A Dialogue Between Two Lifting Theorems

Rodrigo Arocena ^{∗}

En conmemoración de los 20 años del Postgrado en Matemática de la Universidad Central de Venezuela.

Han pasado 20 años desde que formé parte de la primera generación de estudiantes del Postgrado en Matemática de la Facultad de Ciencias de la UCV. Por entonces, la “casa que vence a las sombras” me había ofrecido la oportunidad de volver a estudiar, enseñar y respirar el aire vivificante de una universidad autónoma y democráticamente cogobernada, lo que era imposible en el Cono Sur ensombrecido por las dictaduras. Hace más de una década, al retornar al Uruguay para colaborar en la reconstrucción de una enseñanza devastada, afirmé en la renuncia a mi cargo en la UCV: “pase lo que pase, ésta será para siempre mi Universidad.” Hoy quiero agregar que, cerca o lejos, siempre me he sentido trabajando en el Grupo de Teoría de Operadores de la UCV. Lo que sigue se inscribe en esa labor.

Abstract. The relation between the lifting theorems due to Nagy-Foias and Cotlar-Sadosky is discussed.

PRESENTATION.

The Nagy-Foias commutant lifting theorem is a basic result in Operator Theory and its applications to interpolation problems. Its scope is shown in a fundamental book due to Foias and Frazho where we can read that “the work on the general framework of the commutant lifting theorem continued to grow mainly in Romania, the U.S.A. and Venezuela.” [FF, p. viii]

Now, the “Southamerican” contribution to the subject stems from the purpose of understanding the relations between the Nagy-Foias theorem and the Cotlar-Sadosky theorem on “weakly positive” matrices of measures.

The aim of this note is to recall some aspects of a “dialogue” between those two theorems that ends by showing that they can be seen as alternative ways of describing the same facts: see below, theorems (4) and (7).

∗Universidad de La República, Montevideo, Uruguay.

THE COTLAR-SADOSKY THEOREM.

We shall use the following notation e_{n}(t) = e^{int}, n ∈ ℤ and t ∈ ℝ, P is the space of trigonometric polynomials, i.e. of finite sums ∑ a_{n}e_{n}, with n ∈ ℤ and a_{n} ∈ ℂ, P_{+} = {∑ a_{n}e_{n} ∈ P : a_{n} = 0 if n < 0}, P_{−} = {∑ a_{n}e_{n} ∈ P : a_{n} = 0 if n ≥ 0}; 𝕋 denotes the unit circle on the complex plane ℂ, C(𝕋) is the Banach space of complex continuous functions on 𝕋 and M(𝕋) its dual, i.e., the space of complex Radon measures on 𝕋; for any p ≥ 1, H^{p} = {f ∈ L^{p} ≡ L^{p}(𝕋) : f̂(n) = 0 if n < 0}, where f̂ is the Fourier transform of f.

If μ = {μ_{jk}}_{j,k=1,2} is a matrix with entries in M(𝕋) and f = (f_{1}, f_{2}) ∈ C(𝕋) × C(𝕋), we set

⟨μf, f⟩ = ∑{∫_{𝕋} f_{j}f̄_{k} dμ_{jk} : j, k = 1, 2}.

Then ⟨μf, f⟩ ≥ 0, ∀f = (f_{1}, f_{2}) ∈ C(𝕋) × C(𝕋), iff μ is a positive matrix measure, i.e., {μ_{jk}(Δ)} is a positive matrix for any Borel set Δ ⊂ 𝕋, and the Cotlar-Sadosky theorem [CS. 1] can be stated as follows.

(1) Theorem. If the matrix measure μ = {μ_{jk}}_{j,k=1,2} is such that ⟨μf, f⟩ ≥ 0, ∀f = (f_{1}, f_{2}) ∈ P_{+} × P_{−}, there exists a positive matrix measure σ = {σ_{jk}}_{j,k=1,2} such that ⟨σf, f⟩ = ⟨μf, f⟩, ∀f ∈ P_{+} × P_{−}.

The above statement implies that

σ_{11} = μ_{11}, σ_{22} = μ_{22}, σ_{12} = σ̄_{21} = μ_{11} + h dt,

where dt is the Lebesgue measure in 𝕋 and h ∈ H^{1}.

