels in the standard Hardy space H^{2}

k_{λ}(z) = \frac{(1 − |λ|^{2})^{1/2}}{1 − \overline{λ}z} ,

where σ is a countable subset of the unit disk 𝔻. Note that ∥k_{λ}∥_{2} = 1. Then the corresponding system of one dimensional subspaces E_{λ} = L{k_{λ}} is a Riesz basis (in its closed linear span, not in all H^{2}) if and only if it is uniformly minimal (as it was mentioned above the “only if” part is trivial).

Let us explain that in more details. It is not a difficult exercise to compute the norm of the projection P^{λ}, which is the projection onto L{k_{λ}} with kernel span{k_{μ} : μ ∈ σ \ {λ} }. Namely, cf [1, Lecture VI]

∥P^{λ}∥ = ( ∏_{μ∈σ\{λ}} |b_{μ}(λ)|)^{−1} ,

where b_{λ} is a Blaschke factor, b_{λ}(z) =^{def} (|λ|/λ)(λ − z)(1 − \overline{λ}z)^{−1}. The condition sup_{λ} ∥P^{λ}∥ < ∞ is then nothing but the famous Carleson condition

inf_{λ∈σ} ∏_{μ∈σ\{λ}} |b_{μ}(λ)| =: δ > 0 ,

that is a necessary and sufficient condition for free interpolation in H^{∞} (and δ is exactly the constant of uniform minimality of the system {k_{λ} : λ ∈ σ}, see Section 1.1). Therefore given a finite subset σ_{1} ⊂ σ one can find a function φ ∈ H^{∞}, ∥φ∥_{∞} ≤ C = C(δ), such that φ(λ) = 1 for λ ∈ σ_{1} and φ(λ) = 0 for λ ∈ σ \ σ_{1}. Then

ℙ_{+}\overline{φ}k_{λ} = k_{λ} λ ∈ σ_{1} and ℙ_{+}\overline{φ}k_{λ} = 0 λ ∈ σ \ σ_{1}

Therefore for any f ∈ L{k_{λ} : λ ∈ σ} the projection P_{σ_{1}} onto span{k_{λ} : λ ∈ σ_{1}} with kernel span{k_{λ} : λ ∈ σ \ σ_{1}} is given by the formula

P_{σ_{1}}f = ℙ_{+}\overline{φ}f

and therefore ∥P_{σ_{1}}∥ ≤ ∥φ∥_{∞} ≤ C = C(δ). So by Theorem 1.2 the system {k_{λ} : λ ∈ σ} forms a Riesz basis in its closed linear span.

Let us note that it is possible to obtain directly from the above Carleson interpolation condition that the system {k_{λ} : λ ∈ σ} is a Riesz basis and then get the result about free interpolation in H^{∞} using Sarason’s theorem (a particular case of the Sz.-Nagy–Foiaş commutant lifting theorem). This scheme was realized in [1, Lectures VI, VII].

The connection between free interpolation in H^{∞} and the geometry (an unconditional basis property) of corresponding sequences of reproducing kernels in H^{2} was first independently noticed by Nikolskii and Pavlov [7] and Katsnelson [2] in late 60’s.

Acknowledgment. The author is grateful to the referee who pointed his attention to a gap in the proof in the initial version of the paper.

2. The main result

As everybody familiar with the Sz.-Nagy–Foiaş functional model for a contraction knows, the system of reproducing kernels {k_{λ} : λ ∈ σ} has a very simple operator–theoretic interpretation. It is simply a general form (up to unitary equivalence) of a system of eigenvectors of a completely non-unitary contraction with defect indices 1–1.

Let us remind that a contraction T is called completely non-unitary if there exists no reducing subspace such that the restriction of T on this subspace is a unitary operator. Since the theory of unitary operators is very well developed it seems very natural to “cut off” such unitary part of T (if it exists) and consider only completely non-unitary contractions.

For a contraction T one can consider the so called defect operators D = (I − T^{∗}T)^{1/2} and D_{∗} = (I − T T^{∗})^{1/2} and the corresponding defect indices d = rank D, d_{∗} = rank D_{∗}.

It seems natural to consider general invariant subspaces of a contraction, and also not only a contraction with defect indices 1–1, but with finite defect indices. It was proved by Vasyunin [11] that for an operator with defect indices 1–1 a complete system of invariant subspaces is a Riesz basis if and only if it is uniformly minimal. Then it was proved by the author that the same result holds for a system of eigenvectors of a contraction with finite defects [8, 9], and for a system of invariant subspaces of a contraction whose characteristic function is co-inner [10].

The main result of this paper is the following theorem.

Theorem 2.1. Let T be a completely non-unitary Hilbert space contraction with finite defect indices d, d_{∗}, and {E_{n} : n = 1, 2, … } be a complete system of T -invariant subspaces. Then the system {E_{n} : n = 1, 2, … } is a Riesz basis if and only if it is uniformly minimal

Remark 2.2. Let us note that in the paper we do not discuss the completeness at all. There are many results about when a system of eigenvectors, or of generalized eigenvectors of an operator is complete. We are interested in the question when a system is a basis in its own closed linear span. So in the statement of the above theorem we can omit the assumption that the system {E_{n} : n = 1, 2, … } is complete. In this case the conclusion will be “the system is a Riesz basis in its own linear span”

As it was already said this theorem for an operator with co-inner characteristic function was proved by the author in [10]. The presentation below mostly follows [10]. The main difference is the author now understands how to cope with the outer part of the characteristic function. It turns out that one does not need anything fancy to deal with the outer part: it can be shown that by pretty trivial reasons the influence of the outer part can be eliminated.

Practically all new (comparing with [10]) techniques is contained in Sections 6, 7. However, presenting only these sections and saying that one can easily modify the construction in [10] to get the result, is a sure way to make an unreadable paper that can be understood besides the author by a couple of experts at best. So for the sake of readability and to make the paper self-contained the author needs to repeat the main lines of [10].

3. Idea of the proof of the main result.

The very general idea of the proof is quite simple. We need the following well-known theorem (see [1, 4]) about Riesz bases.

Theorem 3.1. A complete system of subspaces E_{n}, n = 1, 2, … is a Riesz basis if and only if it is uniformly minimal and the following two “embedding theorems”

1. ∑_{n} ∥P_{E_{n}}f∥^{2} ≤ C∥f∥^{2};

2. ∑_{n} ∥P_{E′_{n}}f∥^{2} ≤ C∥f∥^{2}, hold for the system and its biorthogonal (dual) system E′_{n}, n = 1, 2, . . ..

The main part of the proof is to show that if a system of invariant subspaces E_{n}, n = 1, 2, … (of a completely non-unitary contraction T with finite defects — I will omit that in the sequel) is uniformly minimal, then the embedding theorem 1 from the theorem holds. If we do that we are done, because the dual system E′_{n}, n = 1, 2, … is also uniformly minimal (trivially) and E′_{n} are clearly invariant subspaces of the adjoint operator T^{∗}, which is of course a completely non-unitary contraction with finite defects.

According to Sz.-Nagy–Foiaş functional model (see [5]) any (completely non-unitary) contraction in a separable Hilbert space admits the following representation (is unitarily equivalent to the following operator):

Consider two Hilbert spaces E, E_{∗} and construct a space H = H^{2}(E) ⊕ L^{2}(E_{∗}). Let S denote the shift operator in H, i.e. the multiplication by the independent variable z,

S ( \begin{matrix}f(z) \\ g(z) \end{matrix} ) = ( \begin{matrix}zf(z) \\ zg(z) \end{matrix} )

and let S^{∗} be the adjoint of S

S^{∗} ( \begin{matrix}f(z) \\ g(z) \end{matrix} ) = ( \begin{matrix}(f(z) − f(0))/z \\ \overline{z}g(z) \end{matrix} )

Sz.-Nagy–Foiaş functional model theory says that any completely non-unitary contraction can be represented as a restriction of the backward shift S^{∗} to its invariant subspace K ⊂ H^{2}(E) ⊕ L^{2}(E_{∗}), where dim E = d = rank(1 − T^{∗}T), dim E_{∗} = d_{∗} = rank(1 − T T^{∗}),

So we reduced our main theorem (Theorem 2.1) to the following one:

Theorem 3.2. Let E_{n}, n = 1, 2, … be a system of S^{∗}-invariant subspaces in H^{2}(E) ⊕ L^{2}(E_{∗}), dim E < ∞, dim E_{∗} < ∞. Suppose that the system E_{n}, n = 1, 2, …, is uniformly minimal, and let δ be the constant of uniform minimality. Then the embedding theorem

∑_{n} ∥P_{E_{n}}f∥^{2} ≤ C∥f∥^{2} , ∀f ∈ H^{2}(E) ⊕ L^{2}(E_{∗})

holds, where C = C(δ, dim E, dim E_{∗}).

To prove the theorem we first need to remind the reader the description of S^{∗}-invariant subspaces.

4. S- and S^{∗}-invariant subspaces of H^{2}(E) ⊕ L^{2}(E_{∗})

We need the following well known (see [1]) description of S-invariant subspaces in H^{2}(E) ⊕ L^{2}(E_{∗}). Let us represent the space H^{2}(E) ⊕

L^{2}(E_{∗}) as the set ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ) consisting of all columns ( \begin{matrix}f \\ g \end{matrix} ), f ∈

H^{2}(E), g ∈ L^{2}(E_{∗}).

