Nonpositive Curvature: a Geometrical Approach to Hilbert-Schmidt Operators
We give a Riemannian structure to the set of positive invertible unitized Hilbert-Schmidt operators, by means of the trace inner product. This metric makes of a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into . We give an intrinsic algebraic characterization of convex closed submanifolds . We study the group of isometries of such submanifolds: we prove that , the Banach-Lie group generated by , acts isometrically and transitively on . Moreover, admits a polar decomposition relative to , namely as Hilbert manifolds (here is the isotropy of for the action ), and also so is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds . These decompositions are obtained via a nonlinear but analytic orthogonal projection , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism (here stands for the normal bundle of a convex closed submanifold ). Writing down the factorizations for fixed , we obtain with and orthogonal to at . As a corollary we obtain decompositions for the full group of invertible elements .
The aim of this paper is to relate the algebraic and spectral properties of the Banach algebra of unitized Hilbert-Schmidt operators, with the metric and geometrical properties of an underlying manifold . This is a paper on applied nonpositively curved geometry because we first show how the familiar properties of the operator algebra translate into geometrical notions, and then we use the tools of geometry in order to prove new results concerning the operator algebra.
In this paper we study the cone of positive invertible Hilbert-Schmidt operators (extended by the scalar operators) on a separable infinite dimensional Hilbert space . The metric in the tangent space at the identity is given by the trace of the algebra. The local structure induced by the metric is smooth and quadratic; it can be situated in the context of the theory of infinite dimensional Riemann-Hilbert manifolds of nonpositive curvature (cf. Cartan-Hadamard manifolds, as introduced by Lang , McAlpin , Grossman  and others). It is then a paper on Riemannian geometry. On the other hand, since the manifold is clearly not locally compact, some of the standard results for Hadamard manifolds require a different approach. The geometry is then related to the geometry of the metric spaces in the sense of Aleksandrov . It turns out that the notion of convexity (together with the fact that is a simply connected and globally nonpositively curved geodesic length space) plays a key role in our constructions. It is then a paper on metric geometry.
Through the years, several authors have studied the relationship of geometry and algebra in sets of positive operators, with different approaches that led to a variety of results. In his 1955’s paper , G.D. Mostow gave a Riemannian structure to the set of positive invertible matrices; the induced metric makes of a nonpositively curved symmetric space. Mostow showed that the algebraic concept behind the notion of convexity is that of a Lie triple system, which is basically the real part of a given involutive Lie algebra . The geometry of bounded positive operators in an infinite dimensional Hilbert space was studied by G. Corach, H. Porta and L. Recht  among others, using functional analysis techniques. This area of research is currently very active (see  for a list of references).
1.1 Main results
In this paper we study the geometry of a Hilbert manifold which is modeled on the operator algebra of unitized Hilbert-Schmidt operators. In Section 2 we introduce the objects involved and prove some elementary results. The manifold is the set of positive invertible operators of . Let be the classical Banach-Lie group of invertible (unitized) Hilbert-Schmidt operators . The manifold has a natural -invariant metric , which makes it nonpositively curved (we define whenever and are Hilbert-Schmidt operators). Let and stand for the usual analytic exponential, i.e. . This map is injective when restricted to , the set of self-adjoint operators. Let stand for its real analytic inverse. We have , and the exponential map induces a diffeomorphism onto its image, so we identify the tangent space at any point of the manifold with the set of self-adjoint operators , namely for any . In Section 3 we prove
Theorem A: For , the geodesic obtained from Euler’s equation by solving Dirichlet’s problem is the smooth curve , hence
is the Riemannian exponential of , for any . Both and its differential map are -isomorphisms for any , . The curve is the shortest piecewise smooth path joining to , hence
is the distance in induced by the Riemannian metric. The metric space is complete, and it is globally nonpositively curved.
In , N. Grossman proves that the inequality
leads to the minimality of geodesics in a simply connected, complete Hilbert manifold. This approach is also carried out by McAlpin . The following operator inequality involving the differential of the usual exponential map
is the translation to our context of the inequality (1) above. The convexity of Jacobi fields can be deduced from the non positiveness of the sectional curvature, hence the proof of eqn. (2) stems in our context from the Cauchy-Schwarz inequality for the trace inner product. We follow the exposition of Lang  on this subject. On the other hand, (2) can be proved with a direct computation . With this approach the metric completeness of the tangent spaces is not relevant: in Theorem 3.1 of , the authors prove the minimizing property of the geodesics in a non complete manifold. The inequality above, in our context, can be also interpreted as the Hyperbolic Cosine Law (see Corollary 3.12)
Here are the lenghts of the sides of any geodesic triangle in , and is the angle opposite to . From this inequality also follows that the sum of the inner angles of any geodesic triangle in is bounded by .