The matrix μ such that ⟨μf, f⟩ ≥ 0 for every f ∈ P_{+} × P_{−} is called “weakly positive” and the theorem says that the weakly positive form defined by μ on P_{+} × P_{−} can be “lifted” (or, more precisely, extended) to the positive form defined by σ on C(𝕋) × C(𝕋).

THE NAGY-FOIAS THEOREM IMPLIES THE COTLAR-SADOSKY THEOREM

When theorem (1) was proved in 1979, Cotlar said that it was related to the abstract version of Sarason’s generalized interpolation theorem [S.1], i.e., the famous Nagy-Foias commutant lifting theorem proved in 1968 ([NF.1]; see also [NF.2] and [FF]).

In order to recall its statement we fix the following notation. If G, H are Hilbert spaces, L(G, H) is the set of bounded linear operators from G to H and L(G) = L(G, G); if K is a closed subspace of G, P_{K} denotes the orthogonal projection of G onto K, i_{K} the injection of K in G and G θ K the orthogonal complement of K in G. Also, ⋁ means “closed linear span of”. Unless otherwise stated, all spaces are Hilbert spaces and all subspaces are closed subspaces.

If X ∈ L(E_{1}, E_{2}) and E_{j} is a subspace of the space G_{j}, j = 1, 2, then B ∈ L(G_{1}, G_{2}) is a lifting of X if P_{E_{2}}B = XP_{E_{1}}.Nagy’s dilation theorem ([NF. 2], [FF]) says that if T ∈ L(E) is a contraction there exists an essentially unique unitary operator U ∈ L(F) such that E ⊂ T, T^{n} = P_{E}U^{n}|_{E} for every n ≥ 0 and F = ∨{U^{n}E : n ∈ ℤ}; U is called the minimal unitary dilation of T; set G = ∨{U^{n}E : n ≥ 0} and W = U_{|G}, then W ∈ L(G) is the essentially unique minimal isometric dilation of T : W is an isometry that lifts T, P_{E}W = T P_{E}, and G = ∨{W^{n}E : n ≥ 0}. Then:

(2) Theorem. For j = 1, 2 let T_{j} ∈ L(E_{j}) be a contraction in a Hilbert space, W_{j} ∈ L(G_{j}) its minimal isometric dilation and U_{j} ∈ L(F_{j}) its minimal unitary dilation. If X ∈ L(E_{1}, E_{2}) and XT_{1} = T_{2}X, then:

i) ∃B ∈ L(G_{1}, G_{2}) such that BW_{1} = W_{2}B, P_{E_{2}}B = XP_{E_{1}}, ||B|| = ||X||; ii) ∃Y ∈ L(F_{1}, F_{2}) such that Y U_{1} = U_{2}Y, P_{E_{2}}Y_{|E_{1}} = X, ||Y || = ||X||;

In fact, (1) can be proved by means of (2) in the way we now sketch. Let the shift S be given by (Sf)(z) ≡ zf(z). Set F_{j} = L^{2}(μ_{jj}), U_{j} the shift in F_{j}, j = 1, 2, E_{1} (E_{2}) the closure of P_{+} (P_{−}) in F_{1} (F_{2}), T_{1} = U_{1|E_{1}} and T_{2} = P_{E_{2}}U_{2|E_{2}}. Define X ∈ L(E_{1}, E_{2}) by

⟨Xf_{1}, f_{2}⟩ = ∫_{𝕋} f_{1}f̄_{2} dμ_{12}, ∀(f_{1}, f_{2}) ∈ P_{+} × P_{−}.

Then U_{j} is the minimal unitary dilation of T_{j}, j = 1, 2, ||X|| ≤ 1 and XT_{1} = T_{2}X. Any Y as in (2ii) is given by the multiplication by a function u = Y e_{0} so ⟨Y f_{1}, f_{2}⟩ = ∫_{𝕋} f_{1}f̄_{2}u dμ_{22}, ∀(f_{1}, f_{2}) ∈ P×P. Since ||Y || = ||X|| ≤ 1, the matrix measure σ given by σ_{11} = μ_{11}, σ_{22} = μ_{22}, σ_{12} = σ̄_{21} = udμ_{22} is as stated. Remark We obtained the function u because any operator Y that intertwines the shifts, ie., such that Y S_{1} = S_{2}Y, is a multiplication. In this way, the commutant lifting theorem extends Sarason’s method and gives all the solutions of several interpolation problems [FF].