Theorem 4.1. Let M be an S-invariant subspace of ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ),

SM ⊂ M. Then there exists a contractive analytic operator function Θ ∈ H^{∞}(E_{1}→E), an operator function Δ ∈ L^{∞}(E_{1}→E_{∗}) and a function P ∈ L^{∞}(E_{∗}→E_{∗}) whose values are orthogonal projections in E_{∗}, such that Θ^{∗}Θ + Δ^{∗}Δ = I, RangeΔ(ξ) ⊥ RangeP (ξ) a.e. on 𝕋 and

M = M_{Θ,Δ,P} = ( \begin{matrix}Θ \\ Δ \end{matrix} ) H^{2}(E_{1}) ⊕ ( \begin{matrix}0 \\ P L^{2}(E_{∗}) \end{matrix} )

Corollary 4.2. Any S^{∗} invariant subspace K ⊂ ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ) admits

a representation

K = K_{Θ,Δ,P} = ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ) ⊖ M_{Θ,Δ,P}

where M_{Θ,Δ,P} and Θ, Δ, P are from the previous theorem.

Remark 4.3. Clearly if the functions Θ, Δ, P satisfy the assumptions of Theorem 4.1 then the corresponding subspace M_{Θ,Δ,P} is S-invariant. So, Theorem 4.1 and Corollary 4.2 give us a complete description of S-invariant and S^{∗}-invariant subspaces respectively.

Remark 4.4. If E_{∗} = {0} Theorem 4.1 and Corollary 4.2 give a complete description of S and S^{∗}-invariant subspaces of H^{2}(E). In this case Θ is an inner function, i.e. Θ(ξ) is an isometry a.e. on 𝕋, Δ ≡ 0, P ≡ 0 and

M = M_{θ} = ΘH^{2}(E_{1}), K = K_{Θ} = H^{2}(E) ⊖ ΘH^{2}(E_{1})

5. Embedding theorems

For inner functions Θ_{n} the following results were proved in [10].

Theorem 5.1. Let θ_{n}, n ∈ ℕ be a sequence of (scalar valued) inner functions in H^{∞}. Let K_{θ_{n}} := H^{2} ⊖ θ_{n}H^{2} and let P_{θ_{n}} denote the orthogonal projection in H^{2} onto K_{θ_{n}}. Then the embedding theorem

∑_{n} ∥P_{θ_{n}}f∥^{2} ≤ C∥f∥^{2} ∀f ∈ H^{2}  (5.1)

holds if and only if

sup_{λ∈𝔻} ∑_{n}(1 − |θ_{n}(λ)|^{2}) = C < ∞ .  (5.2)

Moreover the norm of the embedding operator can be estimated above by a constant depending only on C.

The following generalization of Theorem 5.1 to the case of matrix valued inner functions also can be found in [10].

Theorem 5.2. Let Θ_{n}, n ∈ ℕ be a sequence of matrix valued inner functions (Θ_{n}(ξ) is an isometry a.e. on 𝕋) in H^{∞}(E_{n}→E). Let K_{Θ_{n}} := H^{2} ⊖ Θ_{n}H^{2} and let P_{θ_{n}} denote the orthogonal projection in H^{2} onto K_{Θ_{n}}. Then the embedding theorem

∑_{n} ∥P_{Θ_{n}}f∥^{2} ≤ C∥f∥^{2} ∀f ∈ H^{2},  (5.3)

holds if and only if

sup_{λ∈𝔻} ∑_{n}(∥e∥^{2} − ∥Θ_{n}(λ)^{∗}e∥^{2}) ≤ C′∥e∥^{2} ∀e ∈ E .  (5.4)

Moreover the norm of the embedding operator can be estimated above by a constant depending only on C and dim E.

If we assume that the determinant of non-square matrix is 0 by definition, then the condition (5.4) immediately follows from

sup_{λ∈𝔻} ∑_{n}(1 − | det Θ_{n}(λ)|^{2}) ≤ C′′ < ∞  (5.5)

and moreover C′ from equation (5.4) is estimated above by C′′ from (5.5). Clearly, if condition (5.5) holds there are only finitely many non-square matrices among Θ_{n}.

Our goal is to show that if the system K_{Θ_{n},Δ_{n},P_{n}}, n ∈ ℕ (the functions Θ_{n} are not necessarily inner) is uniformly minimal, then condition (5.5) is sufficient for the embedding from Theorem 3.2 and then to prove that this condition takes place.

6. Some auxiliary results.

Let K_{Θ,Δ,P} = (M_{Θ,Δ,P})^{⊥} be an S^{∗}-invariant subspace of

( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ), see Section 4, Corollary 4.2.

Let k_{λ} denote the normalized reproducing kernel in H^{2},

k_{λ}(z) := \frac{(1 − |λ|^{2})^{1/2}}{1 − \overline{λ}z}

(note that ∥k_{λ}∥_{H^{2}} = 1).

Lemma 6.1. Let ( \begin{matrix}f \\ g \end{matrix} ) ∈ ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ). Then

P_{M_{Θ,Δ,P}} ( \begin{matrix}f \\ g \end{matrix} ) = ( \begin{matrix}Θ \\ Δ \end{matrix} ) ℙ_{+}(Θ^{∗}f + Δ^{∗}g) + ( \begin{matrix}0 \\ P g \end{matrix} ) .

In particular if f = k_{λ}e, e ∈ E, then

P_{M_{Θ,Δ,P}} ( \begin{matrix}k_{λ}e \\ 0 \end{matrix} ) = ( \begin{matrix}Θ \\ Δ \end{matrix} ) k_{λ} · Θ(λ)^{∗}e .

Proof. Let M := M_{Θ,Δ,P}. Recall that M admits the decomposition

M = M_{Θ,Δ,P} = ( \begin{matrix}Θ \\ Δ \end{matrix} ) H^{2}(E_{1}) ⊕ ( \begin{matrix}0 \\ P L^{2}(E_{∗}) \end{matrix} ) =: M_{1} ⊕ M_{2} ,

see Theorem 4.1. Therefore P_{M} = P_{M_{1}} + P_{M_{2}}.

It is easy to see that

P_{M_{2}} ( \begin{matrix}f \\ g \end{matrix} ) = ( \begin{matrix}0 \\ P g \end{matrix} )

To write down a formula for P_{M_{1}} consider the isometry

V : H^{2}(E_{1}) → ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ),

V f = ( \begin{matrix}Θf \\ Δf \end{matrix} ) , f ∈ H^{2}(E_{1}).

Clearly M_{1} = RangeV, therefore P_{M_{1}} = V V^{∗}. Direct computations yield that

P_{M_{1}} ( \begin{matrix}f \\ g \end{matrix} ) = V V^{∗} ( \begin{matrix}f \\ g \end{matrix} ) == ( \begin{matrix}Θ \\ Δ \end{matrix} ) ℙ_{+}(Θ^{∗}f + Δ^{∗}g)

The statement about k_{λ}e follows immediately from the trivial equality

ℙ_{+}Θ^{∗}k_{λ}e = k_{λ}Θ(λ)^{∗}e

Corollary 6.2. Let f ∈ H^{2}(E). Then

dist\{ ( \begin{matrix}f \\ 0 \end{matrix} ) , K_{Θ,Δ,P} \} = ∥ℙ_{+}Θ^{∗}f∥

Proof. Applying Lemma 6.1 and using the fact that the operator

( \begin{matrix}Θ \\ Δ \end{matrix} ) is an isometry we get

dist\{ ( \begin{matrix}f \\ 0 \end{matrix} ) , K_{Θ,Δ,P} \} = ∥P_{M_{Θ,Δ,P}} ( \begin{matrix}f \\ 0 \end{matrix} )∥ =

= ∥( \begin{matrix}Θ \\ Δ \end{matrix} ) ℙ_{+}Θ^{∗}f∥ = ∥ℙ_{+}Θ^{∗}f∥

Remind that k_{λ} denote the normalized reproducing kernel in H^{2},

k_{λ}(z) := \frac{(1 − |λ|^{2})^{1/2}}{1 − \overline{λ}z}

(note that ∥k_{λ}∥_{H^{2}} = 1).

Corollary 6.3. Let e ∈ E. Then

dist \{( \begin{matrix}k_{λ}e \\ 0 \end{matrix} ) , K_{Θ,Δ,P} \} = ∥Θ(λ)^{∗}e∥

Proof. Trivial corollary of Lemma 6.1.

In the following lemma we assume that for non-square matrix function Θ ∈ H^{∞}(E_{1}→E) the determinant is defined by det Θ ≡ 0 if dim E_{1} < dim E and det Θ ≡ 1 if dim E_{1} > dim E.

Lemma 6.4. Suppose a system of subspaces M_{Θ_{n},Δ_{n},P_{n}}, n ∈ ℕ in

( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ), Θ_{n} ∈ H^{∞}(E_{n}→E), is uniformly minimal with the con-

stant of uniform minimality at least δ. Then there exists a positive constant ε = ε(δ, dim E) such that any point λ ∈ 𝔻 is covered by at most d := dim E sets {z ∈ 𝔻 : | det Θ_{n}(z)| < ε^{d}}. In particular, at most d matrices Θ_{n} satisfy dim E_{n} < dim E.

Proof. Given a point λ ∈ 𝔻, let us suppose there exist more that d := dim E functions Θ_{n} such that | det Θ_{n}(λ)| < ε^{d}. Without loss of generality one can assume that these functions are Θ_{1}, Θ_{2}, …, Θ_{d+1} (part of these functions can be non-square).

