Lobachevskii Journal of Mathematics Vol. 10, 2002, 3-8 \begin{matrix}c \\ ◯ \end{matrix}B. Aqzzouz, M. El Kadiri

B. Aqzzouz, M. El Kadiri

INFINITE DIMENSIONAL EXTENSION OF A.P. CALDERÒN’S

THEOREM ON POSITIVE SEMIDEFINITE BIQUADRARIC FORMS

(submitted by D. Kh. Mushtari)

Abstract. We extend to the infinite dimensional separable real Hilbert spaces a theorem of A. P. Calderòn which says that, if m = 2 or n = 2, then every positive semidefinite biquadratic form on R^{m} × R^{n} is a sum of squares of bilinear forms.

1. Introduction

Let H and K be two real and separable Hilbert spaces. We denote by L(H, K) the space of bounded linear operators from H into K, and Q_{+}(H, K) the convex cone of continuous positive semidefinite biquadratic forms on H × K, that is, the set of real continuous functions F (x, y) on H × K which are quadratic in each variable and such that F (x, y) ≥ 0 forall (x, y) ∈ H ×K. The inner product in the Hilbert space H is always denoted by ⟨, ⟩_{H}. A biquadratic form on H × K can be explicitly defined by

F (x, y) = ∑ a_{jkpq}x_{j}x_{k}y_{p}y_{q}

for all x = ∑ x_{i}e_{i} ∈ H and y = ∑ y_{j}f_{j} ∈ K, where {e_{1}, ..., e_{n}, ...} and {f_{1}, ..., f_{n}, ...} are hilbertian basis of H and K respectively. The following problem is interesting in many of its aspects: Let F ∈ Q_{+}(H, K). Are there any bilinear forms f_{i} on H × K such that F = ∑_{i} f_{i}^{2} ? For H = R^{m} and K = R^{n}, in [1, th.1] A. P. Calderòn answers affirmatively this question when m = 2 or n = 2. In [2] D-M. Choi gives, for m = n = 3, an example of a positive semidefinite biquadratic form which is not the sum of the squares of bilinear forms. This shows that the answer to the above problem is negative in general. The importance of continuous positive semidefinite biquadratic forms on H × K consists in their link with the linear operators on the spaces of self-adjoint operators on H and K. We denote by SL_{2}(H) the space of the self-adjoint Hilbert-Schmidt operators on H, endowed with its natural structure of a Hilbert space defined by the inner product ⟨Φ, Ψ⟩_{tr} = tr(ΦΨ). We let SL_{2}^{+}(H) to be the positive cone of SL_{2}(H) and P(SL_{2}(H), SL_{2}(K)) the cone of positive linear operators from SL_{2}(H) into SL_{2}(K). A linear operator A : SL_{2}(H) → SL_{2}(K) is said to be positive if A(SL_{2}^{+}(H)) ⊂ SL_{2}^{+}(K). In other words, if it transforms the positive self-adjoint Hilbert-Schmidt operators on H into positive self-adjoint Hilbert-Schmidt operators on K. To each biquadratic form F ∈ Q_{+}(H, K) it is associated a unique operator A ∈ P(SL_{2}(H), SL_{2}(K)) such that F (x, y) = ⟨A(x ⊗ x)y, y⟩_{K}. Conversely, each operator A ∈ P(SL_{2}(H), SL_{2}(K)) defines, via the last formula, a unique biquadratic form F ∈ Q_{+}(H, K). If the space H is of finite dimension n, then SL_{2}(H) is just the space S_{n} of real symmetric matrices of order n. Let x, y ∈ H, we denote by x ⊗ y : H ⟶ H the operator defined forall z ∈ H by

Key words and phrases. Biquadratic form, Positive operator, Hilbert-Schmidt operator.

x ⊗ y(z) = ⟨y, z⟩_{H}x.