If is a set of operators, we use to denote the set of positive operators of ; note that . In Section 4 we show that a submanifold is geodesically convex if and only if its tangent space at the identity is a Lie triple system. Clearly any such submanifold is nonpositively curved, and Theorem 4.18 states:
Theorem B: For any geodesically convex, closed submanifold there exists a connected Banach-Lie group which acts isometrically and transitively on . Moreover, the polar decomposition of the elements of reduces to in the sense that . Let be the isotropy of for the action; then is a connected Banach-Lie subgroup of and there is an isomorphism . In particular any convex submanifold of is an homogeneous space for a suitable Banach-Lie group, which is an analytical subgroup of . The submanifold is flat if and only if is an abelian Banach-Lie subgroup of .
The existence of smooth polar decompositions for the involutive Banach-Lie groups can be obtained from the general results of Neeb (, Theorem 5.1). Neeb introduces the notion of seminegative curvature (SNC) on Banach-Finsler manifolds , given by the condition of inequality (1) above, plus the condition that should be invertible for any (the metric of sould be invariant under parallel transport along geodesics). Neeb proves (Theorem 1.10 of ) that in a connected, geodesically complete manifold with SNC, the exponential map is a covering map and is metrically complete, a result which extends that of Grossman and McAlpin mentioned above to the Banach-Finsler context.
The manifold can be decomposed by means of any convex closed submanifold . Let be the normal bundle of . In Section 5 we prove
Theorem C: For any convex closed submanifold there is a nonlinear, real analytic projection , which is is contractive for the geodesic distance
The point is the (unique) point of closest to . It can also be viewed as the unique point in such that there exists a geodesic through orthogonal to at . The exponential map induces an analytic Riemannian isomorphism .
Since is the point in closest to , one can prove the existence of such a point using a metric argument valid in any nonpositively curved geodesic length space . We choose to give a differential-geometry argument here.
In Section 6 we exhibit a decomposition for the submanifold of positive diagonal operators, which is a maximal abelian subalgebra of . This decomposition theorem (Theorem 6.2) takes the form of a factorization , where has null diagonal and is an invertible diagonal operator. We stress that there is no known algorithm that allows to compute explicitly (not even if we reduce the problem to matrices, that can be thought of as a particular case of the general theory). As a corollary to the decomposition theorems we obtain
Theorem D: Any invertible operator admits a unique polar decomposition relative to a fixed closed convex submanifold . Namely where , and is a unitary operator. The map is an analytic bijection which gives the isomorphism
This isomorphism generalizes the decomposition of given in .
In Section 7 we show that the manifold can be decomposed by means of a foliation of totally geodesic submanifolds, namely
There is a Riemannian isomorphism induced by the projection of Theorem C above. As an application, we show a decompositon relative to the algebra of positive invertible matrices: fix an -dimensional subspace , let be the orthogonal projection to and the orthogonal projection to . Let stand for the algebra of bounded linear operators of . Let , and consider the set
Let be the Banach-Lie subgroup of unitary operators in .
Theorem E: For any there is a unique factorization where , is a unitary operator, and . In particular
The manifold can be regarded as a universal model for the symmetric spaces of the noncompact type, namely
Theorem F: For any finite dimensional real symmetric manifold of the noncompact type (i.e. with no Euclidean de Rham factor, simply connected and with nonpositive sectional curvature), there is an embedding which is a diffeomorphism between and a closed geodesically convex submanifold of . If we pull back the inner product on to , this inner product is a positive constant multiple of the inner product of on each irreducible de Rham factor.
The proof of the theorem is straightforward fixing an orthonormal basis of (see Section 7.1) and recalling the well known result  that for any such space there is an almost isometric embedding of into , where is the Lie algebra of the Lie group (the connected component of the identity of the group of isometries of ).
2 Background and definitions
Let be the set of bounded operators acting on a complex, infinite dimensional and separable Hilbert space , and let be the bilateral ideal of Hilbert-Schmidt operators of . Recall that is a Banach algebra (without unit) when given the norm (see  for a detailed exposition on trace-class ideals). We will use to denote the closed subspace of self-adjoint Hilbert-Schmidt operators. In we define
the complex linear subalgebra consisting of Hilbert-Schmidt perturbations of scalar multiples of the identity (the closure of this algebra in the operator norm is the set of compact perturbations of scalar multiples of the identity). There is a natural Hilbert space structure for this subspace (where scalar operators are orthogonal to Hilbert-Schmidt operators) which is given by the inner product
The algebra is complete with this norm. The model space that we are interested in is the real part of ,
which inherits the structure of (real) Banach space, and with the same inner product, becomes a real Hilbert space.
By virtue of trace properties, for any , and also for and .
Let be the subset of positive invertible operators in . It is clear that is an open set of (for instance, using the lower semi continuity of the spectrum).
For , we identify with , and endow this manifold with a (real) Riemannian metric by means of the formula
Throughout, let . Equivalently, .