For k = 1, 2, …, dim E + 1, one can find vectors e_{k} ∈ E such that ∥Θ_{k}(λ)^{∗}e_{k}∥ ≤ ε, k = 1, 2, …, dim E + 1. Note that if a function Θ_{n} is non-square and dim E_{n} < dim E, one can find a vector e_{n} such that

Θ_{n}(λ)^{∗}e_{n} = 0. By Corollary 6.3 dist \{( \begin{matrix}k_{λ}e_{k} \\ 0 \end{matrix} ) , K_{Θ_{k},Δ_{k},P_{k}}\} < ε.

The subspaces K_{Θ_{k},Δ_{k},P_{k}} are uniformly minimal, so for small enough ε = ε(δ, d) the vectors k_{λ}e_{k}, k = 1, 2, …, dim E + 1 are uniformly minimal as well, and so are the vectors e_{k}. In particular the vectors e_{k} are linearly independent. But this is impossible because we have more that dim E vectors.

Lemma 6.5. Let f ∈ L^{2}(E_{∗}). Then

dist\{ ( \begin{matrix}0 \\ f \end{matrix} ) , K_{Θ,Δ,P} \} = (∥P f∥^{2} + ∥ℙ_{+}Δ^{∗}f∥^{2})^{1/2}

Proof. Trivial Corollary of Lemma 6.1.

The following lemma can be viewed as a “boundary analogue” of Lemma 6.4. Let us remind that for a subspace K_{Θ,Δ,P}, RangeP (ξ) ⊥ RangeΔ(ξ) a.e.

Lemma 6.6. Assume that a system K_{Θ_{n},Δ_{n},P_{n}}, n ∈ ℕ is uniformly minimal. Let σ_{k} denote a Borel support of Δ_{k}, and τ_{k} be a Borel set where RangeP_{k}(ξ) ⊕ RangeΔ_{k}(ξ) ≠ E_{∗}. Then almost all points ξ ∈ 𝕋 are covered by at most d_{∗} := dim E_{∗} sets σ_{k} and by at most d_{∗} sets τ_{k}. In particular, at most d_{∗} functions Θ_{n} ∈ H^{∞}(E_{n}→E) satisfy dim E_{n} > dim E.

Proof. Let σ_{k} denote a Borel support of Δ_{k}. Consider all intersections of d_{∗} + 1 sets σ_{k} (there are countably many such intersections). If all such intersections have measure 0. the conclusion of the lemma is true. Suppose that there exists d_{∗} + 1 sets σ_{k} such that their intersection has positive measure. Without loss of generality we can assume that these sets are σ_{1}, σ_{2}, …, σ_{d_{∗}+1}.

Take functions f_{k} ∈ L^{2}(E_{∗}) such that f_{k}(ξ) ∈ RangeΔ_{k}(ξ), ∥f_{k}(ξ)∥ = 1 on σ_{k} and f_{k}(ξ) = 0 for ξ ∉ σ_{k}.

Let us approximate the functions f_{k} uniformly by simple functions f̃_{k}, f̃_{k} = ∑_{j} χ_{σ_{k}^{j}}e_{k}^{j}, where σ_{k}^{j}, j = 1, 2, … is a disjoint measurable covering of σ_{k}, e_{k}^{j} ∈ E_{∗}, ∥e_{k}^{j}∥ = 1 and

∥f_{k}(ξ) − f̃_{k}(ξ)∥ ≤ δ/(20(d_{∗} + 1)) a.e.,  (6.1)

where δ is the constant of uniform minimality of the system K_{Θ_{n},Δ_{n},P_{n}}, n ∈ ℕ.

Since for each k = 1, 2, …, d_{∗} + 1 the sets σ_{k}^{j} cover σ_{k}, one can find sets σ_{k}^{j_{k}}, k = 1, 2, …, d_{∗} + 1 with intersection σ of positive Lebesgue measure. Define functions g_{k} = m(σ)^{−1/2}χ_{σ}e_{k}^{j_{k}} =: m(σ)^{−1/2}χ_{σ}e_{k} (for the brevity of notation we skip the index j_{k}). Note that ∥g_{k}∥ = 1.

Lemma 6.5 implies that that for big enough n ∈ ℕ

dist \{\overline{z}^{n} ( \begin{matrix}0 \\ g_{k} \end{matrix} ) , K_{Θ_{k},Δ_{k},P_{k}}\} ≤ δ/(10(d_{∗}+1)) , k = 1, 2, …, d_{∗}+1 ,

where δ is the constant of uniform minimality of the system K_{Θ_{n},Δ_{n},P_{n}}, n ∈ ℕ (we can get as close to δ/(20(d_{∗} + 1)) as we want, see (6.1)). We know that the system of subspaces K_{Θ_{k},Δ_{k},P_{k}} is uniformly minimal. Therefore, for big enough n the system of functions \overline{z}^{n}g_{k} = m(σ)^{−1/2}\overline{z}^{n}χ_{σ}e_{k}, k = 1, 2, …, d_{∗} + 1 is uniformly minimal. That implies that the system of vectors e_{k} ∈ E_{∗}, k = 1, 2, …, d_{∗} + 1 = dim E_{∗} + 1 is uniformly minimal, and that is impossible.

To prove the statement about τ_{k} let us first notice that if f(ξ) ⊥ RangeP (ξ) ⊕ RangeΔ(ξ) a.e., then by Lemma 6.5

( \begin{matrix}0 \\ f \end{matrix} ) ∈ K_{Θ,Δ,P} .

Suppose that there exist d_{∗} +1 = dim E_{∗} +1 sets τ_{k} with intersection τ of positive measure. Again, as before, we can assume without loss of generality that these sets are τ_{1}, τ_{2} …, τ_{d_{∗}+1}. Take functions f_{k} ∈ L^{∞}(E_{∗}) such that f(ξ) ⊥ RangeP (ξ) ⊕ RangeΔ(ξ), ∥f(ξ)∥ = 1 a.e. on τ_{k}, and f_{k}(ξ) = 0, ξ ∉ τ_{k}. Then we approximate the functions f_{k} uniformly by simple functions f̃_{k}, f̃_{k} = ∑_{j} χ_{σ_{k}^{j}}e_{k}^{j}, where σ_{k}^{j}, j = 1, 2, …

is a disjoint measurable covering of σ_{k}, e_{k}^{j} ∈ E_{∗}, ∥e_{k}^{j}∥ = 1, and f̃_{k} satisfy (6.1).

Again as above we find sets τ_{k}^{j_{k}}, k = 1, 2, …, d_{∗} + 1, such that τ := ∩_{k=1}^{d_{∗}+1}τ_{k}^{j_{k}} has positive measure. Define g_{k} := (m(τ))^{−1/2}χ_{τ}e_{k}^{j_{k}} =: (m(τ))^{−1/2}χ_{τ}e_{k}. Then

dist \{( \begin{matrix}0 \\ g_{k} \end{matrix} ) , K_{Θ_{k},Δ_{k},P_{k}}\} ≤ δ/(10(d_{∗} + 1)) , k = 1, 2, …, d_{∗} + 1 ,

and we can conclude that the system g_{k}, k = 1, 2, …, d_{∗}+1 is uniformly minimal. So the vectors e_{k} form a uniformly minimal system, which is impossible.

Let us remind that P_{Θ,Δ,P} denotes the orthogonal projection onto K_{Θ,Δ,P}.

Lemma 6.7. Let g = ( \begin{matrix}0 \\ f \end{matrix} ) ∈ ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ), and let the system of

subspaces K_{Θ_{n},Δ_{n},P_{n}} be uniformly minimal. Then the following embedding holds,

∑_{n} ∥P_{Θ_{n},Δ_{n},P_{n}}g∥^{2} ≤ 2 dim E_{∗} · ∥g∥^{2}, ∀g = ( \begin{matrix}0 \\ f \end{matrix} ) ∈ ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ) .

Proof. Let σ_{k}, τ_{k} be the sets from Corollary 6.6. By Lemma 6.5

∥P_{Θ_{n},Δ_{n},P_{n}}g∥^{2} = ∥g∥^{2} − (∥P_{n}f∥^{2} + ∥ℙ_{+}Δ_{n}^{∗}f∥^{2}) =

= ∥f∥^{2} − ∥P_{n}f∥^{2} − ∥ℙ_{+}Δ_{n}^{∗}f∥^{2} =

= ∫_{σ_{n}∪τ_{n}} ∥f(ξ)∥^{2}dm(ξ)+ ∫_{𝕋\(σ_{n}∪τ_{n})} ∥f(ξ)∥^{2}dm(ξ)−∥P_{n}f∥^{2}−∥ℙ_{+}Δ_{n}^{∗}f∥^{2} ≤

≤ ∫_{σ_{n}} ∥f(ξ)∥^{2}dm(ξ) + ∫_{τ_{n}} ∥f(ξ)∥^{2}dm(ξ) + ∫_{𝕋\(σ_{n}∪τ_{n})} ∥f(ξ)∥^{2}dm(ξ)

− ∫_{𝕋\(σ_{n}∪τ_{n})} ∥P_{n}(ξ)f(ξ)∥^{2}dm(ξ) = ∫_{σ_{n}} ∥f(ξ)∥^{2}dm(ξ) + ∫_{τ_{n}} ∥f(ξ)∥^{2}dm(ξ)

Here in the last equality we used the fact that RangeP_{n}(ξ) = E_{∗} on 𝕋 \ (σ_{n} ∪ τ_{n}).