THE EXTENDED COTLAR-SADOSKY THEOREM IMPLIES THE NAGY-FOIAS THEOREM

The following [CS.2] is an extension of theorem (1).

(3) Theorem. For j = 1, 2 let V_{j} be a vector space, L_{j} a subspace and τ_{j} : V_{j} → V_{l} a linear isomorphism such that τ_{1}L_{1} ⊂ L_{1} and τ_{2}^{−1}L_{2} ⊂ L_{2}, α_{j} : V_{j} ×V_{j} → ℂ is a positive form such that α_{j}(τ_{j}v, τ_{j}w) ≡ α_{j}(v, w), and β′ : L_{1} × L_{2} → ℂ a sesquilinear form such that β′(τ_{1}w_{1}, w_{2}) = β′(w_{1}, τ_{2}^{−1}w_{2}) and |β′(w_{1}, w_{2})|^{2} ≤ α_{1}(w_{1}, w_{1})α_{2}(w_{2}, w_{2}), ∀(w_{1}, w_{2}) ∈ L_{1} × L_{2}. Then β′ can be extended to a sesquilinear form β : V_{1} × V_{2} → ℂ such that β(τ_{1}v_{1}, τ_{2}v_{2}) = β(v_{1}, v_{2}) and |β(v_{1}, v_{2})|^{2} ≤ α_{1}(v_{1}, v_{1})α_{2}(v_{2}, v_{2}), ∀(v_{1}, v_{2}) ∈ V_{1} × V_{2}.

Set V_{j} = L^{2}(μ_{jj}), τ_{j} the shift in V_{j}, α_{j}(v, w) ≡ ∫_{𝕋} vw̄dμ_{jj}, L_{1} = P_{+}, L_{2} = P_{−} and β′ ≡ ∫_{𝕋} vw̄dμ_{12}; apply (3), then β is given by an operator that intertwines the shifts and (1) follows as above.

When β′(τ_{1}w_{1}, w_{2}) ≡ β′(w_{1}, τ_{2}^{−1}w_{2}) it is said that β′ is a generalized Hankel form, and when β(τ_{1}v_{1}, τ_{2}v_{2}) ≡ β(v_{1}, v_{2}) it is said that β is a generalized Toeplitz form; thus, (3) is a result concerning the extension of Hankel forms to Toeplitz forms.

Theorem (1) was presented as a property of a class of “modified Toeplitz kernels” [CS.1]. That result was extended to vector valued “generalized Toeplitz kernels” in [AC], where an extension of the famous Naimark dilation for Toeplitz kernels was proved by the method of unitary extensions of isometries. The dilation theorem for generalized Toeplitz kernels gives a proof of the Nagy-Foias theorem ([A.1]; see also [FF], VII.8) so the last and (an extension of) theorem (1) are in fact closely related.

But the story of the dialogue between these two lifting theorems is much longer. For example, theorem (3) was first proved as a consequence of the Nagy-Foias theorem and then an independent proof was given by the method of unitary extensions of isometries ([CS.3]), a method by means of which a direct proof of the Nagy-Foias theorem can be given ([A.2]; see also [S.2] and [F]).

We shall now show that theorem (3) implies (2). With notations as before, assume ||X|| = 1 and set V_{j} = F_{j}, τ_{j} = U_{j}, L_{1} = G_{1}, L_{2} = G′_{2} := ∨{U_{2}^{n}E_{2} : n ≤ 0}, α_{j} the scalar product in F_{j}; let β′ : L_{1} × L_{2} → ℂ be given by β′(w_{1}, w_{2}) = ⟨XP_{E_{1}}w_{1}, w_{2}⟩. Thus

