Matrix norm inequalities and their applications in matrix geometry

Transcription

1 Matrix norm inequalities and their applications in matrix geometry Who? From? When? Gabriel Larotonda Instituto Argentino de Matemática (CONICET) and Universidad Nacional de General Sarmiento Advanced School and Workshop on Matrix Geometries and Applications. ICTP Trieste - July 2013 Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 1 / 38

2 Introduction Overview 1 The manifold P of positive invertible operators on Hilbert space. 2 Convex submanifolds and projection maps onto them. 3 Extending the projection map theorem: several approaches. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 2 / 38

3 Introduction Why extend? Disgression 1 In some settings, the most adequate norm to tackle a problem is not necessarilly the Frobenius norm. 2 Many physical models can only be described with infinite dimensional manifolds, e.g. the famous problem on the motions of an incompressible fluid in a bounded region of space. 3 To learn which parts of the rich structure that is underlying are essential to some geometrical results, and which ones are not! (is it relevant that the objects of P act on a Hilbert space like C n? Why not operators acting on a Banach space?) Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 3 / 38

4 Introduction Disgression Approach via inequalities Matrix inequalities Finsler spaces are point spaces of an essentially new character which we must approach directly without being prejudiced by euclidean or Riemannian methods... H. Busemann, in The geometry of geodesics, All those wonderful matrix inequalities contain essential geometrical properties and ideas, ideas which can be used in other variety of settings with no underlying associative algebras like M n (C) or no underlying Hilbert space. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 4 / 38

5 Overview of the manifold of positive invertible operators Matrices, operators Notation In this talk, we will deal with several Banach spaces, starting with X = C n, or H =infinite dimensional Hilbert space; which are both specific examples with certain additional structure (Riemannian structure). Layers We will denote B(X) =bounded linear operators acting on X (which is also a Banach space). So eventually we will be dealing with X = B(E) for certain Banach spaces E. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 5 / 38

6 Overview of the manifold of positive invertible operators Finsler manifolds M = differentiable manifold with a given metric : TM [0, + ) Finsler metrics The metric is not necessarilly Riemannian. For instance, the uniform norm for an operator or a matrix T: T = sup{tx X : x X 1} Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 6 / 38

7 Overview of the manifold of positive invertible operators Lenght of curves and geodesic distance Rectifiable length and distance For γ M, and a given tangent metric L(γ) = b a γ γ. dist(p, q) =inf{l(γ) : γ joins p to q}. Non-smooth manifolds? (M, d) metric space, γ continuous, L(γ) =sup d(γ(ti+1 ),γ(t i )). Sup is taken over all {t i } partitions of [a, b]. Inner metric space d(p, q) =dist(p, q) Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 7 / 38

8 Overview of the manifold of positive invertible operators Sectional curvature Riemannian manifold For orthonormal tangent vectors x, y TM, Sec(x, y) =R(x, y)x, y g where R is the curvature tensor of M. It is a measure of how far away is from zero. 2 x y 2 y x Non- Riemannian? Only if the norm comes from an inner product (Riemannian metric on M), the definition of Sec above makes sense. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 8 / 38

9 Overview of the manifold of positive invertible operators Geodesics Sectional curvature Define geodesics as curves of minimal length on M (for a given metric). Let x, y T p M, consider the unique geodesics α, β starting at p with initial speed x, y respectively, α(0) =x, β(0) =y. Milnor (1963) Consider the quantity s(x, y, t) = tx ty p dist(α(t),β(t)) t 2 dist(α(t),β(t)) Equivalent expression Sec p (x, y) = lim 6 s(x, y, t) t 0 + Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 9 / 38

10 Overview of the manifold of positive invertible operators The meaning of s(x, y, t) Comparing triangles Sectional curvature s(x, y, t) = tx ty p dist(α(t),β(t)) t 2 dist(α(t),β(t)) compares divergence of geodesics in M with divergence of flat geodesics (i.e. segments): Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 10 / 38

11 Overview of the manifold of positive invertible operators Non-positive curvature Triangle comparison inequality We now focus on the condition Sec(x, y) 0, i.e. tx ty p dist(α(t),β(t)) 0 or equivalently dist(α(1),β(1)) x y p. Exponential map The exponential map is defined as exp p (x) =α(1), where α is the unique geodesic of the connection starting at p with initial speed x. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 11 / 38

12 Overview of the manifold of positive invertible operators Non-positive curvature EMI Then Sec 0 if and only if dist(exp p (x), exp p (y)) x y p, p M, x, y T p M. This is known as the Exponential Metric Increasing property Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 12 / 38