Taking the sum over all n and applying Corollary 6.6 we get the required embedding theorem.

7. Reduction to a scalar embedding theorem

In this section we want to prove the following theorem. Recall that P_{Θ,Δ,P} stands for the orthogonal projection onto K_{Θ,Δ,P}

Theorem 7.1. Let dim E < ∞, dim E_{∗} < ∞. Suppose that the system of subspaces K_{Θ_{n},Δ_{n},P_{n}} is uniformly minimal and

sup_{λ∈𝔻} ∑_{n}(1 − | det Θ_{n}(λ)|^{2}) =: C < ∞ .  (7.1)

Then the embedding

∑_{n} ∥P_{Θ_{n},Δ_{n},P_{n}}f∥^{2} ≤ C′∥f∥^{2}  (7.2)

holds for all f ∈ ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ) and, moreover,

C′ = C′(C, dim E, dim E_{∗}).

First of all let us remind that by Lemmas 6.4 and 6.6 at most dim E+ dim E_{∗} matrices Θ_{n} are non-square, so one always can omit this nonsquare terms without influencing the embedding theorem.

If we would be able to show that the uniform minimality of the system of subspaces K_{Θ_{n},Δ_{n},P_{n}} implies the condition (7.1) the main result would be proved. Indeed, Theorem 7.1 then implies the embedding (7.2), which is exactly the first embedding in Theorem 3.1. (Formally, for the embedding 1 in Theorem 3.1 we need only (7.2) for f ∈ span{K_{Θ_{n},Δ_{n},P_{n}} : n = 1, 2, … }, but of course it is the same.)

The dual system to K_{Θ_{n},Δ_{n},P_{n}}, n = 1, 2, … is a system of invariant subspaces of T^{∗}, and T^{∗} is a completely non-unitary contraction with finite defects. It can be represented as (unitarily equivalent to) a system

of subspaces K_{Θ′_{n},Δ′_{n},P′_{n}} in a space ( \begin{matrix}H^{2}(E_{∗}) \\ L^{2}(E) \end{matrix} ), where Θ′_{n}, Δ′_{n}, P′_{n} are

some operator valued functions. It is possible to write some formulas to relate Θ_{n}, Δ_{n}, P_{n} and Θ′_{n}, Δ′_{n}, P′_{n}, but we do not need that!. We know that the dual system is of course also uniformly minimal, so we can apply Theorem 7.1 without even knowing anything about functions Θ′_{n}, Δ′_{n}, P′_{n}. We get another embedding from Theorem 3.1 and we are done.

To prove Theorem 7.1 we need the following result. Let ℙ_{−} denote the orthogonal projection from L^{2} onto H^{2}.

Theorem 7.2. Let F_{n} be a sequence of functions, F_{n} ∈ H^{∞}. Then the embedding

∑_{n} ∥ℙ_{−}\overline{F}_{n}f∥_{2}^{2} ≤ C∥f∥^{2} ∀f ∈ H^{2}

holds if and only if

sup_{λ∈𝔻} ∑_{n}(|F_{n}|^{2}(λ) − |F_{n}(λ)|^{2}) =: C′ < ∞;  (7.3)

here |F_{n}|^{2}(λ) denotes the harmonic extension of |F_{n}|^{2} |𝕋 at the point λ ∈ D. Moreover, the constants C and C′ are equivalent, i.e. A^{−1}C ≤ C′ ≤ AC where A is some absolute constant.

Proof. Consider the Hankel operator H\overline{F} : H^{2} → H_{−}^{2}(ℓ^{2}) = H_{−}^{2} ⊗ ℓ^{2}

(H\overline{F} f)_{k} := H_{\overline{F}_{k}}f = ℙ_{−}\overline{F}_{k}f .

In other words the symbol \overline{F} of the Hankel operator H\overline{F} is the column with entries \overline{F}_{k}, k = 1, 2, …

Then the sum in (7.3) is the square of an equivalent BMO norm (the so-called Garsia norm) of the function ℙ_{−}\overline{F}. It is well known that for Hankel operators whose symbol is a column \overline{F} the norm of the corresponding Hankel operator and the BMO norm of ℙ_{−}\overline{F} are equivalent. For the scalar case this fact is contained in all textbooks, and for general case the proof follows the lines there. Note that the result does not hold for Hankel operators with general (operator-valued) symbols. It is essential that our symbol has only one column.

Corollary 7.3. Let F_{n} be a sequence of operator-valued functions, F_{n} ∈ H^{∞}(E_{n}→E). Then the embedding

∑_{n} ∥ℙ_{−}F_{n}^{∗}f∥_{2}^{2} =: C∥f∥_{2}^{2} ∀f ∈ H^{2}(E)

holds if and only if

sup_{e∈E, ∥e∥=1} sup_{λ∈𝔻} ∑_{n}(∥F_{n}^{∗}e∥^{2}(λ) − ∥F_{n}(λ)^{∗}e∥^{2}) =: C′ < ∞;  (7.4)

here ∥F_{n}^{∗}e∥^{2}(λ) denotes the harmonic extension of the function ∥F_{n}^{∗}e∥^{2} defined the unit circle 𝕋 to a point λ ∈ 𝔻. Moreover, the constants C and C′ are equivalent, i.e. A^{−1}C ≤ C′ ≤ AC where A depends only on dim E.

Proof. Consider a restriction of the embedding to the functions f of form f = φe where φ is a scalar H^{2} function and e ∈ E is a fixed vector, ∥e∥ = 1. The condition (7.4) guarantees that the embedding holds on those functions. Consider an orthonormal basis {e_{n}}_{n=1}^{d}, d := dim E in E. Then the space H^{2}(E) is an orthogonal sum of subspaces H_{k}, H_{k} consists of all functions of form f = φe_{k} where φ is a scalar H^{2} function. We have embedding on each H_{k} and since there are only finitely many of them, the embedding holds on H^{2}(E).

Proof of Theorem 7.1. By Lemma 6.7 it is enough to check the embed-

ding on vectors ( \begin{matrix}f \\ 0 \end{matrix} ) ∈ ( \begin{matrix}H^{2}(E) \\ L^{2}(E) \end{matrix} ). By Corollary 6.2

∥P_{Θ,Δ,P} ( \begin{matrix}f \\ 0 \end{matrix} )∥^{2} = ∥f∥^{2} − ∥ℙ_{+}Θ^{∗}f∥^{2}

= ∥(I − ΘΘ^{∗})^{1/2}f∥^{2} + ∥Θ^{∗}f∥^{2} − ∥ℙ_{+}Θ^{∗}f∥^{2}
= ∥(I − ΘΘ^{∗})^{1/2}f∥^{2} + ∥ℙ_{−}Θ^{∗}f∥^{2}

So

∑_{n} ∥P_{Θ_{n},Δ_{n},P_{n}} ( \begin{matrix}f \\ 0 \end{matrix} )∥^{2} = ∑_{n} ∥(I − Θ_{n}Θ_{n}^{∗})^{1/2}f∥^{2} + ∑_{n} ∥ℙ_{−}Θ_{n}^{∗}f∥^{2} (7.5)

First of all, by Lemmas 6.4 and 6.6 at most dim E + dim E_{∗} matrices Θ_{n} are non-square. So we can assume that in the sum we have only square matrices, in other words that Θ_{n} ∈ H^{∞}(E→E) and det Θ_{n} are well defined.

The condition (7.1) clearly implies (because Θ_{n}(ξ), ξ ∈ 𝕋 are contractions)

sup_{e∈E, ∥e∥=1} sup_{λ∈𝔻} ∑_{n}(1 − ∥Θ_{n}(λ)^{∗}e∥^{2}) < ∞ ,

and the last condition implies

sup_{e∈E, ∥e∥=1} sup_{λ∈𝔻} ∑_{n}(∥Θ_{n}^{∗}e∥^{2}(λ) − ∥Θ_{n}(λ)^{∗}e∥^{2}) < ∞ .

By Corollary 7.3 that implies

∑_{n} ∥ℙ_{−}Θ_{n}^{∗}f∥^{2} ≤ C∥f∥^{2} ∀f ∈ H^{2}(E) ,

so we have estimated the second sum in (7.5).

To estimate the first sum, let us denote by σ_{n} a Borel support of Δ_{n}, see Corollary 6.6. For ξ ∈ 𝕋 \ σ_{n} the operators Θ_{n}(ξ) are isometries, and since we assumed Θ_{n} ∈ H^{∞}(E→E), dim E < ∞, it follows that (I − Θ_{n}(ξ)Θ_{n}(ξ)^{∗}) = 0 for ξ ∈ 𝕋 \ σ_{n}. Therefore

∥(I − Θ_{n}Θ_{n}^{∗})^{1/2}f∥^{2} ≤ ∫_{σ_{n}} ∥f(ξ)∥^{2}dm(ξ) .

By Corollary 6.6 almost all points of 𝕋 are covered by at most dim E_{∗} sets σ_{n} and therefore

∑_{n} ∥(I − Θ_{n}Θ_{n}^{∗})^{1/2}f∥^{2} ≤ dim E_{∗}∥f∥^{2}

so we have estimated the first sum in (7.5).