β′(U_{1}w_{1}, w_{2}) ≡ ⟨XP_{E_{1}}W_{1}w_{1}, w_{2}⟩ ≡ ⟨T_{2}XP_{E_{1}}w_{1}, w_{2}⟩ = β′(w_{1}, U_{2}^{−1}w_{2})

and |β′(w_{1}, w_{2})|^{2} ≤ ⟨w_{1}, w_{1}⟩⟨w_{2}, w_{2}⟩ ∀(w_{1}, w_{2}) ∈ L_{1} × L_{2}. Then (3) says that there exists an extension β of β′ such that ||β|| ≤ 1 and β(U_{1}v_{1}, U_{2}v_{2}) = β(v_{1}, v_{2}). Consequently, there exists Y ∈ L(F_{1}, F_{2}) such that β(v_{1}, v_{2}) ≡ ⟨Y v_{1}, v_{2}⟩ and that (2.ii) holds. Moreover, P_{G′_{2}}Y_{|G_{1}} = XP_{E_{1}}; since G′_{2}θE_{2} = F_{2}θG_{2}, we see that Y G_{1} ⊂ G_{2}; setting B = Y_{|G_{1}}, (2.i) follows.

THE FIXED POINT PROOF OF AN EXTENDED NAGY-FOIAS THEOREM

As an illustration of the approach to lifting problems developed in [AADM.1,2] and [G], and related with [TV], we shall sketch the proof of a particular case of the results obtained by means of a fixed point theorem.

It is said that W ∈ L(G) is an expansive operator if ||v|| ≤ ||W v|| for every v ∈ G. The following is a slightly extended version of the Nagy-Foias theorem.

(4) Theorem. Let W_{1} ∈ L(G_{1}) be an expansive lifting of T_{1} ∈ L(E_{1}) and W_{2} ∈ L(G_{2}) be a contracting lifting of T_{2} ∈ L(E_{2}); if X ∈ L(E_{1}, E_{2}) and XT_{1} = T_{2}X there exists B ∈ L(G_{1}, G_{2}) such that BW_{1} = W_{2}B, P_{E_{2}}B = XP_{E_{1}} and ||B|| = ||X||.

The theorem can be proved in two steps which we now sketch. We may assume ||X|| = 1. Assertion (i) Set β = {B ∈ L(G_{1}, G_{2}) : P_{E_{2}}B = XP_{E_{1}}, ||B|| = ||X||}; for any B ∈ β there exists B^{♯} ∈ β such that B^{♯}W_{1} = W_{2}B.

Set X′ = XP_{E_{1}} ∈ L(G_{1}, G_{2}); then X′ ∈ β and T_{2}X′ = X′W_{1}. Thus, B ∈ β iff B = X′ + K(I − X^{′∗}X′)^{1/2} with K a contraction in L(G_{1}, G_{2}θE_{2}). With obvious notation, B^{♯}W_{1} = W_{2}B iff K^{♯}(I − X^{′∗}X′)^{1/2}W_{1} = P_{G_{2}θE_{2}}W_{2}B. Now,

||P_{G_{2}θE_{2}}W_{2}Bw||^{2} = ||W_{2}Bw||^{2} − ||T_{2}X′w||^{2} ≤ ||w||^{2} − ||X′W_{1}w||^{2}
≤ ||W_{1}w||^{2} − ||X′W_{1}w||^{2} = ||(I − X^{′∗}X′)^{1/2}W_{1}w||^{2}

for every w ∈ G_{1}; let L be the closure of (I −X^{′∗}X′)^{1/2}W_{1}G_{1}; a unique contraction K^{♯} ∈ L(G_{1}, G_{2}θE_{2}) is defined by K^{♯} = K^{♯}P_{L} and K^{♯}(I −X^{′∗}X′)^{1/2}W_{1}w ≡ P_{G_{2}θE_{2}}W_{2}Bw. Assertion (i) follows. Assertion (ii) Set Σ = {K ∈ L(G_{1}, G_{2}θE_{2}) : ||K|| ≤ 1}; the map λ : Σ → Σ given by λ(K) ≡ K^{♯} has a fixed point.

With the operator topology in L(G_{1}, G_{2}θE_{2}), Σ is compact and λ is continuous: if K_{t} → K in Σ then, for every w ∈ G_{1} and w ∈ G_{2}θE_{2},

⟨λ(K_{t})[(I − X^{′∗}X′)^{1/2}W_{1}w], v⟩ = ⟨W_{2}[X′ + K_{t}(I − X^{′∗}X′)^{1/2}]w, x⟩
→ ⟨W_{2}[X′ + K(I − X^{′∗}X′)^{1/2}]w, x⟩
= ⟨λ(K)[(I − X^{′∗}X′)^{1/2}W_{1}w], v⟩

so λ(K_{t}) → λ(K). Thus, (ii) follows from the Schauder-Tychonov fixed point theorem [DS].