It is clear that, for all x ∈ H, we have x ⊗ x ∈ SL_{2}^{+}(H). On the other hand, it follows from the classical spectral theory that each operator Φ ∈ SL_{2}(H) can be written as

Φ = ∑_{i≥1} λ_{i}x_{i} ⊗ x_{i},

where x_{i}’s are the eigenvectors of Φ associated with the eigenvalues λ_{i}’s and forming an orthonormal basis of H. In particular, each Φ ∈ SL_{2}^{+}(H) can be written as follows

Φ = ∑_{i≥1} x_{i} ⊗ x_{i},

where x_{1}, x_{2}, ... are mutually orthogonal vectors in H. For every Σ ∈ L(H, K) and Φ ∈ SL_{2}(H), we have ΣΦΣ^{∗} ∈ SL_{2}(K), where Σ^{∗} is the adjoint operator of Σ. This allows us to define an operator ρ(Σ) ∈ L(SL_{2}(H), SL_{2}(K)) by

ρ(Σ)(Φ) = ΣΦΣ^{∗}, ∀Σ ∈ L(H, K).

It is easy to see that this operator belongs to P(SL_{2}(H), SL_{2}(K)). In fact, this is due to the spectral decomposition and the identity

ρ(Σ)(x ⊗ x) = Σx ⊗ Σx

for all x ∈ H. Hence ⟨ρ(Σ)(x ⊗ x)y, y⟩_{K} = ⟨Σx, y⟩_{K}^{2} for all x ∈ H and y ∈ K. In terms of operators, the theorem of A. P. Calderòn says that, if m = 2 or n = 2, every operator A ∈ P(S_{m}, S_{n}) is the sum of operators of the form ρ(Σ), where Σ ∈ L(R^{n}, R^{m}). Let us recall that the simple convergence topology on L(H, K) is the topology associated with the family of semi-norms p_{x}, x ∈ H, defined by

p_{x}(A) = ||A(x)||_{K}, ∀A ∈ L(H, K).

Endowed with this topology, L(H, K) is a locally convex vector space. The main result of this paper is: Theorem 1. If the dimension of H or K is two, then for every operator A ∈ P(SL_{2}(H), SL_{2}(K)), there exists a measure μ ≥ 0, on the space L(H, K) endowed with its Borel σ-algebra associated with the simple convergence topology such that

A = ∫ ρ(Σ)dμ(Σ)

in that weak sense forall x ∈ H and y ∈ K we have

⟨A(x ⊗ x)y, y⟩_{K} = ∫ ⟨Σx, y⟩_{K}^{2}dμ(Σ).

This result extends a theorem of A. P. Calderòn [1, th.1] to the spaces R^{2} ×H and H × R^{2}, where H is a real separable infinite dimensional Hilbert space.

2. Proof of the Theorem 1 If K is a convex cone of vertex 0 in a real vector space E, we denote by Êxt(K) the set of extremal rays of K. We say that an element x of K is extremal if it belongs to an extremal ray of K, and we write x ∈ Êxt(K). Let us also point out that for every x ∈ H we have x ⊗ x ∈ Êxt(SL_{2}^{+}(H)).

Theorem 2.1. For every Σ ∈ L(H, K), the operator ρ(Σ) belongs to Êxt(P(SL_{2}(H), SL_{2}(K))).