So we made one more reduction. Now to prove the main result (Theorem 2.1) it remains to show that the uniform minimality of the system of subspaces K_{Θ_{n},Δ_{n},P_{n}}, n = 1, 2, … implies the condition (7.1)

8. Geometry of vector reproducing kernels.

Consider in a Hilbert space H a system of subspaces E_{n}, n = 1, 2, . . .. Suppose that the system is a Riesz basis, and let R be a normalized orthogonalizer for this system, see Section 1.1. Let us remind that an orthogonalizer is a bounded invertible operator that maps the system E_{n}, n = 1, 2, … into an orthogonal one, and the “normalized” means that R|E_{n} is an isometry for all n.

It follows from the definition of the normalized orthogonalizer that for any (finite) set of vectors f_{n} ∈ E_{n}

∥R∥^{−2} ∑_{n} ∥f_{n}∥^{2} ≤ ∥ ∑_{n} f_{n}∥^{2} ≤ ∥R^{−1}∥^{2} ∑_{n} ∥f_{n}∥^{2} .  (8.1)

We need the following simple lemma.

Lemma 8.1. Let a system of subspaces E_{n}, n = 1, 2, … in a Hilbert space H be a Riesz basis, and let R be its normalized orthogonalizer. Then for a Hilbert space E the system E_{n} ⊗E, n = 1, 2, … in H ⊗E is a Riesz basis and R⊗I_{E} is a normalized orthogonalizer for this system.

We apply this Lemma in the following situation. Consider a countable subset σ ⊂ 𝔻, and let {k_{λ}}_{λ∈σ} be a system of normalized reproducing kernels in H^{2},

k_{λ}(z) := \frac{(1 − |λ|^{2})^{1/2}}{1 − \overline{λ}z} ,

∥k_{λ}∥_{2} = 1.

Suppose the system {k_{λ}}_{λ∈σ} is a Riesz basis (i.e. the system of corresponding one-dimensional subspaces is a Riesz basis), and let R be its (normalized) orthogonalizer. In our case it is simply an operator that maps the system {k_{λ}}_{λ∈σ} into an orthonormal system. The following result is a simple corollary of Lemma 8.1

Lemma 8.2. Let the system {k_{λ}}_{λ∈σ} be a Riesz basis, let R be its orthogonalizer, and let E be a Hilbert space. Then for any finite set of vectors f_{λ} ∈ E

∥R∥^{−2} ∑_{λ∈σ} ∥f_{λ}∥^{2} ≤ ∥ ∑_{λ∈σ} k_{λ}f_{λ}∥^{2} ≤ ∥R^{−1}∥^{2} ∑_{λ∈σ} ∥f_{λ}∥^{2} .

Proof. The statement of the lemma is simply the inequality (8.1), because by Lemma 8.1 R ⊗ I is a normalized orthogonalizer of the system {span{k_{λ}} ⊗ E}_{λ∈σ}, and clearly ∥R ⊗ I∥ = ∥R∥, ∥(R ⊗ I)^{−1}∥ = ∥R^{−1}∥.

We need several well known facts about the geometry of a system of (scalar) reproducing kernels. The following theorem can be found for example in [1].

Theorem 8.3. The system of reproducing kernels {k_{λ}}_{λ∈σ} is a Riesz basis if and only if the set σ is an interpolating set,

inf_{λ∈σ} ∏_{μ∈σ\{λ}} |b_{μ}(λ)| =: δ > 0

where b_{λ} is a Blaschke factor, b_{λ}(z) =^{def} (|λ|/λ)(λ − z)(1 − \overline{λ}z)^{−1}.

Moreover, the measure of nonorthogonality ∥R∥ · ∥R^{−1}∥ of the system (R is a normalized orthogonalizer) admits the estimate above by a constant C(δ) depending only on δ.

Let θ_{n} be scalar valued inner functions in H^{∞}, and let K_{θ_{n}} denote the corresponding S^{∗}-invariant subspace, K_{θ_{n}} := H^{2} ⊖ θ_{n}H^{2}.

We need the following theorem that was proved by Vasyunin in [11], see also [1, Lecture IX]

Theorem 8.4 (Vasyunin). Let the system of subspaces K_{θ_{n}}, where θ_{n} are scalar inner functions in H^{∞}, be uniformly minimal and let δ be its constant of uniform minimality. Then the system is a Riesz basis, and there exists a constant CV (δ) such that the measure of nonorthogonality ∥R∥·∥R^{−1}∥ (recall that R is the normalized orthogonalizer of the system) admits the following estimate

∥R∥ · ∥R^{−1}∥ ≤ CV (δ)

Remark 8.5. The above theorem can be also proved using Theorem 5.1 above. To do this it is enough to notice that the uniform minimality of the system of subspaces K_{θ_{n}} implies

inf_{n} inf_{λ∈𝔻}{|θ_{n}(λ)| + ∏_{k≠n} |θ_{k}(λ)|} =: δ′ > 0

and that δ′ ≤ 2δ, see [1, Lecture IX]. The above condition implies (5.2) from Theorem 5.1, and therefore the embedding (5.1). The dual system has the same geometry, so the embedding for the dual system holds automatically.

9. Carleson measures and bases of reproducing kernels.

For an arc I ⊂ 𝕋 let S(I) denote the Carleson square,

S(I) := {z ∈ 𝔻 : z/|z| ∈ I, 1 − |I| ≤ |z| < 1} .

A measure μ on the disk 𝔻 is called Carleson if

sup_{I⊂𝕋} \frac{μ(S(I))}{|I|} =: K < ∞;

here supremum is taken over all arcs I ⊂ 𝕋.

The constant K is called the Carleson norm of the measure μ.

The following theorem is well known, see for example monographs [16, 1].

Theorem 9.1. Let μ be a measure in 𝔻. The following statements are equivalent 1. The measure μ is Carleson;

2. sup_{λ∈𝔻} ∫ |k_{λ}(z)|^{2}dμ(z) =: K′ < ∞, where k_{λ} is the normalized reproducing kernel of H^{2}, k_{λ}(z) = (1 − |λ|^{2})^{1/2}(1 − \overline{λ}z)^{−1};

3. the embedding

∫_{𝔻} |f(z)|^{2}dμ(z) ≤ K′′∥f∥^{2} , f ∈ H^{2}

holds. Moreover, the Carleson norm K of μ, the constant K′ and the best possible K′′ are equivalent in the sense of two-sided estimates with some absolute constants.

The following description of interpolating sets is well known and again can be found for example in [16, 1]

Theorem 9.2. The set σ ⊂ 𝔻 satisfies the Carleson interpolation condition

inf_{λ∈σ} ∏_{μ∈σ\{λ}} |b_{μ}(λ)| =: δ > 0

if and only if

1. The points of σ are separated in hyperbolic metric, i.e.

inf {|b_{λ}(μ)| : λ, μ ∈ σ, λ ≠ μ } =: α > 0, and

2. the measure ∑_{λ∈σ}(1 − |λ|^{2})δ_{λ}, where δ_{λ} stands for the unit mass

at λ, is Carleson. Moreover the interpolating constant δ can be estimated by the constants depending only on α and the Carleson norm K of the measure

δ ≥ δ(α, K),

and vice versa K and α can be estimated by constant depending only on δ,

K ≤ K(δ), α ≥ α(δ)

In [16] the estimates are not stated, but they are contained in the proofs.

10. Carleson contours

We need the following result

Theorem 10.1. Given ε > 0 there exists ε′ = ε′(ε) > 0, ε′ < ε such that for any φ ∈ H^{∞}, ∥φ∥_{∞} ≤ 1 one can find a region O satisfying

{z ∈ 𝔻 : |φ(z)| < ε′} ⊂ O ⊂ {z ∈ 𝔻 : |φ(z)| ≤ ε}

and such that the arc-length on γ := ∂O ∩ 𝔻 is a Carleson measure with the Carleson norm bounded by a constant K = K(ε).

This famous theorem was proved by L. Carleson [3].

In [17] the same theorem was proved with the constant K independent on ε. The paper deals with Blaschke products: standard approximation reason implies immediately the result for inner functions as well.

The statement we need can be found for example in [16], Chapter 8. The boundary ∂O ∩ 𝔻 of O is usually called a Carleson contour for

φ.

We will need the following lemma.

Lemma 10.2. Let Φ ∈ H^{∞}, ∥Φ∥_{∞} ≤ 1. Given a small ε > 0 let γ be a Carleson contour for Φ. There exist interpolating Blaschke products B_{k}, k = 1, 2, … m, m = m(ε) with zeroes on γ such that

1. ∏_{k=1}^{m} |B_{k}(z)|^{M} ≤ |Φ(z)|, for z ∈ γ, where M = M(ε);

2. the interpolating constant (see Theorem 9.2) of each B_{k} is bounded below by an absolute constant δ_{0}.

The idea of putting zeroes of an interpolating Blaschke product on a Carleson contour is now a commonplace in analysis. To the author’s knowledge it was first used (in a different context) in [18]

Proof. Let us recall that the hyperbolic distance between two points λ and μ in 𝔻 is defined as ρ(λ, μ) := \frac{1}{2} log\{(1 + |b_{λ}(μ)|)/(1 − |b_{λ}(μ)|)\}, where b_{λ} denotes a Blaschke factor with zero at λ. It is well known, see [12, 13] that ρ is indeed a metric, and it is easy to see that if |b_{λ}(μ)| is small, then ρ(λ, μ) ≈ |b_{λ}(μ)|.

