versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007
Esteban Andruchow and Alejandro Varela
Abstract We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold of positive invertible operators of a Hilbert space , 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 .
Key words and phrases. positive operator, Hilbert-Schmidt class
2000 Mathematics Subject Classification. 58B20, 47B10, 47B15
The space of positive definite (invertible) matrices is a differentiable manifold, in fact an open subset of the real euclidean space of hermitian matrices. Let be hermitian matrices and positive definite, the formula
endows 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 by means of the identification
where and are, respectively, the linear and unitary groups of . This example has a universal property: every symmetric space of noncompact type can be realized isometrically as a complete totally geodesic submanifold of .
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 , , in partial differential equations , and in mathematical physics , , . They have also been generalized to infinite dimensions, i.e. Hilbert spaces and operator algebras: , ,  , .
The purpose of this note is to introduce a Riemannian metric in the space of infinite positive definite matrices, that is the set of positive and invertible operators on an infinite dimensional Hilbert space , which are of the form
where and , the class of Hilbert-Schmidt operators, i.e. elements such that . We shall regard as an infinite dimensional manifold, in fact as an open subset of an apropriate infinite dimensional euclidean space, and introduce a Riemannian metric in , whichs looks formally identical to the metric for finite matrices. It will be shown that with this metric becomes a non positively curved, complete Riemannian manifold, which contains in a natural (isometric, flat) manner all spaces .
Therefore 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 and by the Hilbert-Schmidt norm: . Denote by
Note that since is infinite dimensional, the scalars and the operators in are linearly independent. In particular, one has that if and only if and . Formally, and , where denotes the real Hilbert space of selfadjoint Hilbert-Schmidt operators. Let us define
Clearly this inner product makes , , respectively, complex and real Hilbert spaces, where the scalars and the operators (in ) are orthogonal. The space will be considered with the relative topology induced by this inner product norm. It follows that the maps , and , are orthogonal projections and their adjoints are the inclusions, which are therefore isometric.
Note that means that and the spectrum of is a subset of . Indeed, the first assertion follows from the fact that , and the second is obvious.
In what follows, we denote by the norm of . No confusion should arise with the norm of , because the former extends the latter.
Let us prove some elementary facts concerning the topology of .
Proof. The fact that is convex is apparent. Let . Since is positive and invertible, it follows that the eigenvalues of are bounded from below by , and do not aproach . Then there exists such that , or in other words, is positive and invertible. Consider the ball
We claim that if , then . Indeed, . Then and the operator norm . Then and , and therefore
which is positive and invertible. It follows that . □
The following elementary estimations will be useful.
Proof. Let be the singular values of . Then . On the other hand, . Since the singular values accumulate eventually only at , clearly one has or for some . In either case
which proves the first assertion. For the second, . Since , . Then
By an argument similar to the one given above, , and the second assertion follows. □
If , one has the usual inequalities and . As a consequence of 2.2, one has that the product is continuous, and therefore smooth, as a map from to .
Next we show that the inversion map , is smooth. The second inequality in 2.2, shows that can be renormed in order to become a Banach algebra. Indeed, putting , one obtains
Proof. The map is the restriction of the inversion map of the regular group of the Banach algebra , which is an analytic map , to the smooth submanifold . □
Note that in fact is real analytic in (, being open in , has in fact real analytic structure).
We finish this section establishing certain identities which are satisfied by the inner product of . 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 (selfadjoint) and , then may not be equal to .
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 . □
For , consider the following inner product on (regarded as the tangent space ):
First note that in fact it is a positive definite form, which varies smoothly with , because the inversion map is smooth. Also note that it looks formally similar to the nonpositively curved metric for the space of positive definite finite matrices. However there are significant differences. For instance, if is finite dimensional, clearly is ( dimension of ), but the inner product defined on Is not the same as the trace inner product. An evidence of this is that in general for , .
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
where is a tangent vector at , and is a vector field. Here denotes derivation of the field in the direction, performed in the ambient space .
Proof. The formula (3.2) defines a connection in . It clearly takes values in , which is the tangent space of 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 is a smooth curve in and are tangent vector fields along , then
where as is usual notation, . On one hand, using that , and that , one has
On the other hand
This is the same as the second identity in Lemma 2.4, with and . □
for , . Here denotes the usual commutator for operators.
Again we may use the same identity from Lemma 2.4 as follows:
The other terms above can be modified likewise. Let us denote and . Then equals
Let us compare and . Note that , let , . Then
After cancellations, in order to compare and it suffices to compare and . By the Cauchy-Schwarz inequality for the trace, one has
Analogously one proves that
It follows that .
Remark 3.3. As was stated above, the fact that the formula to compute the Riemannian connection looks formally equal for and for the space of positive definite finite matrices (in fact, also for positive invertible operators of an abstract -algebra , ) implies that one knows the explicit form of the geodesic curves. Let . Then the curve
is a geodesic, which is defined for all , and joins and .
Then is a simply connected (in fact convex) manifiold on non positive sectional curvature. It follows ,  that the geodesic (3.6) is the unique geodesic joining and . In fact one has the following:
Corollary 3.4. The curve given in (3.6) is the unique geodesic joining and , and it realizes the geodesic distance. The manifold is complete with the geodesic distance.
The geodesic distance of a non positively curved simply connected manifold has also the following property : if and are two geodesics of , then the map
is convex. As in , we obtain the following consequence of this fact:
Proof. The proof follows as in Thm. 3 of . We outline the argument. Let and . These are geodesics of which start at and verify and . The function is convex, with . Then for . Note that . Then
If is small, then is close to , and therefore one has the usual power series for the logarithm, . Using also the power series of the involved exponentials and taking limit , one obtains
The map provides a symmetry for . It is clearly a diffeomorphism with . Note that it is isometric. Indeed, if is a curve in with and , then . Then
which equals by the first identity in Lemma 2.4.
Fix a positive integer and let be an -dimensional subspace of . Let be the orthogonal projection onto , and . The space of positive definite (invertible) matrices identifies naturally with the space of positive invertible operators of . We shall consider the manifold with the Riemannian metric
There is also a natural map from into ,
Note that is well defined: has finite rank and therefore is a finite rank perturbation of the indentity.
Proof. Clearly, it is injective. Let and be hermitian elements of , regarded as tangent vectors of at . Apparently, . In particular, the range of is which is complemented in . One has
Note that , where denotes the inverse of in . Also , and then is a finite rank operator, in particular, . Then
Remark 4.2. Another implication of the fact that the connections of and look formally identical, is that the maps are flat inclusions. The spaces regarded as submanifolds of , are not curved in . In particular these submanifolds are geodesically complete, or in other words, geodesics of are also geodesics of the ambient space . One may fix an orthonormal basis for , and consider the span of . Let . Then via this family of imbeddings, one may think of as an ambient for all spaces of positive definite matrices, of all possible sizes.
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Instituto de Ciencias,
Universidad Nacional de Gral. Sarmiento,
J. M. Gutierrez 1150,
(1613),Los Polvorines, Argentina
Instituto de Ciencias,
Universidad Nacional de Gral. Sarmiento,
J. M. Gutierrez 1150,
(1613),Los Polvorines, Argentina
Recibido: 19 de mayo de 2005
Aceptado: 30 de noviembre de 2006