Proof. The case where dimH = dimK < +∞ was treated in [4]. For x ∈ H we have ρ(Σ)(x ⊗ x) = (Σx) ⊗ (Σx) ∈ Êxt(SL_{2}^{+}(K)), and if A ∈ P(SL_{2}(H), SL_{2}(K)) is such that ρ(Σ) − A ∈ P(SL_{2}(H), SL_{2}(K)), then ρ(Σ)(x ⊗ x) − A(x ⊗ x) ∈ SL_{2}^{+}(K), hence there exists a real α(x) ∈ [0, 1] such that A(x ⊗ x) = α(x).ρ(Σ)(x ⊗ x) since ρ(Σ(x ⊗ x) = (Σx ⊗ Σx) is an extremal element of SL_{2}^{+}(K). We are going to prove that we can choose the real α(x) not depending on x, this fact will prove the theorem. Let {e_{1}, e_{2}, ...} and {f_{1}, f_{2}, ...} be Hilbertian basis of H and K respectively. Let us denote by (a_{ij}(x)) the matrix of the operator A(x ⊗ x) in the basis {f_{1}, f_{2}, ..} of K, and let (b_{ij}) be the matrix of the operator Σ in the basis {e_{1}, e_{2}, ...} and {f_{1}, f_{2}, ...}. Since the matrix (a_{ij}(x)) is positive not identically 0, there exists an integer i such that the quadratic form on H, say P (x) = a_{ii}(x), is not identically 0. Set also Q(x) = ∑_{j≥1} b_{ij}x_{j}. Thus P (x) = α(x).Q(x)^{2} for all x ∈ H. The quadratic form P is positive semi-definite, hence it can be written P (x) = ∑_{i≥1} P_{i}(x)^{2} for every x ∈ H, where for each i ≥ 1, P_{i} is a continuous linear form on H. The relation P (x) = α(x).Q^{2}(x) for all x ∈ H implies that the linear forms P_{i}, 1 ≤ i, and Q have the same kernel and thus are proportional. This allows to choose α(x) not depending on x and completes the proof of the theorem. The following trivial lemma allows us to restrict ourselves in the proof of the main result to the case where dimK = 2. Let us recall that in this case SL_{2}(K) can be identified with the space S_{2}. Lemma 2.2. Let A ∈ P(SL_{2}(H), SL_{2}(K)), then A ∈ Êxt(P(SL_{2}(H), SL_{2}(K))) if and only if A^{∗} ∈ Êxt(P(SL_{2}(K), SL_{2}(H))). In the sequel we will fix a Hilbertian basis {e_{1}, e_{2}..., } of H and denote by {f_{1}, f_{2}} the canonical basis of R^{2}. The space L(SL_{2}(H), SL_{2}(K)) is endowed with the topology defined by the family of seminorms

p_{Φ,Ψ}(A) = |tr(A(Φ).Ψ)|, Φ ∈ SL_{2}(H), Ψ ∈ L_{2}(K).

We recall that a subset C of a convex cone S in a Hausdorff topological vector space X is called a cap if C is compact convex and its complement in S is convex. A cap C of S is said to be universal if it generates S, that is if S = R_{+}.C.

Proposition 2.3. The set

C = {A ∈ P(SL_{2}(H), S_{2}); ∑_{i≥1,j=1,2} f_{j}^{T} A(e_{i} ⊗ e_{i})f_{j} ≤ 1}

is an universal cap of P(SL_{2}(H), S_{2}). Proof. It is clear that C and P(SL_{2}(H), S_{2}) \ C are convex. Let us prove that C is compact. This will complete the proof, because C is universal follows from the fact that for each A ∈ P(SL_{2}(H), S_{2}) and j = 1, 2, A^{∗}(f_{j} ⊗ f_{j}) is a Hilbert-Schmidt operator on H, and hence

∑_{i≥1,j=1,2} f_{j}^{T} A(e_{i} ⊗ e_{i})f_{j} = ∑_{i≥1,j=1,2}⟨A^{∗}(f_{j} ⊗ f_{j})e_{i}, e_{i}⟩_{H}

is finite. To prove that C is compact, let U be an ultrafilter on C, and Φ ∈ SL_{2}(H). It follows from the spectral theory that

Φ = ∑_{i≥1} λ_{i}e_{i} ⊗ e_{i} + ∑_{1≤i<k} λ_{ik}(e_{i} ⊗ e_{k} + e_{k} ⊗ e_{i})

in operators norm. The real sequences (λ_{i}) and (λ_{ik}) are bounded, thus there exists a real M > 0 such that |λ_{i}| ≤ M and |λ_{ik}| ≤ M for all i ≥ 1 and k > i. We then have for j = 1, 2 and A ∈ C:

|f_{j}^{T} A(Φ)f_{j}| ≤ M(∑_{i≥1} f_{j}^{T} A(e_{i} ⊗ e_{i})f_{j} + ∑_{1≤i<k≤n} |f_{j}^{T} A(e_{i} ⊗ e_{k} + e_{k} ⊗ e_{i})f_{j}|).