Fix a constant α, 0 < α < 0.1. Pick on the contour γ a discrete set σ such that the points in this set are hyperbolically separated, |b_{λ}(μ)| > α > 0, λ, μ ∈ σ, λ ≠ μ, but are dense enough, namely that for any point z ∈ γ one can find a point λ ∈ σ for which |b_{λ}(z)| < α.

It can be done by putting points λ on the contour in such a fashion that hyperbolic discs of (hyperbolic) radius ρ = \frac{1}{2} log\{(1 + α)/(1 − α)\} centered at those points cover the contour γ_{n}, but that every next point (center) is chosen outside the union of hyperbolic disk of radius ρ with centers in the points already constructed.

To show that we indeed can cover the contour this way, let us split the disk 𝔻 into “layers” D_{m}, D_{m} := {z ∈ 𝔻 : 1 − 2^{−m} ≤ |z| < 1 − 2^{−m−1}}, m = 0, 1, 2, …, and put points λ on the contour first in the “layer” D_{0} while it is possible, then in the “layer” D_{2}, and so on. By the construction the hyperbolic disks of radius ρ/2 centered at points λ ∈ σ are disjoint. Since each “layer” D_{m} has a finite hyperbolic area, and the hyperbolic area of a hyperbolic disk depends only of radius, we can put only finitely many points of σ_{n} in each “layer”. So it is indeed possible to cover all the contour γ_{n} by countably many hyperbolic disks.

Note that the hyperbolic disks of radius ρ/2 are disjoint. Note also that each of this disks contain a part of γ of length comparable with the Euclidean diameter of the disk (and so with 1 − |λ|^{2}, where λ is the hyperbolic center of the disk). Indeed, the above statement is trivial if γ intersects the boundary of the disk. And if the disk contain a connected component of γ, there must be a zero μ of Φ surrounded by the component. So the disk {z : |b_{μ}(z)| < ε′} must be surrounded by the component, so it length must be at least the diameter of the disk.

This implies that the measure ∑_{λ∈σ}(1 − |λ|^{2})δ_{λ} is Carleson and its Carleson norm can be estimated as

∥ ∑_{λ∈σ_{n}}(1 − |λ|^{2})δ_{λ}∥_{Carl} ≤ B(ε, ε′, α) ∥|dz||γ_{n}∥_{Carl}

Now we apply the following simple lemma

Lemma 10.3. Let σ be a countable subset of 𝔻 such that the measure ∑_{λ∈σ}(1 − |λ|^{2})δ_{λ} is Carleson and its Carleson norm is estimated above by m (m ∈ ℕ). Then one can split the set σ into m sets σ_{k} such that σ = ∪_{k=1}^{m}σ_{k} and the measures ∑_{λ∈σ_{k}}(1 − |λ|^{2})δ_{λ}, k = 1, 2, …, m has a Carleson norm bounded by an absolute constant.

Applying the lemma we split σ into m = m(ε, ε′, α) sets σ_{k}, such that the measures ∑_{λ∈σ_{k}}(1 − |λ|^{2})δ_{λ}, k = 1, 2, …, m have Carleson norms bounded by an absolute constant. Since the hyperbolic distance between two points of σ is at lest ρ, we can easily see that the Blaschke products B_{k} := ∏_{λ∈σ_{k}} b_{λ} are interpolating and their interpolating constants are bounded below by an absolute constant.

By the construction the Blaschke product B = ∏_{k=1}^{m} B_{k} = ∏_{λ∈σ} b_{λ} satisfy |B(z)| ≤ α on γ, so for sufficiently large M

|B(z)|^{M} = ∏_{k=1}^{m} |B_{k}(z)|^{M} ≤ ε′ ≤ |Φ(z)|, z ∈ γ.

So the lemma is proved modulo Lemma 10.3.

Proof of lemma 10.3. Let split the disk 𝔻 into dyadic “squares” Q_{n,l}, n = 0, 1, 2, …, l = 1, 2, …, 2^{n},

Q_{n,l} := \{z = re^{2πiθ} : 1 − 2^{−n} ≤ r < 1 − 2^{−(n+1)}, \frac{l − 1}{2^{n}} ≤ θ < \frac{l}{2^{n}} \} .

Let us enumerate the set σ in the following way: take the lexicographic ordering of Q_{n,l}, i.e. Q_{0,1}, Q_{1,1}, Q_{1,2}, Q_{2,1}, Q_{2,2}, Q_{2,3}, … and fix an arbitrary ordering of each set σ ∩ Q_{n,l}. Since the sets σ ∩ Q_{n,l} are finite, we get an enumeration of σ, σ = {λ_{1}, λ_{2}, λ_{3}, … }. Split σ into the sets the sets σ_{k}, k = 1, 2, …, m in a natural way

σ_{k} = {λ_{k+mj} : j = 1, 2, … }

Let us check the estimate of Carleson norms.

Take an arc I ⊂ 𝕋, and let 2^{−n−1} < |I| ≤ 2^{−n}. Any such arc can be covered by at most two dyadic arcs of length 2^{−n}, so it is enough to check the Carleson norms only on dyadic squares S_{n,l},

S_{n,l} := \{z = re^{2πiθ} : r ≥ 1 − 2^{−n}, \frac{l − 1}{2^{n}} ≤ θ < \frac{l}{2^{n}} \} .

We want to estimate ∑_{λ∈σ_{k}∩S_{n,l}}(1 − |λ|^{2}). Since 1 − |λ|^{2} = (1 − |λ|)(1 + |λ|) ≤ 2(1 − |λ|), it is enough to estimate ∑_{λ∈σ_{k}∩S_{n,l}}(1 − |λ|). Let us fix a layer 1 − 2^{−j} ≤ |z| ≤ 1 − 2^{−j−1}, j ≥ n of S_{n,l} and let us estimate the number of point of σ_{k} in this layer. It follows from the construction that if there are M points of σ in this layer, then there are at most M/m + 1 points of σ_{k} there.

Therefore

∑_{λ∈σ_{k}∩S_{n,l}}(1 − |λ|) ≤ \frac{2}{m} ∑_{λ∈σ∩S_{n,l}}(1 − |λ|^{2}) + 2^{−j} ,

and taking the sum over all j ≥ n we get

∑_{λ∈σ_{k}}(1 − |λ|) ≤ \frac{1}{m} ∑_{λ∈σ}(1 − |λ|^{2}) + ∑_{j≥n} 2^{−j} ≤ \frac{2m}{m}2^{−n} + 2 · 2^{−n} ≤ 4 .

and the lemma is proved

11. The main construction

11.1. Approximation by interpolating Blaschke products. In what follows we will need the following elementary inequality. Let 0 < α_{1}, α_{2} ≤ 1. Then

1 − α_{1}α_{2} = α_{2}(1 − α_{1}) + (1 − α_{2}) ≤ (1 − α_{1}) + (1 − α_{2}) ,

and by induction we get

1 − α_{1}α_{2} … α_{n} ≤ (1 − α_{1}) + (1 − α_{2}) + … + (1 − α_{n}) ,  (11.1)

for 0 < α_{1}, α_{2}, …, α_{n} ≤ 1.

Let us pick a small ε and for each n construct a Carleson contour γ_{n} for det Θ_{n} (all contours with same ε′). The constant ε will be chosen later. For now we only assume that it is less than ε from Lemma 6.4. As it was said above we can assume without loss of generality that all matrix-functions Θ_{n} are d × d matrix-functions, so determinants are well defined.

So we have open sets O_{n} with rectifiable boundary ∂O_{n}, such that

{z ∈ 𝔻 : | det Θ_{n}(z)| < ε′^{d}} ⊂ O_{n} ⊂ {z ∈ 𝔻 : | det Θ_{n}(z)| ≤ (ε)^{d}}

and the arc-length on γ_{n} := ∂O_{n} ∩ 𝔻 is a Carleson measure with Carleson norm at most K = K(ε).

By Lemma 10.2 there exist interpolating Blaschke products B_{n,k}, 1 ≤ k ≤ m(ε) such that their interpolating constants are bounded below by an absolute constant δ_{0} and

∏_{k=1}^{m} |B_{n,k}(z)|^{M} ≤ det Θ_{n}(z) ∀z ∈ γ_{n} .  (11.2)

The following lemma allows us to replace det Θ_{n} by the constructed above Blaschke products B_{n,k}.

Lemma 11.1. If for all k = 1, 2, …, m

sup_{λ∈𝔻} ∑_{n}(1 − |B_{n,k}(λ)|^{2}) ≤ K < ∞  (11.3)

then the condition

sup_{λ∈𝔻} ∑_{n}(1 − | det Θ_{n}(λ)|^{2}) ≤ C = C(d, d_{∗}, K, α, ε, ε′) < ∞ (11.4)

holds.

Proof. Let h_{n} be outer functions in H^{∞} such that

|h_{n}(ξ)| = max{| det Θ_{n}(ξ)|, ε′^{d}} , ξ ∈ 𝕋 .

Clearly ∥h_{n}∥_{∞} ≤ 1.

Notice that by (11.2)

∏_{k=1}^{m} |B_{n,k}(z)|^{N} ≤ | det Θ_{n}(z)| , z ∈ γ_{n} .

