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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007

 

Non Positively Curved Metric in the Space of Positive Definite Infinite Matrices

Esteban Andruchow and Alejandro Varela

Abstract We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ ∞ of positive invertible operators of a Hilbert space H , which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive definite complex matrices. Moreover, these spaces of finite matrices are naturally imbedded in Σ ∞ .

Key words and phrases. positive operator, Hilbert-Schmidt class

2000 Mathematics Subject Classification. 58B20, 47B10, 47B15

1. INTRODUCTION

The space M +n (ℂ ) of positive definite (invertible) matrices is a differentiable manifold, in fact an open subset of the real euclidean space of hermitian matrices. Let X, Y be hermitian matrices and A positive definite, the formula

 -1 - 1 < X, Y >A= tr(XA Y A )

endows M +n (ℂ) with a Riemannian metric, which makes it a non positively curved, complete symmetric space. This metric is natural: it is the Riemannian metric obtained by pushing the usual trace norm on matrices to M + (ℂ) n by means of the identification

 + Gl(n )∕U (n) ≃ M n (ℂ ),

G + U(n ) ↦→ G *G,

where Gl(n) and U(n) are, respectively, the linear and unitary groups of ℂn . This example has a universal property: every symmetric space of noncompact type can be realized isometrically as a complete totally geodesic submanifold of  + M n (ℂ ) [7].

These facts are well known, have been used in a variety of contexts, and have motivated several extensions. For example, in the interpolation theory of Banach and Hilbert spaces [6], [16], in partial differential equations [15], and in mathematical physics [12], [17], [9]. They have also been generalized to infinite dimensions, i.e. Hilbert spaces and operator algebras: [17], [3], [2] [5], [1].

The purpose of this note is to introduce a Riemannian metric in the space Σ∞ of infinite positive definite matrices, that is the set of positive and invertible operators on an infinite dimensional Hilbert space H , which are of the form

A = λI + A0 = λ + A0,

where λ ∈ ℂ and A0 ∈ B2 (H ) , the class of Hilbert-Schmidt operators, i.e. elements B ∈ B(H ) such that tr(B *B ) < ∞ . We shall regard Σ∞ as an infinite dimensional manifold, in fact as an open subset of an apropriate infinite dimensional euclidean space, and introduce a Riemannian metric in Σ ∞ , whichs looks formally identical to the metric for finite matrices. It will be shown that with this metric Σ ∞ becomes a non positively curved, complete Riemannian manifold, which contains in a natural (isometric, flat) manner all spaces M n+(ℂ) .

Therefore Σ ∞ can be regarded as a universal model, containing isometric and totally geodesic copies of all finite dimensional symmetric spaces of non compact type.

Let us fix some notation. We shall denote by ∥ ∥ the usual norm of B(H ) and by ∥ ∥2 the Hilbert-Schmidt norm: ∥B ∥2 = tr(B *B)1∕2 . Denote by

H = {λ + X ∈ B(H ) : λ ∈ ℂ, X ∈ B2 (H )},

and

 * H ℝ = {λ + B ∈ H : (λ + B ) = λ + B }.

Note that since H is infinite dimensional, the scalars λ and the operators X, B in B2 (H ) are linearly independent. In particular, one has that λ + B ∈ H ℝ if and only if λ ∈ ℝ and B* = B . Formally, H = ℂ ⊕ B (H ) 2 and H = ℝ ⊕ B (H ) ℝ 2 h , where B2 (H )h denotes the real Hilbert space of selfadjoint Hilbert-Schmidt operators. Let us define

 -- * < λ + X, μ + Y >= λμ + tr(Y X ).

Clearly this inner product makes H , H ℝ , respectively, complex and real Hilbert spaces, where the scalars λ and the operators (in B2(H ) ) are orthogonal. The space Σ ∞ ⊂ H ℝ will be considered with the relative topology induced by this inner product norm. It follows that the maps H → ℂ , λ + X ↦→ λ and H → B (H ) 2 , λ + X ↦→ X are orthogonal projections and their adjoints are the inclusions, which are therefore isometric.

Note that λ + A ≥ 0 means that λ ≥ 0 and the spectrum of A is a subset of [- λ,+ ∞ ) . Indeed, the first assertion follows from the fact that 0 ∈ σ(A ) , and the second is obvious.

In what follows, we denote by ∥ ∥ 2 the norm of H . No confusion should arise with the norm of B2(H ) , because the former extends the latter.

2. BASIC PROPERTIES OF Σ∞

Let us prove some elementary facts concerning the topology of Σ ∞ .