The covariant derivative in (for the metric introduced in Remark 2.2) is given by
Here denotes derivation of the vector field in the direction of performed in the linear space .
Note that is clearly symmetric and verifies all the formal identities of a connection; the proof that it is the Levi-Civita connection relays on the compatibility condition between the connection and the metric, (see for instance  Chapter VIII, Theorem 4.1). Here is a smooth curve in and are tangent vector fields along . This identity is straightforward from the definitions and the properties of the trace. ∎
Let (here ). The exponential is given by the usual series; note that any positive invertible operator has a real analytic logarithm, which is the inverse of the exponential in the Banach algebra. Note that whenever and also whenever and .
Euler’s equation for the covariant derivative introduced above reads , and it is not hard to see that the (unique) solution of this equation with , is given by the smooth curve
We will use to denote the exponential map of . Differentiating at the curve above, we obtain , hence
Note that by the construction above the map is surjective (for given take , then ). Rearranging the exponential series we get the expressions .
The metric in is invariant under the action of the group of invertible elements: if is an invertible operator in , then is an isometry of .
First note that for any we have assuming and invertible, so maps into itself. Also note that for any , hence
where the third equality in the above equation follows from Remark 2.1.∎
3 Local and global structure
We start showing that curvature in this manifold is a measure of noncommutativity, and then give a few definitions, which are necessary because of the infinite dimensional setting. Let stand for the usual commutator of operators, .
The curvature tensor for the manifold is given by:
This follows from the usual definition . The formula for given in Lemma 2.3. ∎
A Riemannian submanifold is flat at if the sectional curvature vanishes for any 2-subspace of . The manifold is flat if it is flat at any . The manifold is geodesic at if geodesics of the ambient space starting at with initial velocity in are also geodesics of . The manifold is a totally geodesic manifold if it is geodesic at any . Equivalently, is totally geodesic if any geodesic of is also a geodesic of .
The manifold has nonpositive sectional curvature.
Let . Let , . We may assume that are orthonormal at . A straightforward computation shows that
Since , and for and . The equation reduces to
Note that is an inner product on , so we have the Cauchy-Schwarz inequality . Putting , we obtain
Let be a submanifold. Assume that is flat and geodesic at . If , then commutes with .
Since is geodesic at , the curvature tensor is the restriction of the curvature tensor of , so in equation (6) above the right hand term must be zero if is flat at . But the Cauchy-Schwarz inequality is an equality only if the vectors are linearly dependent; in the notation of the previous theorem, we have for some ; replacing this in the above equation we obtain , namely . Recalling the definitions for and we obtain the assertion. ∎
3.2 Convexity of Jacobi fields
Let be a Jacobi field along a geodesic of , i.e. is a solution of the differential equation
where is the covariant derivative along . We may assume that is non vanishing, hence
The third term is clearly positive and the first two terms add up to a nonnegative number by the Cauchy-Schwarz inequality: . In other words, the smooth function is convex, exactly as in the finite dimensional setting.
3.3 The exponential map
We present two theorems that, in this infinite dimensional setting, stem from McAlpin’s PhD. Thesis (for a proof see  or Theorem 3.7 of Chapter IX in ). First, if one identifies the Riemannian exponential with a suitable Jacobi lift, one obtains
The map has an expansive differential:
This result implies that the differential of the exponential map is injective and has closed range. Playing with the Hilbert structure of the tangent bundle and using the well known identity for operators , it can be proved that this map is surjective, moreover
The differential of the Riemannian exponential is a linear isomorphism for any . Hence, is a -diffeomorphism.
The last assertion is due to the fact that the map is a bijection (see Remark 2.4 above).
3.4 The shortest path and the geodesic distance
The following inequality is the key to the proof of the fact that geodesics are minimizing. It was proved by R. Bhatia  for matrices, and his proof can be translated almost verbatim to the context of operator algebras with a trace, see . However since the Riemannian metric in is complete, the inequality can be easily deduced from the fact that the norm of a Jacobi field is a convex map (in Theorem 3.5 put , and ):
If denotes the differential at of the usual exponential map, then for any
As usual, one measures length of curves in using the norms in each tangent space,
We define the distance between two points as the infimum of the lengths of piecewise smooth curves in joining to ,
Recall (Remark 2.4 and the paragraph above it) that for any pair of elements , we have the smooth curve joining to , which is the unique solution of Euler’s equation in . Computing the derivative, we get
The minimality of these (unique) geodesics joining two points can be deduced from general considerations , we present here a direct proof.
Let . Then the geodesic is the shortest curve joining and in , if the length of curves is measured with the metric defined above (8).