A straightforward computation shows that the operator e_{i} ⊗ e_{i} + e_{k} ⊗ e_{k} − e_{i} ⊗ e_{k} − e_{k} ⊗ e_{i} belongs to SL_{2}^{+}(H), thus we have for every A ∈ P(SL_{2}(H), S_{2})

|f_{j}^{T} A(e_{i} ⊗ e_{k} + e_{k} ⊗ e_{i})f_{j}| ≤ (f_{j}^{T} A(e_{i} ⊗ e_{i})f_{j}) + (f_{j}^{T} A(e_{k} ⊗ e_{k})f_{j}).

Hence |f_{j}A(Φ)f_{j}| ≤ ∑_{i≤n} |λ_{i}| ≤ 3, thus lim_{U} f_{j}^{T} A(Φ)f_{j} is finite and, consequently, lim_{U} A(Φ) exists in S_{2} and we have lim_{U} A(Φ) ∈ S_{2}^{+} if Φ ∈ SL_{2}^{+}(H). In an analogous way we may check that lim_{U} A^{∗}(Ψ) exists in SL_{2}(H) and that lim_{U} A^{∗}(Ψ) ∈ SL_{2}(H)^{+}if Ψ ∈ S_{2}^{+}. The operator B defined for Ψ ∈ S_{2} by lim_{U} A^{∗}(Ψ) is linear and continuous on S_{2} since it is positive. On the other hand, we have

⟨lim_{U} AΦ, Ψ⟩_{tr} = lim_{U}⟨AΦ, Ψ⟩_{tr} = ⟨B^{∗}Φ, Ψ⟩_{tr}.

It is clear that B^{∗} ∈ C. We then deduce that the ultrafilter U converges in P(SL_{2}(H), S_{2}) to an element of C. Hence C is compact. The proposition is proved. For every integer n ≥ 1 we identify R^{n} with the subspace of H generated by the vectors e_{1}, e_{2}, ..., e_{n}, and denote by i_{n} the canonical embedding from R^{n} into H and by π_{n} the orthogonal projection operator from H into R^{n}. We have

π_{n}(i_{n}(x)) = x, ∀x ∈ R^{n}

and

i_{n}(π_{n}(x)) = π_{n}(x), ∀x ∈ H.

To each operator A ∈ P(SL_{2}(H), S_{2}), we associate the operator A_{n} ∈ P(S_{n}, S_{2}) defined by

A_{n}(x ⊗ x) = A(i_{n}(x) ⊗ i_{n}(x)), ∀x ∈ R^{n}.

Conversely, to each operator A_{n} ∈ P(S_{n}, S_{2}) there corresponds an operator Ã_{n} ∈ P(SL_{2}(H), S_{2}) defined by

Ã_{n}(x ⊗ x) = A_{n}(π_{n}(x) ⊗ π_{n}(x)), ∀x ∈ H.

For every n ≥ 1, let

C_{n} = {A ∈ P(S_{n}, S_{2}); ∑_{j=1,2,1≤i≤n} f_{j}^{T} A(u_{i} ⊗ u_{i})f_{j} ≤ 1},

where {u_{1}, u_{2}, ..., u_{n}} is the canonical basis of R^{n}. If A ∈ P(SL_{2}(H), S_{2}), then A ∈ C if and only if A_{n} ∈ C_{n} for every n. Remark. We can replace S_{2} by S_{m} in the previous results and similar conclusions hold. We also need the following lemmas. Lemma 2.4. For every A ∈ C, we have Ã_{n} ∈ C for all n. Lemma 2.5. Let Σ̃ be the operator of L(H, R^{2}) defined by Σ̃ = Σ ∘ π_{n} then for each integer n and Σ ∈ L(R^{n}, R^{2}), we have ρ̃(Σ) = ρ(Σ̃). Lemma 2.6. The set F = {Σ ∈ L(H, R^{2}); ρ(Σ) ∈ C} is compact with respect to the simple convergence topology of L(H, R^{2}). Proof. It is easy to see that F is closed for the simple convergence topology of L(H, R^{2}) and that for each x ∈ H, the set {Σ(x); Σ ∈ F } is bounded in R^{2}. The lemma follows then easily from these observations. Lemma 2.7. For each A ∈ C and integer n > 0, there exists a probability measure μ_{n} on the compact {Σ ∈ L(H, R^{2}); ρ(Σ) ∈ C} of L(H, R^{2}) endowed with the simple convergence topology such that