Clearly, if λ(K) = K, B = X′ + K(I − X^{′∗}X′)^{1/2} is as in (II.1). Remark The lifting problem can have no solution: set ([FF], p.100) E_{1} = E_{2} = ℂ, T_{1} = T_{2} = 0, X = 1, G_{1} = G_{2} = ℂ^{2}, W_{1} = [w_{jk}^{(1)}] with w_{11} = w_{12} = w_{21} =

0, w_{22} = 1, W_{2} = [w_{jk}^{(2)}] with w_{11} = w_{12} = w_{22} = 0, w_{21} = 1. Then an operator B as in (II.1) does not exist. Note that W_{2} is a contractive lifting of T_{2} and that W_{1} is a lifting of T_{1} but W_{1} is not expansive.

(5) Corollary. For j = 1, 2 let S_{j} ∈ L(E_{j}) be a contraction with minimal isometric dilation V_{j} ∈ L(G_{j}) such that P_{E_{j}}R_{j} = R_{j}P_{E_{j}}. If R_{1} is expansive, R_{2} is contractive and X ∈ L(E_{1}, E_{2}) is such that XS_{1}R_{1|E_{1}} = R_{2}S_{2}X, then, there exists B ∈ L(G_{1}, G_{2}) such that BV_{1}R_{1} = R_{2}V_{2}B, P_{E_{2}}B = XP_{E_{1}} and ||B|| = ||X||. \underline{Proof}. Set W_{1} = V_{1}R_{1}, T_{1} = S_{1}R_{1|E_{1}}, W_{2} = R_{2}V_{2}, T_{2} = R_{2}S_{2}. Then W_{1} is expansive, W_{2} is contractive, P_{E_{1}}W_{1} = T_{1}P_{E_{1}}, P_{E_{2}}W_{2} = T_{2}P_{E_{2}} and XT_{1} = T_{2}X. The result follows from (4).

The corollary above was suggested by the following result due to Sebestyén [Se].

(6) Theorem. Let S ∈ L(E) be a contraction with minimal isometric dilation V ∈ L(G) and R ∈ L(G) a contraction that commutes with the orthogonal projection P_{n} of G onto ∨{V^{j}E : 0 ≤ j ≤ n} for n = 0, 1, … If X ∈ L(E) satisfies XS = RSX there exists B ∈ L(G) such that BV = RV B, P_{E_{2}}B = xP_{E_{1}} and ||B|| = ||X||.

Since P_{0} = P_{e}, (6) is a particular case of (5).

A REFORMULATION A LA COTLAR-SADOSKY OF THE EX-TENDED NAGY-FOIAS THEOREM

As we shall see, the following result is not only quite similar but also equivalent to the extended Nagy-Foias theorem (4) and it gives an extension of the Cotlar-Sadosky theorem (7).

(6) Theorem. Let E_{j} be a Hilbert space and T_{j} ∈ L(E_{j}), j = 1, 2 and γ : E_{1} × E_{2} → ℂ a sesquilinear bounded form such that γ(T_{1}e_{1}, e_{2}) ≡ γ(e_{1}, T_{2}e_{2}). If W_{1} ∈ L(G_{1}) is an expansive lifting of T_{1} and W_{2} ∈ L(G_{2}) a contractive extension of T_{2}, there exists a sesquilinear bounded extension λ : G_{1} × G_{2} → ℂ of γ such that:

λ(W_{1}g_{1}, g_{2}) ≡ λ(g_{1}, W_{2}g_{2})  (1)
λ(g_{1}, e_{2}) = γ(P_{E_{1}}g_{1}, e_{2}) for every g_{1} ∈ G_{1} and e_{2} ∈ E_{2}  (2)
||λ|| = ||γ||  (3)

Assertion (i) Theorem 4 implies theorem 7.

Let X ∈ L(E_{1}, E_{2}) be such that γ(e_{1}, e_{2}) ≡ ⟨Xe_{1}, e_{2}⟩, then XT_{1} = T_{2}^{∗}X and W_{2}^{∗} is a contractive lifting of T_{2}^{∗}, so there exists B ∈ L(G_{1}, G_{2}) such that BW_{1} = W_{2}B, P_{E_{2}}B = XP_{E_{1}} and ||B|| = ||X||. Setting λ(g_{1}, g_{2}) ≡ ⟨Bg_{1}, g_{2}⟩ the result follows.