Let be a smooth curve in with and . We must compare the length of with the length of . Since the invertible group acts isometrically, it preserves the lengths of curves. Thus we may act with , and suppose that both curves start at , or equivalently that . Therefore , with . The length of is then . The proof follows easily from the inequality of Corollary 3.7. Indeed, since is a smooth curve in , it can be written as , with . Then is a smooth curve of self-adjoint operators with and . Moreover,
On the other hand, by the mentioned inequality,
The geodesic distance induced by the metric is given by
Hence the unique geodesic joining to is also the shortest path joining to . This means that is a (not locally compact) geodesic length space in the sense of Aleksandrov and Gromov . These curves look formally equal to the geodesics between positive definite matrices, when this space is regarded as a symmetric space.
If , are geodesics, the map , is convex.
The distance between the points and is given by the geodesic , which is obtained as the variable ranges in a geodesic square with vertices (the starting and ending points of and ). Taking the partial derivative along the direction of gives a Jacobi field along the geodesic and it also gives the speed of . Hence
This equation states that can be written as the limit of a convex combination of convex functions , so must be convex itself. ∎
In a recent paper (Corollary 8.7 of ), the authors prove this property of convexity of the geodesic distance in a general setting concerning nonpositively curved symmetric spaces given by a quotient of Banach-Lie groups.
For any we have
Take , and as in the previous corollary; we may assume that . Note that , hence for any ; hence . Now
The inner angles of any geodesic triangle in add up to at most .
Here (i=1,2,3) are the sides of any geodesic triangle and is the angle opposite to . These inequalities put together show that one can construct a comparison Euclidean triangle in the affine plane with sides . For this triangle with angles (opposite to the side ) we have . This equation together with inequality (10) imply that the angle is bigger than for . Adding the three angles we have . ∎
The metric space is complete with the distance induced by the minimizing geodesics.
Consider a Cauchy sequence . Again by virtue of inequality (9) of Lemma 3.11, is a Cauchy sequence in . Since Hilbert-Schmidt operators are complete with the trace norm, there is a vector such that in the trace norm. Since the inverse map, the exponential map, the product and the logarithm are all analytic maps with respect to the trace norm, when . ∎
4 Geodesically convex submanifolds
A set is geodesically convex if for any two given points , the unique geodesic of joining to lies entirely in . A Riemannian submanifold is complete at if is defined in the whole tangent space and maps onto . The manifold is complete if it is complete at any point.
These previous notions are strongly related; it is not hard to see that for any Riemannian submanifold of , is geodesically convex if and only if is complete and totally geodesic. On the other hand, it should be clear from the definitions that whenever is a convex submanifold of , is nonpositively curved.
4.1 An intrinsic characterization of convexity
From now on the term convex stands for the longer geodesically convex. As before denotes the usual commutator of operators in . To deal with convex sets the following definition will be useful; assume is a real linear space.
We say that is a Lie triple system if for any . Equivalently, whenever .
Note that whenever are self-adjoint operators, is also a self-adjoint operator. So, for any involutive Lie subalgebra of operators (in particular: for any associative Banach subalgebra), is a Lie triple system in .
Assume is a submanifold such that , and is geodesic at . Then is a Lie triple system, because the curvature tensor at is the restriction to of the curvature tensor of , and . In particular, if is geodesically convex, must be a Lie triple system. This weak condition on the tangent space turns out to be strong enough to obtain convexity:
As P. de la Harpe pointed out, the proof of G. D. Mostow for matrices in  can be translated to Hilbert-Schmidt operators without any modification: we give a sketch of the proof here. Assume , , and consider the curve . Then it can be proved that with a Lipschitz map that sends into (this is nontrivial). Since and is a Lipschitz map by the uniqueness of the solutions of ordinary differential equations we have . Hence and the claim follows.∎
Assume , and is as in the above theorem. Then is a closed convex submanifold.
Take . Then , with . If we put , then because and are in . Moreover, . But the unique geodesic of joining to is , hence . ∎
Assume is a closed, commutative associative Banach subalgebra of . Then the manifold is a closed, convex and flat Riemannian submanifold. Moreover, is an open subset of and an abelian Banach-Lie group.
The first assertion follows from the fact that is a Lie triple system. Curvature is given by commutators, hence is flat. Since is a closed subalgebra, for any , so . That is open follows from the fact that is a isomorphism (Corollary 3.6).∎
If is flat and geodesic at , is abelian (by Proposition 3.4), therefore
Assume is closed and flat. If is geodesic at , then is a convex submanifold. Moreover, is an abelian Banach-Lie group and an open subset of .
We adopt the usual definition of a symmetric space :
A Riemann-Hilbert manifold is called a globally symmetric space if each point is an isolated fixed point of an involutive isometry . The map is called the geodesic symmetry.
Assume is closed and convex. Then is a symmetric space; the geodesic symmetry at is given by for any . In particular, is a symmetric space.
Observe that, for , , ; this shows that maps into . To prove that is an isometry, for any vector consider the geodesic of such that