Proposition 2.1. Σ ∞ is open and convex in Hℝ .

Proof. The fact that Σ ∞ is convex is apparent. Let X0 = λ0 + A0 ∈ Σ∞ . Since X0 is positive and invertible, it follows that the eigenvalues of A0 are bounded from below by - λ0 , and do not aproach - λ0 . Then there exists r > 0 such that - λ0 + r < A0 , or in other words, λ - r + A 0 0 is positive and invertible. Consider the ball

D (X ) = {X = μ + B ∈ H : ∥X - X ∥ < r∕2 }. r∕2 0 ℝ 0 2

We claim that if X ∈ Dr ∕2(X0 ) , then X ∈ Σ ∞ . Indeed, ∥X - X0 ∥22 = (λ - λ0)2 + ∥B - A0 ∥22 . Then |λ - λ0| < r∕2 and the operator norm ∥B - A0 ∥ ≤ ∥B - A0∥2 < r∕2 . Then λ > λ0 - r∕2 and B - A0 ≥ - r ∕2 , and therefore

λ + B > λ0 - r∕2 + B = λ0 - r∕2 + (B - A0 ) + A0 ≥ λ + A0 - r,

which is positive and invertible. It follows that X ∈ Σ ∞ . □

The following elementary estimations will be useful.

Lemma 2.2. Let X = λ + B, Y = μ + C ∈ H , then

  1.  - ∥X ∥ ≥ √2∥X ∥. 2 2
  2. ∥XY ∥2 ≤ 2∥X ∥2∥Y ∥2

Proof. Let βn be the singular values of B . Then  2 2 ∑ 2 ∥X ∥2 = λ + n≥1 βn . On the other hand, ∥X ∥ = sup {|λ + βn | : n ≥ 1} . Since the singular values accumulate eventually only at 0 , clearly one has ∥X ∥ = |λ| or ∥X ∥ = |λ + βk| for some k . In either case

 ∑ 2(λ2 + β2n) ≥ 2(λ2 + β2k) ≥ (|λ| + |βk|)2 ≥ (λ + βk)2, n≥1

which proves the first assertion. For the second, ∥(λ + B )(μ + C )∥2 ≤ |λ ||μ| + |μ |∥B ∥2 + |λ|∥C ∥2 + ∥BC ∥2 . Since B, C ∈ B2 (H ) , ∥BC ∥2 ≤ ∥B ∥∥C ∥2 ≤ ∥B ∥2∥C ∥2 . Then

∥(λ + B )(μ + C )∥2 ≤ (|λ| + ∥B ∥2)(|μ| + ∥C ∥2).

By an argument similar to the one given above,  √ -- |λ| + ∥B ∥2 ≤ 2∥λ + B∥2 , and the second assertion follows. □

If X, Y ∈ B2 (H ) , one has the usual inequalities ∥X ∥ ≤ ∥X ∥2 and ∥XY ∥2 ≤ ∥X ∥2∥Y ∥2 . As a consequence of 2.2, one has that the product (X, Y ) → XY is continuous, and therefore smooth, as a map from H × H to H .

Next we show that the inversion map σ : Σ → Σ ∞ ∞ , σ(A ) = A-1 is smooth. The second inequality in 2.2, shows that H can be renormed in order to become a Banach algebra. Indeed, putting ∥X ∥0 = 2∥X ∥2 , one obtains

∥XY ∥0 ≤ ∥X ∥0∥Y ∥0, X, Y ∈ H.

Corollary 2.3. The map

 -1 σ : Σ∞ → Σ ∞, σ (A) = A

is  ∞ C .

Proof. The map Σ∞ is the restriction of the inversion map of the regular group of the Banach algebra (H, ∥ ∥0) , which is an analytic map [14], to the smooth submanifold Σ ∞ . □

Note that in fact A ↦→ A -1 is real analytic in Σ∞ (Σ ∞ , being open in H ℝ , has in fact real analytic structure).

We finish this section establishing certain identities which are satisfied by the inner product of H . Because it is defined in terms of the trace, this inner product inherits certain symmetries. But not others: for example, it is easy to see that if A ∈ H ℝ (selfadjoint) and X, Y ∈ H , then < AX, Y > may not be equal to < X, AY > .

Lemma 2.4. Let X, Y ∈ H and A, B ∈ H ℝ , then the following hold:

  1. < AX, Y A >= < XA, AY > .
  2. < AX, Y B > + < BX, Y A >= < XA, BY > + < XB, AY > .