Assertion (ii) Theorem 7 implies theorem 4.

Set γ(e_{1}, e_{2}) ≡ ⟨Xe_{1}, e_{2}⟩, then γ(T_{1}e_{1}, e_{2}) ≡ γ(e_{1}, T_{2}^{∗}e_{2}) and W_{2}^{∗} is a contractive extension of T_{2}^{∗}, so there exists λ as in (7); let B ∈ L(G_{1}, G_{2}) be such that λ(g_{1}, g_{2}) ≡ ⟨Bg_{1}, g_{2}⟩ . Then ||B|| = ||λ|| = ||γ|| = ||X||; also,

⟨BW_{1}g_{1}, g_{2}⟩ ≡ λ(W_{1}g_{1}, g_{2}) ≡ λ(g_{1}, W_{2}^{∗}g_{2}) ≡ ⟨Bg_{1}, W_{2}^{∗}g_{2}⟩,

so BW_{1} = W_{2}B; finally,

⟨P_{E_{2}}Bg_{1}, e_{2}⟩_{E_{2}} ≡ λ(g_{1}, e_{2}) ≡ γ(P_{E_{1}}g_{1}, e_{2}) ≡ ⟨XP_{E_{1}}g_{1}, e_{2}⟩

so P_{E_{2}}B = XP_{E_{1}}.

Assertion (iii) Theorem 7 implies theorem 3.

For j = 1, 2 let F_{j} be the Hilbert space generated by the vector space V_{j} and the positive form α_{j}: there exists a linear operator π_{j} : V_{j} → F_{j} such that π_{j}(V_{j}) is dense in F_{j} and ⟨π_{j}v, π_{j}v′⟩ = α_{j}(v, v′) for every v, v′ ∈ V_{j}. Let U_{j} ∈ L(F_{j}) be the unitary operator given by U_{j}π_{j} = π_{j}τ_{j} and G_{j} be the closure in F_{j} of π_{j}L_{j}. Let γ be a sesquilinear form γ : G_{1} × G_{2} → ℂ such that γ(U_{1}g_{1}, g_{2}) ≡ γ(g_{1}, U_{2}^{−1}g_{2}) and ||γ|| ≤ 1 is defined by setting γ(π_{1}v_{1}, π_{2}v_{2}) = β′(v_{1}, v_{2}) for every (v_{1}, v_{2}) ∈ L_{1} × L_{2}.

Since U_{2}^{−1} ∈ L(F_{2}) is a contractive extension of U_{2}^{−1}_{|G_{2}}, there exists a sesquilinear form λ : G_{1} × F_{2} → ℂ that extends γ and is such that ||λ|| ≤ 1 and that λ(U_{1}g_{1}, U_{2}f_{2}) = λ(g_{1}, f_{2}) holds for every (g_{1}, f_{2}) ∈ G_{1} × F_{2}.

Now set F′_{1} = ∨{U_{1}^{−n}G_{1} : n ≥ 0} and extend λ to a sesquilinear form λ_{1} : F_{1} × F_{2} → ℂ by setting, for any n ≥ 0 and (g_{1}, f_{2}) ∈ G_{1} × F_{2}, λ_{1}(U_{1}^{−n}g_{1}, f_{2}) = λ(g_{1}, U_{2}^{n}f_{2}), then ||λ_{1}|| = ||λ|| and λ_{1}(f_{1}, f_{2}) ≡ λ_{1}(U_{1}f_{1}, U_{2}f_{2}).

Setting β(v_{1}, v_{2}) = λ_{1}(P_{F′_{1}}π_{1}v_{1}, π_{2}v_{2}) for every (v_{1}, v_{2}) ∈ V_{1} × V_{2}, the result follows. Final Remark.

Summing up, as Cotlar anticipated, theorems 1 and 2 are in fact very closely related.

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Rodrigo Arocena, Centro de Matemática, Facultad de Ciencias, Universidad de la República. José M. Montero 3006 Montevideo URUGUAY. Fax: (5982)402954 E-mail: rarocena@cmat.edu.uy