On the other hand for the points ξ ∈ 𝕋 that are angular limits of points in 𝔻 \ O_{n} the inequality | det Θ_{n}(ξ)| ≥ ε′^{d} holds, thus | det Θ_{n}(ξ)| ≥ |h_{n}(ξ)| for almost all such points. So by the maximum modulus principle (we have to use a generalization of a classical maximum modulus principle, see Lemma 11.2 below)

|h_{n}(z)| · ∏_{k=1}^{m} |B_{n,k}(z)|^{N} ≤ | det Θ_{n}(z)| , z ∈ 𝔻 \ O_{n} .

Applying the inequality (11.1) several times we get for a point z ∈ 𝔻

∑_{n: z∉O_{n}}(1 − | det Θ_{n}(z)|^{2}) ≤ ∑_{n}(1 − |h_{n}(z)|^{2} ∏_{k=1}^{m} |B_{n,k}(z)|^{2N}) ≤

≤ ∑_{n}(1 − |h_{n}(z)|^{2}) + ∑_{n}(1 − ∏_{k=1}^{m} |B_{n,k}(z)|^{2N}) ≤

≤ ∑_{n}(1 − |h_{n}(z)|^{2}) + N ∑_{k=1}^{m} ∑_{n}(1 − |B_{n,k}(z)|^{2}) .

By Lemma 6.4 any point z ∈ 𝔻 is covered by at most d = dim E sets {z ∈ 𝔻 : | det Θ_{n}(z)| < ε^{d}} ⊃ O_{n} (we assume that ε is small enough), hence

∑_{n}(1−| det Θ_{n}(z)|^{2}) ≤ ∑_{n}(1−|h_{n}(z)|^{2})+N ∑_{k=1}^{m} ∑_{n}(1−|B_{n,k}(z)|^{2})+d .

To complete the proof of the lemma we need to estimate ∑_{n}(1 − |h_{n}(z)|^{2}). Let s_{k} denote a Borel subset of 𝕋 such that |h_{k}(ξ)| < 1, ξ ∈ s_{k} and |h_{k}(ξ)| = 1, ξ ∈ 𝕋 \ s_{k}. By Corollary 6.6 almost all points ξ ∈ 𝕋 are covered by at most d_{∗} = dim E_{∗} sets s_{k}. Therefore the product ∏_{n} |h_{n}| converges on 𝕋 and moreover

∏_{n} |h_{n}(ξ)| ≥ (ε′)^{d_{∗}} , ξ ∈ 𝕋 .

Therefore, if we normalize all h_{n} by the condition h_{n}(0) > 0, then clearly the product ∏_{n} h_{n} converges in 𝔻 and moreover, by maximum modulus principle

|∏_{n} h_{n}(z)| ≥ (ε′)^{d_{∗}} , z ∈ 𝔻 .

It follows that

(ε′)^{d_{∗}} ≤ ∏_{n} |h_{n}(z)| = exp (\frac{1}{2} ∑_{n} log |h_{n}(z)|^{2}) ≤

≤ exp ( − \frac{1}{2} ∑_{n}(1 − |h_{n}(z)|^{2})) .

Therefore

∑_{n}(1 − |h_{n}(z)|^{2}) ≤ 2d_{∗} log \frac{1}{ε′} .

The following lemma is the maximum modulus principle we have used in the proof of Lemma 11.1.

Lemma 11.2. Let Ω ⊂ 𝔻 be an open set, f be a bounded analytic in Ω function, continuous on clos Ω ∩ 𝔻. Suppose that |f| ≤ 1 on 𝔻 ∩ ∂Ω and lim sup_{z→ξ} |f(z)| ≤ 1 for all points ξ ∈ 𝕋 that are angular limits of points in Ω (lim sup also means angular upper limit). Then |f| ≤ 1 on Ω.

Such results are for sure familiar to specialists. The lemma stated above (even a stronger version) can be found for example in [15], see Lemma 63 there.

11.2. Choice of ε. According to Lemma 11.1 to prove the theorem it is enough to show that for all k = 1, 2, …, m

sup_{λ∈𝔻} ∑_{n}(1 − |B_{n,k}(λ)|^{2}) ≤ K < ∞

(K does not depend on k). For the brevity of notation let us skip the index k in what follows.

Let σ_{n} be zero set of the Blaschke product B_{n}. Since B_{n} are interpolating Blaschke products with the interpolating constant bounded below by an absolute constant δ_{0}, Theorem 8.3 implies that the system {k_{λ}}_{λ∈σ_{n}} is a Riesz basis, and its measure of non-orthogonality ∥R∥ · ∥R^{−1}∥ admits the estimate

∥R∥ · ∥R^{−1}∥ ≤ C .  (11.5)

where C is an absolute constant.

Let δ be the constant of uniform minimality of the system E_{n} = K_{Θ_{n},Δ_{n},P_{n}}, n = 1, 2, …, and let CV (δ/2) be the constant from Theorem 8.4. Let C be the constant from (11.5). Recall that we fixed the small constant α. We will chose ε smaller than ε from Lemma 6.4 and such that

εC · CV (δ/2) < δ/10  (11.6)

The choice will be clear from what follows below.

11.3. Reduction to “almost scalar” case. Consider a point λ ∈ σ_{n}. Recall that σ_{n} = σ_{n,k} is the zero set of the Blaschke product B_{n,k} constructed above. From now on we forget about index k, and just skip it. In what follows k has a completely different meaning.

Since by the construction for λ ∈ σ_{n} we have | det Θ_{n}(λ)| < ε^{d}, there exists a vector e_{λ} ∈ E, ∥e_{λ}∥ = 1, such that ∥Θ(λ)^{∗}e_{λ}∥ < ε.

Consider a system of vector reproducing kernels {k_{λ}e_{λ}}_{λ∈σ} in H^{2}(E), where k_{λ}(z) = (1 − |λ|^{2})^{1/2}(1 − \overline{λ}z)^{−1} is the normalized reproducing kernel in H^{2}.

Let ε be the constant chosen above in Section 11.2. Let e^{1}, e^{2}, …, e^{N} be a finite ε-net for the unit sphere in E.

For each n split the zero set σ_{n} of B_{n} into N disjoint sets σ_{n}^{1}, σ_{n}^{2}, …, σ_{n}^{N} such that for any λ ∈ σ_{k} the corresponding vector e_{λ} is close to e^{k}

∥e_{λ} − e^{k}∥ < ε ,  λ ∈ σ_{n}^{k} .  (11.7)

Let B_{n}^{k} be a Blaschke product with zero set σ_{n}^{k} (if σ_{n}^{k} = ∅ we put B_{n}^{k} := 1). Clearly B_{n} = ∏_{k=1}^{N} B_{n}^{k}.

Lemma 11.3. For Blaschke products B_{n} and B_{n}^{k} constructed above

sup_{λ∈𝔻} ∑_{n}(1 − |B_{n}(λ)|^{2}) ≤ ∑_{k=1}^{N} sup_{λ∈𝔻} ∑_{n}(1 − |B_{n}^{k}(λ)|^{2}) .

Proof. This lemma admits a simple geometrical interpretation and a simple proof in terms of embedding theorems, but also can be proved by applying the inequality (11.1).

12. The main Lemma

Lemma 12.1. Let K_{Θ_{n},Δ_{n},P_{n}}, n = 1, 2, … be a uniformly minimal

system of subspaces in ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ), and let δ be its constant on uniform

minimality. Let B_{n} be interpolating Blaschke products with zero sets σ_{n} and let for all n the measure of non-orthogonality (see Introduction) ∥R∥ · ∥R^{−1}∥ of the family of reproducing kernels {k_{λ}}_{λ∈σ_{n}} be at most C. Let ε > 0 be a small such that

C · CV (δ/2) · ε < δ/10 ,

where CV is the constant from Theorem 8.4. Suppose that there exists a vector e ∈ E, ∥e∥ = 1 such that for all n and for all λ ∈ σ_{n}

∥Θ_{n}(λ)^{∗}e∥ < 2ε .

Then the system of subspaces K_{B_{n}} ⊂ H^{2} (scalar), n = 1, 2, … is uniformly minimal with constant of uniform minimality at least δ/2.

This lemma clearly implies the main result. For a fixed k the constructed above Blaschke products B_{n}^{k} and matrix functions Θ_{n} satisfy the assumptions of the lemma with σ_{n} := σ_{n}^{k}, e := e^{k} because

∥Θ_{n}(λ)^{∗}e^{k}∥ ≤ ∥Θ_{n}(λ)^{∗}e_{λ}∥ + ∥Θ_{n}(λ)^{∗}(e_{λ} − e^{k})∥ < 2ε

for λ ∈ σ_{n}^{k}. Therefore the system of subspaces K_{B_{n}^{k}} is uniformly minimal. By theorem 8.4 the system is an unconditional basis, so the embedding

∑_{n} ∥P_{K_{B^{k}_{n}}}f∥^{2} ≤ C∥f∥^{2} ∀f ∈ H^{2}

holds. By Theorem 5.1

∑_{n}(1 − |B_{n}^{k}(λ)|^{2}) ≤ K = K(δ) < ∞ , λ ∈ 𝔻.

As it was shown above in Section 11 this is enough to prove the main theorem.