Proof. The proof is a simple verification, and is left to the reader. The only issues here are the properties of the trace and the fact that scalars are orthogonal to operators in B (H ) 2 . □

3. NON POSITIVELY CURVED METRIC ON Σ∞

For A ∈ Σ∞ , consider the following inner product on H ℝ (regarded as the tangent space (T Σ∞ )A ):

< X, Y > = < A- 1X, Y A -1 > . A
(3.1)

First note that in fact it is a positive definite form, which varies smoothly with A , because the inversion map is smooth. Also note that it looks formally similar to the nonpositively curved metric for the space M +n (ℂ ) of positive definite finite matrices. However there are significant differences. For instance, if H is finite dimensional, clearly Σ ∞ is M +n (ℂ ) (n = dimension of H ), but the inner product defined on H Is not the same as the trace inner product. An evidence of this is that in general < AX, Y >⁄= < X, AY > for A ∈ Σ ∞ , X, Y ∈ H .

Nevertheless, the known formulas for the geodesics and curvature from the finite dimensional case, can be extended in this context. The reason for this is that the covariant derivative has the same formula as in the matrix case.

Proposition 3.1. The Riemannian connection of the metric defined in (3.1) is given by

 1- -1 -1 ∇X Y = X {Y } - 2(XA Y + YA X ),
(3.2)

where X is a tangent vector at A ∈ Σ ∞ , and Y is a vector field. Here X {Y} denotes derivation of the field Y in the X direction, performed in the ambient space H .

Proof. The formula (3.2) defines a connection in Σ∞ . It clearly takes values in Hℝ , which is the tangent space of Σ∞ at any point, and also verifies the formal identities of a connection. Also it is apparent that it is a symmetric connection. Therefore, in order to prove that it is the Riemannian connection of the metric from (3.1), it suffices to show that the connection and the metric are compatible. This amounts to proving that if γ is a smooth curve in Σ∞ and X, Y are tangent vector fields along γ , then

d- < X, Y >γ= < DX--,Y > γ + < X, DY--> γ, dt dt dt

where as is usual notation, DdXt-= ∇ ˙γX . On one hand, using that (X ˙γ- 1) = X˙γ -1 + X (γ˙-1) , and that  ˙- 1 - 1 -1 (γ ) = - γ ˙γγ , one has

 d -- < X, Y > γ=< γ-1X˙, Y γ-1 > - < γ- 1˙γγ-1X, Y γ- 1 > + dt
+ < γ-1X, Y˙γ-1 > - < X γ- 1,Yγ -1˙γγ- 1 > .
(3.3)

On the other hand

 DX DY 1 < ----,Y > γ + < X, ---- >γ= < γ-1X˙, Y γ-1 > - --< γ-1˙γγ -1X, Y γ-1 > - dt dt 2
 1- - 1 -1 - 1 -1 ˙ -1 1- -1 - 1 -1 - 2 < γ X γ ˙γ,Y γ > + < γ X, Y γ > - 2 < γ X, ˙γγ Y γ > -
 1 - --< γ -1X, Yγ -1˙γγ -1 > . 2
(3.4)

In the expression above, one may use the first identity in Lemma 2.4 to replace < γ -1X γ-1γ˙,Y γ -1 > by < X γ-1˙γγ -1,γ-1Y > and < γ- 1X, γ˙γ-1Y γ-1 > by  -1 -1 -1 < X γ ,γ γ˙γ Y > . Then proving the equality of (3.3) and (3.4) is equivalent to prove that

 -1 -1 -1 -1 -1 - 1 < γ γ˙γ X, Y γ > + < γ X, Yγ ˙γγ > = = < X γ -1˙γγ- 1,γ -1Y > + < X γ-1,γ- 1˙γγ-1Y > .

This is the same as the second identity in Lemma 2.4, with  -1 -1 A = γ γ˙γ and  -1 B = γ . □

As a consequence, one obtains that the curvature tensor has the same expression as in the matrix case [3] [5]:

R (X, Y )Z = - 1-A [[A -1X, A -1Y],A -1Z ], A 4
(3.5)

for A ∈ Σ ∞ , X, Y, Z ∈ H ℝ . Here [ , ] denotes the usual commutator for operators.

Theorem 3.2. Σ ∞ has non positive sectional curvature.

Proof. Compute

< R (X, Y )Y,X > = - 1 < [[A -1X, A- 1Y],A -1Y ],XA -1 > A A 41 -1 - 1 2 -1 = - 4{< A-1 X (A-1 Y )- ,1XA ->1 - - 2 < A Y A XA Y,XA > + + < (A -1Y )2A -1X, XA -1 >}.