13 Overview of the manifold of positive invertible operators Positive invertible matrices Famous example, Mostow (1955) Geodesics P = {a > 0 B(H), i.e. a = a,σ(a) (0, + )} Given a, b P, σ(a 1 b)=σ(b 1 2 a 1 b 1 2 ) (0, + ). a 1 b has a logarithm, and the one-parameter group joins a to b, and in fact γ a,b (t) =a exp(t log(a 1 b)) γ a,b (t) =a 1 2 a 1/2 ba 1/2 t a 1 2 > 0 Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 13 / 38

14 Overview of the manifold of positive invertible operators Tangent norms Gauge norms A norm φ on B(H) is symmetric if xyz φ x y φ z. In particular, it is unitarily invariant, for u, v unitary operators. uxv φ = x φ Uniform norm x = sup{xξ : ξ 1} Frobenius norm x F = Tr(x x) also known as the 2-norm. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 14 / 38

15 Overview of the manifold of positive invertible operators Symmetric norms Infinitesimal EMI Tangent norms on P For a P, v T a P, let v a := a 1/2 va 1/2 φ. x, y T 1 P, exp : T 1 P P, then D exp x (y) T e xp. Remarkably (any φ) D exp x (y) e x y φ = y 1 P. Frobenius norm: Mostow (1955) - Symmetric norms: Hiai-Kosaki (1999) This is the Infinitesimal EMI property It can be written as an inequality involving the logarithmic mean 1 0 e (1/2 s)x ye (s 1/2)x ds = sinh(adx) y φ adx y φ, φ where adx(y) =[x, y] =xy yx. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 15 / 38

16 Overview of the manifold of positive invertible operators Parenthesis: Riemann-Hilbert manifolds McAlpin- Grossmann, 60 s The infinitesimal EMI (in Riemannian manifolds M of any dimension) D(exp p ) x (y) expp (x) y p is equivalent to Sec 0 on M. Note that it is in fact the infinitesimal version of Sec 0 according to Milnor, i.e. dist(exp p (v), exp p (w)) v w p. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 16 / 38

17 Overview of the manifold of positive invertible operators Minimality of one-parameter groups Theorem: γ a,b is short in P Why? dist(a, b) =L(γ a,b )= log(a 1/2 ba 1/2 ) φ The IEMI property holds in P (any φ), and this enables comparison of curves in P with segments (short geodesics on the vector space H). Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 17 / 38

18 Overview of the manifold of positive invertible operators Convexity of the geodesic distance Equivalent to Sec 0 Two geodesics starting at p M, with (M, dist) inner metric space: dist(α(t),β(t)) t dist(α(1),β(1)), nonpositive curvature in the sense of Busemann Matrix/op. inequality in P Does log(e tx/2 e ty e tx/2 ) φ t log(e x/2 e y e x/2 ) φ hold for any φ? Lawson-Lim (2007) Proof in a general setting, and in particular for P with any φ. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 18 / 38

19 Overview of the manifold of positive invertible operators Action of GL (the group of invertibles) The action of the full linear group Consider I : GL(H) P P given by I(g, a) =I a (g) =g ag. This is a transitive action (the orbit of any a is full). Isometric action The action is isometric: if v T a P and g GL(H), then g vg g ag = v a =: a 1/2 va 1/2 φ The group of isometries and this holds for any φ. GL can be regarded as a (sufficiently large part of) Iso(P). Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 19 / 38

20 Overview of the manifold of positive invertible operators Homogeneous spaces P as homogeneous space From the action I 1 : GL(H) P given by g g g, the isotropy group is {g GL(H) :g g = 1} = U(H), the group of unitary matrices. Then P GL/U(H). Involutive Lie subgroups G = G GL(H), consider P G = {g g : g G} P. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 20 / 38

21 Convex Submanifolds and projection maps Convex submanifolds Lie algebra Let L(G) stand for the Lie algebra of G, then L(G) =p k where p B(H) h and k = L(U G ) B(H) ah. T 1 P G = p It is easy to check that exp(p) =P G. Lie Triple Systems Theorem (Mostow, 1955) p is not a Lie algebra, but [p, [p, p]] p. P G = exp(p) is geodesically convex, that is γ a,b P G whenever a, b P G. Equivalently, ab 1 a stays in P G. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 21 / 38