Idea of the proof of the main lemma. The idea is very simple and looks very naïve. Suppose that the constant of the uniform minimality of the system K_{B_{n}}, n = 1, 2, … is less than δ/2. That means that for a vector f_{n} ∈ K_{B_{n}}, ∥f_{n}∥ = 1 there exist vectors f_{k} ∈ K_{B_{k}} such that ∥f_{n} − ∑_{k≠n} f_{k}∥ < δ/2.

Consider the spaces K_{B_{k}} ⊗ e ⊂ H^{2}(E) embedded in ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ). It

can be shown that the subspaces K_{B_{k}} ⊗e are close to K_{Θ_{k},Δ_{k},P_{k}}, and we are going to obtain a similar approximation for some f̃_{n} ∈ K_{Θ_{n},Δ_{n},P_{n}}, ∥f̃_{n}∥ = 1

∥f̃_{n} − ∑_{k≠n} f̃_{k}∥ < δ, f_{k} ∈ K_{Θ_{k},Δ_{k},P_{k}} ,

that contradicts the fact that the constant of uniform minimality of the original system is δ.

The main trick here is that one has to take not too many summands f̃_{k} in the approximation, so that the accumulated “round-off” error caused by the deviation from the scalar case does not worsen the estimate obtained for this case by more that δ/2. Quantitatively this “not too many” is expressed by the following lemma, see [9, 8, 10].

Here and in what follows δ({E_{k}}) denotes the constant of uniform minimality of the system of subspaces {E_{n}}. We assume that δ({E_{k}}) = 0 if the system is not uniformly minimal.

Lemma 12.2. Let δ({E_{k}}) < δ < 1. There exists a finite subset N of indices such that

δ({E_{k}}_{k∈N}) < δ  (12.1)

but for any n ∈ N

δ({E_{k}}_{k∈N \{n}}) ≥ δ.  (12.2)

Proof. Chose a finite set of indices N satisfying (12.1) (such choice is always possible because δ({E_{k}}) < δ). If (12.2) holds, the lemma is proved. Otherwise remove from N a point n for which δ({E_{k}}_{k∈N \{n}}) < δ, check (12.2) again, and so on. Obviously, we eventually obtain the desired set N, since the constant of uniform minimality of a system consisting of one subspace is 1.

Proof of the main lemma (Lemma 12.1). Suppose the conclusion of the lemma does not hold and δ({K_{B_{k}}}_{k∈ℕ}) < δ/2. Let us apply Lemma 12.2 with constant δ/2 to the system {K_{B_{k}}}_{k∈ℕ}. We obtain a finite set N of indices such that

δ({K_{B_{k}}}_{k∈N}) < δ/2,  (12.3)

and

δ({K_{B_{k}}}_{k∈N \{n}}) ≥ δ/2,  ∀n ∈ N .  (12.4)

By (12.3) there exists a vector f_{n} ∈ K_{B_{n}}, ∥f_{n}∥ = 1 (n ∈ N), that can be approximated by a vector f^{n} ∈ H^{2}, ∥f^{n}∥ ≤ 1 of the form

f^{n} = ∑_{k∈N \{n}} f_{k} , f_{k} ∈ K_{B_{k}} ,

so that

∥f_{n} − f^{n}∥ < δ/2 .

Recall that σ_{k} denotes the zero set of the Blaschke product B_{k}. The vectors f_{k} ∈ K_{B_{k}} can be represented in the form

f_{k} = ∑_{λ∈σ_{k}} c_{λ}k_{λ} , c_{λ} ∈ ℂ, k_{λ}(z) = (1 − |λ|^{2})^{1/2}(1 − \overline{λ}z)^{−1} .

By the hypotheses of the lemma the system of vectors {k_{λ}}_{λ∈σ_{k}} is an unconditional basis and its measure of non-orthogonality ∥R∥ · ∥R^{−1}∥ is at most C. By (12.4) the system of subspaces {K_{B_{k}}}_{k∈N \{n}} is a Riesz basis and its measure of non-orthogonality is at most CV (δ/2). Therefore the family of vectors {k_{λ}}_{λ∈σ}, σ = ∪_{k∈N \{n}}σ_{k} is a Riesz basis in span(K_{B_{k}} : k ∈ N \ {n}) with the measure of non-orthogonality at most C · CV (δ/2).

So, if R = R({k_{λ}}_{λ∈σ}) is a normalized orthogonalizer of {k_{λ}}_{λ∈σ} we have

∥R∥ · ∥R^{−1}∥ ≤ C · CV (δ/2).  (12.5)

Let f_{k} = ∑_{λ∈σ_{k}} c_{λ}k_{λ} be the expansion of f_{k} in the basis {k_{λ}}_{λ∈σ_{k}}. Since we assume that the family of subspaces {K_{B_{k}}}_{k∈N \{n}} is a Riesz basis, the zero sets σ_{k} are disjoint, and we can write

f^{n} = ∑_{λ∈σ} c_{λ}k_{λ}

(recall that σ = ∪_{k∈N \{n}}σ_{k}).

Let K := span{K_{Θ_{k},Δ_{k},P_{k}} : k ∈ N \ {n} }. Define

f̃_{n} := P_{K_{Θn,Δn,Pn}}f_{n} · e,  f̃^{n} := P_{K}f^{n} · e,

where e is the vector from the assumptions of the main lemma. By the triangle inequality

∥f̃_{n} − f̃^{n}∥ ≤ ∥f̃_{n} − f_{n} · e∥ + ∥f̃^{n} − f^{n} · e∥ + ∥f_{n}e − f^{n}e∥.

The last summand is equal to

∥f_{n} − f^{n}∥ < δ/2.

The first two terms are also easy to estimate. Let us for example estimate the second one. The subspace K defined above is an S^{∗}-

invariant subspace of ( \begin{matrix}H^{2}(E) \\ L^{2}(E_{∗}) \end{matrix} ), so it can be represented as K =

K_{Θ,Δ,P}. Hence by Corollary 6.2

∥f̃^{n} − f^{n}e∥ = dist\{( \begin{matrix}f^{n}e \\ 0 \end{matrix} ) , K_{Θ,Δ,P} \} = ∥ℙ_{+}Θ^{∗}f^{n}e∥ =

= ∥ℙ_{+}Θ^{∗} ∑_{λ∈σ} c_{λ}k_{λ}e∥ = ∥ ∑_{λ∈σ} c_{λ}k_{λ}Θ(λ)^{∗}e∥ .

Since by the hypothesis of the lemma ∥Θ_{k}(λ)^{∗}e∥ < 2ε for λ ∈ σ_{k}, and since by Corollary 6.3

∥Θ(λ)^{∗}e∥ = dist\{( \begin{matrix}k_{λ}e \\ 0 \end{matrix} ) , K_{Θ,Δ,P} \} ≤

≤ dist\{( \begin{matrix}k_{λ}e \\ 0 \end{matrix} ) , K_{Θ_{k},Δ_{k},P_{k}}\} = ∥Θ_{k}(λ)^{∗}e∥

one can conclude that

∥Θ(λ)^{∗}e∥ < 2ε ∀λ ∈ σ.

Applying both inequalities from Lemma 8.2 we can get

∥ ∑_{λ∈σ} c_{λ}k_{λ}Θ(λ)^{∗}e∥^{2} ≤ ∥R^{−1}∥^{2} ∑_{λ∈σ} |c_{λ}|^{2}∥Θ(λ)^{∗}e∥^{2} ≤

≤ (2ε)^{2}∥R^{−1}∥^{2} ∑_{λ∈σ} |c_{λ}|^{2} ≤ (2ε)^{2}∥R^{−1}∥^{2}∥R∥^{2}∥∑_{λ∈σ} c_{λ}k_{λ}∥^{2}

So

∥f̃^{n} − f^{n}e∥ ≤ 2ε · ∥R∥ · ∥R^{−1}∥ ≤ 2ε · C · CV (δ/2) ≤ δ/5.

Similarly

∥f̃_{n} − f_{n}e∥ ≤ 2ε · C ≤ 2ε · C · CV (δ/2) ≤ δ/5.

(the inequality in the middle holds because the measure of non-orthog-onality is always at least 1). Gathering estimates for all three summands together we have

∥f̃_{n} − f̃^{n}∥ < \frac{9}{10}δ,

∥f̃_{n}∥ = (1 − ∥f̃_{n} − f_{n}e∥^{2})^{1/2} ≥ (1 − (δ/5)^{2})^{1/2} > \sqrt{24/25} > 9/10,

and so

∥∥f̃_{n}∥^{−1} · f̃_{n} − ∥f̃_{n}∥^{−1} · f̃^{n}∥ < δ ,

which contradicts the definition of δ, since f̃_{n} ∈ K_{Θ_{n},Δ_{n},P_{n}} and f̃^{n} ∈ K = span{K_{Θ_{k},Δ_{k},P_{k}} : k ≠ n}.

References

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[15] S. V. Khrusčev and N. K. Nikolskii, Functional model and some problems of spectral theory of functions, Trudy Mat. Inst. Steklov, 176 (1987), 97–209 (Russian); English traslation: Proc. Steklov Inst. Math.

[16] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981.

[17] J. Bourgain, On finitely generated closed ideals in H^{∞}(𝔻), An. Institute Fourier, Grenoble, 35(1985), No 4, 163–174.

[18] S. Ziskind, Interpolating sequences and the Shilov boundary of H^{∞}(Δ), J. Funct. Anal, 21 (1976), No. 4, 380–388.

Department of Mathematics, Michigan State University, East Lansing, MI 48823

E-mail address: treil@math.msu.edu