Again we may use the same identity from Lemma 2.4 as follows:

< A- 1X (A -1Y )2,A- 1X >= < A -1∕2XA - 1∕2(A -1∕2Y A -1∕2)2,A -1∕2XA -1∕2 > .

The other terms above can be modified likewise. Let us denote X¯ = A -1∕2XA -1∕2 and ¯Y = A -1∕2Y A- 1∕2 . Then < R (X, Y)Y, X >A equals

 1 - --{< X¯(Y¯)2, ¯X > - 2 < ¯Y ¯X ¯Y,X¯ > + < (¯Y )2 ¯X, ¯X > }. 4

Let us compare  2 < X¯(¯Y ) , ¯X > and < Y¯X¯¯Y ,X¯ > . Note that X¯, Y¯ ∈ H ℝ , let X¯ = λ + X ′ , ¯Y = μ + Y′ . Then

 ¯ ¯ 2 ¯ 2 2 ′ ′ ′ 2 ′ 2 ′′2 < X (Y) ,X > = (λ μ + 2λ μ tr(Y X ) + λ tr((Y ) X ) + μ tr((X ))+ +2 μ tr(X ′′X ′) + tr((X ′)2(Y ′)2).

Analogously

< ¯Y ¯X ¯Y,X¯ > = λ2 μ2 + 2λμ tr(Y ′X ′) + μ2 tr((X ′)2) + μ tr(X ′Y ′X ′)+ + λ tr((Y ′)2X ′) + μ tr(Y ′(X ′)2) + tr(Y ′X ′Y ′X ′).

After cancellations, in order to compare < X¯(¯Y )2, ¯X > and < ¯Y X¯¯Y , ¯X > it suffices to compare tr(X ′X ′Y ′Y ′) and tr(Y ′X ′Y ′X ′) . By the Cauchy-Schwarz inequality for the trace, one has

 ′ ′ ′ ′ ′ ′* ′ ′ tr(Y X Y X ) = tr ((X Y ′)Y′* X′) ′1∕2 ′ ′ * ′ ′1∕2 ≤ tr((X Y ) X Y ) tr((Y X ) Y X ) = tr(X ′X ′Y ′Y ′).

Therefore

< Y¯X¯Y¯,X¯ > ≤ < X¯(¯Y )2, ¯X > .

Analogously one proves that

< ¯Y X¯¯Y , ¯X > ≤ < (Y¯)2 ¯X, ¯X > .

It follows that < R (X, Y )Y,X >A ≤ 0 .

Remark 3.3. As was stated above, the fact that the formula to compute the Riemannian connection looks formally equal for Σ∞ and for the space of positive definite finite matrices (in fact, also for positive invertible operators of an abstract C * -algebra [3], [5]) implies that one knows the explicit form of the geodesic curves. Let A, B ∈ Σ ∞ . Then the curve

 1∕2 -1∕2 - 1∕2 t 1∕2 γA,B (t) = A (A BA ) A
(3.6)

is a geodesic, which is defined for all t ∈ ℝ , and joins A and B .

Then Σ ∞ is a simply connected (in fact convex) manifiold on non positive sectional curvature. It follows [10], [11] that the geodesic (3.6) is the unique geodesic joining A and B . In fact one has the following:

Corollary 3.4. The curve given in (3.6) is the unique geodesic joining A and B , and it realizes the geodesic distance. The manifold Σ∞ is complete with the geodesic distance.

The geodesic distance of a non positively curved simply connected manifold has also the following property [10]: if γ 1 and γ 2 are two geodesics of Σ ∞ , then the map

f : ℝ → ℝ ≥0, f(t) = d (γ1 (t),γ2(t))

is convex. As in [5], we obtain the following consequence of this fact:

Corollary 3.5. Let X,Y ∈ H ℝ and  X Y A = e ,B = e ∈ Σ ∞ . Then

 -1∕2 -1∕2 ∥X - Y ∥2 ≤ d(A, B) = ∥log(A BA )∥2.