Ã_{n} = ∫ ρ(Σ)dμ_{n}(Σ)

in the sense of the Theorem 1. Proof. According to A. P. Calderòn’s theorem we have,

A_{n} = ∑_{i=1}^{n(n+1)} ρ(Σ_{i})

where Σ_{i} ∈ L(R^{n}, R^{2}) for every i. Then, following Lemma 2.5,

Ã_{n} = ∑_{i=1}^{n(n+1)} ρ(Σ̃_{i}).

Since Ã_{n} ∈ C, we have ρ(Σ̃_{i}) ∈ C for all i such that 1 ≤ i ≤ n(n + 1). We then deduce that

Ã_{n} = ∫ ρ(Σ)dμ_{n}(Σ)

where μ_{n} is the probability measure ∑_{i=1}^{n(n+1)} \frac{l(ρ(Σ_{i}))}{l(Ã_{n})}ϵ_{Σ_{i}∘π_{n}}, l is the gauge of C and ϵ_{Σ_{i}} is the Dirac measure at Σ_{i}. Lemma 2.8. For every A ∈ P(SL_{2}(H), S_{2}), the sequence of operators Ã_{n} converges to A for the simple convergence topology in L(SL_{2}(H), S_{2}). Proof. For all n, S_{n} can be identified with a subspace of SL_{2}(H) by means of the embedding j_{n} defined by

j_{n}(x ⊗ x) = i_{n}(x) ⊗ i_{n}(x), ∀x ∈ R^{n}.

Let Π_{n} the orthogonal projection operator from SL_{2}(H) on S_{n}, we then have Ã_{n} = A ∘ Π_{n} and this easily yields the lemma. We can now prove the Theorem 1. Proof of the Theorem 1. Let A ∈ P(SL_{2}(H), S_{2}). Since the cap C is universal, then by multiplying A by a real number t > 0, we may assume that A ∈ C. It follows from Lemma 2.7 that there exists a sequence (μ_{n}) of probability measures on the compact set K = {Σ; ρ(Σ) ∈ C} such that Ã_{n} = ∫ ρ(Σ)dμ_{n}(Σ) for all n. On the other hand, we can extract from (μ_{n}) a subsequence (μ_{n′}) which converges weakly to a probability measure μ on K. So we get A = ∫ ρ(Σ)dμ(Σ) by passing to the weak limit in the relation Ã_{n′} = ∫ ρ(Σ)dμ_{n′}(Σ). Corollary. If dimH = 2 or dimK = 2, we have

Êxt(P(SL_{2}(H), SL_{2}(K))) = {ρ(Σ); Σ ∈ L(K, H)}.

References

[1] A. P. Calderón, A note on biquadratic forms, Linear Algebra Appl. 7 (1973), 175-177.

[2] M.-D. Choi, Positive semidefinite biquadratic forms, Linear Algebra Appl. 12 (1975), 95-100.

[3] M. El Kadiri, Sur les génératrices extrémales de certains cônes de formes quadratiques doublement positives, C.R. Acad. Sci. Paris, Ser. I, 324 (1997), 421-425.

[4] M. El Kadiri, Opérateurs positifs sur certains espaces de formes quadratiques, Linear Algebra Appl., 283 (1998), 273-288.

Belmesnaoui Aqzzouz, Département de Mathématiques, Faculté des Sciences, Université Ibn Tofail, B.P. 133, Kénitra, Morocco E-mail address: aqzzouz@hotmail.com Mohamed El Kadiri, B.P. 726 Salé-Tabriquet, Salé, Morocco E-mail address: elkadiri@fsr.ac.ma

Received December, 20, 2001; Revised version February, 25, 2002