22 Convex Submanifolds and projection maps Theorem: Mostow (1955) Metric Projections H = C n, φ = F. Given a P, there exists a unique π G (a) P G dist(a, P G )=dist(a,π G (a)). Moreover, π G : P P G is a contraction: dist(π G (A),π G (A)) dist(a, A) Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 22 / 38

23 Convex Submanifolds and projection maps Proof Uniform Convexity Metric Projections Combine dist(e X, e Y ) X Y φ with 1 4 X Y2 F = 1 2 X2 F Y2 F X + Y 2 to obtain a semi-paralellogram law in P: 1 4 L(γ)2 1 2 dist(a,γ 0) dist(a,γ 1) 2 dist(a,γ1) 2, 2 2 F Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 23 / 38

24 Convex Submanifolds and projection maps Metric Projections Proof {B n } P G a minimizing sequence, dist(b n, A) d := dist(p G, A) Let γ P with γ(0) =B n, γ(1) =B m. Since P G is convex, dist(a,γ1) d 2 and 1 4 dist(b n, B m ) dist(a, B n) dist(a, B m) 2 d 2 Hence {B n } is a Cauchy sequence in P G. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 24 / 38

25 Convex Submanifolds and projection maps Another equivalence for Sec 0 CAT(0) (or Bruhat-Tits) inequality Inner metric spaces 1 4 L(γ)2 1 2 dist(x,γ 0) dist(x,γ 1) 2 dist(x,γ1) 2. 2 CAT is for Cartan, Alexandrov and Toponogov (M, dist) verifies CAT(0) nonpositive curvature in the sense of Alexandrov Hierarchy Alexandrov Busemann But not the other way around. Counterexample: P with the uniform norm Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 25 / 38

26 Convex Submanifolds and projection maps Projections to closed convex subsets Essential for in Mostows s approach Porta-Recht decompositions (1994) The parallelogram law for Hilbert spaces. Consider P with the uniform norm, and a convex submanifold as before (A is a subalgebra of B(H)) P G = {gg : g GL(A)}. With the uniform norm, strong convexity properties are not available (no semi-parallelogram law). However... Theorem Each a P is reached by a unique normal geodesic with foot in P G. Moreover, the projection map π A : P P G is smooth. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 26 / 38

27 Extending the projection map theorem: several approaches Do all these count? First naive approach What is essential here? There is an interplay between norms, geodesic convexity, associative algebra structure, topology, dimension. Study a straightforward extension of Mostow s: HS = {a B(H) :a is compact,σ(a) 2 } P = {gg : g GL(H), g 1 HS} = {e X : X HS h }, a typical tangent vector v T a P is an operator in HS h, and v a = a 1/2 va 1/2 2 as before, where X ei X 2 = Tr X 2 =, X e i. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 27 / 38

28 Extending the projection map theorem: several approaches Projections to closed convex subsets L (2007) Expected theorems on minimality of geodesics, strict convexity of the geodesic distance, and Projection theorem to convex closed submanifolds. The projection map π here is not only a contraction, but also smooth and in fact real analytic. The technique of the proof is a combination of the ideas mentioned before due to Mostow, Porta and Recht (mixed proof). Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 28 / 38

29 Extending the projection map theorem: several approaches p-uniform convexity Second naive approach The p-norm X p = p Tr X p obeys Clarkson s inequality (p 2) 1 2 p X Yp p 1 2 Xp p Yp p X + Y 2 and there is a similar one for 1 < p 2. p p p-alexandrov spaces Conde-L (2010) Combining Clarkson with the EMI, 1 2 p L(γ)p 1 2 dist(a,γ 0) p dist(a,γ 1) p dist(a,γ1) p, 2 Metric projections to convex submanifolds (here the algebraic underlying structure is not relevant, it does not play any role). Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 29 / 38

30 Extending the projection map theorem: several approaches Abstract setting The Lie group approach G a Lie group, L(G) its Lie algebra, σ an involutive (σ 2 = id) automorphism of G. Consider the homogeneous space M = G/K where K is the fixed point set of σ (and, in fact, a Lie subgroup). The example to have in mind here is, of course G = B(H), σ(g) =(g ) 1, K = U(H), so M = P. Helgason s book or Neeb s paper σ induces a direct sum decomposition L(G) =p L(K) and we identify TM p. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 30 / 38