Proof. The proof follows as in Thm. 3 of [5]. We outline the argument. Let γ1(t) = etX and γ2(t) = etY . These are geodesics of Σ ∞ which start at I and verify γ1(1) = A and γ2(1) = B . The function f(t) = d(γ1(t),γ2(t)) is convex, with f (0) = 0 . Then f (t)∕t ≤ f(1) for t ∈ [0,1] . Note that  tX tY - tX∕2 tY -tX∕2 d(e ,e ) = ∥log(e e e )∥2 . Then

f(t) 1 ---- = ∥--log(e- tX∕2etYe- tX∕2)∥2. t t

If t > 0 is small, then e-tX ∕2etY e-tX ∕2 is close to I , and therefore one has the usual power series for the logarithm, log(1 + T ) = I - T + 12T 2 - 13T 3 + ... . Using also the power series of the involved exponentials and taking limit t → 0 , one obtains

 1- -tX∕2 tY -tX∕2 d(A, B ) = f(1) ≥ ltim→0∥ t log(e e e )∥2 = ∥Y - X ∥2.

The map σ : Σ ∞ → Σ ∞ provides a symmetry for Σ ∞ . It is clearly a diffeomorphism with  2 σ = id . Note that it is isometric. Indeed, if γ is a curve in Σ ∞ with γ(0) = A and γ˙(0 ) = X , then  -1 - 1 (σ ˙(γ )) = - γ ˙γγ . Then

dσA (X ) = - A -1XA -1.

Then

< dσA(X ),dσA (X ) > σ(A)=< A- 1XA - 1,A -1XA -1 >A -1=< XA -1,A- 1X >

which equals < X, X >A by the first identity in Lemma 2.4.

4. THE INMERSION OF M n+(ℂ) IN Σ∞ .

Fix a positive integer n and let Hn be an n -dimensional subspace of H . Let P be the orthogonal projection onto Hn , and P¯ = 1 - P . The space M +n (ℂ) of n × n positive definite (invertible) matrices identifies naturally with the space Σ Hn of positive invertible operators of Hn . We shall consider the manifold ΣHn with the Riemannian metric

 -1 -1 < X, Y >A= tr(XA Y A ), A ∈ ΣHn, X, Y ∈ B (Hn )h.

This metric is well known in differential geometry ([15],[16]), and has been thoroughly studied and generalized to various infinite dimensional contexts ([3], [5]).

There is also a natural map from ΣH n into Σ ∞ ,

iHn : ΣHn → Σ ∞, iHn(A ) = A + P¯.

Note that iHn is well defined: A has finite rank and therefore  ¯ A + P is a finite rank perturbation of the indentity.

Proposition 4.1. The map iHn is an isometric imbedding.

Proof. Clearly, it is injective. Let A ∈ ΣHn and X, Y be hermitian elements of B (Hn ) , regarded as tangent vectors of ΣHn at A . Apparently, d(iHn)A(X ) = X + 0P¯ = X . In particular, the range of d(iH )A (X ) n is B(Hn )h ⊕ 0 which is complemented in B (H )h . One has

 -1 - 1 < d (iHn )A (X ),d(iHn )A(Y ) >A+ P¯=< (A + P¯) X, Y (A + ¯P ) > .

Note that (A + P¯)-1 = A -1 + ¯P , where A- 1 denotes the inverse of A in B (HN ) . Also X P¯ = 0 , and then X (A + P¯)-1 = XA -1 is a finite rank operator, in particular,  ¯ -1 X (A + P) ∈ B2 (H ) . Then

< (A + ¯P )-1X, Y(A + ¯P)- 1 >= tr(XA -1Y A- 1).

Remark 4.2. Another implication of the fact that the connections of ΣHn and Σ ∞ look formally identical, is that the maps iHn are flat inclusions. The spaces ΣH n regarded as submanifolds of Σ ∞ , are not curved in Σ∞ . In particular these submanifolds are geodesically complete, or in other words, geodesics of Σ Hn are also geodesics of the ambient space Σ∞ . One may fix {ek : k ≥ 1 } an orthonormal basis for H , and consider Hn the span of {e1,...,en} . Let in = iHn . Then via this family of imbeddings, one may think of Σ ∞ as an ambient for all spaces M +n (ℂ ) of positive definite matrices, of all possible sizes.

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Esteban Andruchow
Instituto de Ciencias,
Universidad Nacional de Gral. Sarmiento,
J. M. Gutierrez 1150,
(1613),Los Polvorines, Argentina
eandruch@ungs.edu.ar

Alejandro Varela
Instituto de Ciencias,
Universidad Nacional de Gral. Sarmiento,
J. M. Gutierrez 1150,
(1613),Los Polvorines, Argentina
avarela@ungs.edu.ar

Recibido: 19 de mayo de 2005
Aceptado: 30 de noviembre de 2006