31 Extending the projection map theorem: several approaches The Lie group approach Finsler manifolds of semi-negative curvature An adequate Ad K invariant norm on p makes of M a space of nonpositive curvature, in the following sense: K.-H. Neeb (2003) adopted the infinitesimal EMI D(exp p ) X (Y) expp (X) Y p as the definition of non-positive curvature. By the discussion above, our P is such a manifold (any φ). In the abstract setting, L(G) is not an associative algebra. Hence, majorization techniques, continuous functional calculus techniques are out of reach. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 31 / 38

32 Extending the projection map theorem: several approaches The Lie group approach However, L(G) is a Lie algebra, acting on itself via ad x (y) =[x, y]. And moreover, B(L(G)) is a Banach algebra, so analytic functional calculus is in reach. We can produce inequalities. L (2008) We needed a series of well-known inequalities for matrix norms (or at least, some adequate replacements) in order to tackle the study of the homogeneous space, and in particular to control projections (or even existence) to convex closed submanifolds. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 32 / 38

33 Extending the projection map theorem: several approaches The Lie group approach Let H G be a Lie subgroup, involutive (i.e. σ(h) H). Then, if we recall M = G/K, let C = H/(H K). Theorems: Conde-L (2010) C M is a geodesically convex submanifold, and there is a reductive structure TC S = TM. S acts as a natural normal bundle to TC, and NB = {exp p (v) :p C, v S} called the normal bundle fills M. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 33 / 38

34 Extending the projection map theorem: several approaches Corollaries The Lie group approach Each a M is reached by a unique geodesic with normal speed v S and foot p C. The normal bundle is isomorphic to M. We obtain Corach-Porta-Recht factorizations of M: a = exp p (v), M TC S. Sketch of proof Consider the map Θ:NB NB given by Θ:exp p (v) exp p (2v), p C, v S. This map is expansive, i.e. DΘ Z (W) W for any Z. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 34 / 38

35 Extending the projection map theorem: several approaches The Lie group approach Then Θ is a local iso NB open. Moreover π C : NB C given by taking the foot (goes in the opposite direction of Θ) is contractive NB is closed. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 35 / 38

36 Extending the projection map theorem: several approaches An inequality CPR The inequality DΘ Z (W) W is essentially cosh(adz)w W, which in turn can be rewritten as e Z We Z + e Z We Z 2W, Corach-Porta-Recht inequality. As shown by Kittaneh (1994), in the setting of Hilbert space operators it is equivalent to the arithmetic-geometric mean inequality: AA X + XBB 2A XB. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 36 / 38

37 Application Application: complexification Complexif. Theorem: Miglioli (2013) Via polar decomposition, GL(H) can be though of as the complexification of the real Lie group U(H), in a very precise sense: A complexification of a real manifold M is a complex Banach manifold Y endowed with an anti-holomorphic involutive diffeomorphism σ such that the fixed point submanifold Y 0 = {y Y : σ(y) =y} is a strong deformation retract of Y, and also Y 0 is diffeomorphic to X. Let G be a complex Lie group with involution, K its (real!) fixed point subgroup, and H an involutive Lie subgroup as before. Assume that M = G/K is of semi-negative curvature. Then G/H is a complexification of K/K H. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 37 / 38

38 THANKS Thank you all for your kind attention! Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38

39 THANKS Andruchow, Esteban; Corach, Gustavo; Stojanoff, Demetrio. GEOMETRICAL SIGNIFICANCE OF LÖWNER-HEINZ INEQUALITY. Proc. Amer. Math. Soc. 128 (2000), no. 4, R. Bhatia. ON THEEXPONENTIALMETRIC INCREASING PROPERTY Linear Algebra Appl. 375 (2003), H. Busemann. SPACES WITH NON-POSITIVE CURVATURE. Acta Math. 80, (1948) H. Busemann. GEOMETRY OF GEODESICS. Academic Press, Cartan, Élie. LA GÉOMÉTRIE DES GROUPES SIMPLES. (French) Ann. Mat. Pura Appl. 4 (1927), no. 1, Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38

40 THANKS Conde, Cristian; Larotonda, Gabriel. MANIFOLDS OF SEMI-NEGATIVE CURVATURE. Proc. Lond. Math. Soc. (3) 100 (2010), no. 3, Corach, Gustavo; Maestripieri, Alejandra. DIFFERENTIAL AND METRICAL STRUCTURE OF POSITIVE OPERATORS. Positivity 3 (1999), no. 4, Corach, Gustavo; Porta, Horacio; Recht, Lázaro. GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS. Internat. J. Math. 4 (1993), no. 2, Corach, Gustavo; Porta, Horacio; Recht, Lázaro. THE GEOMETRY OF THE SPACE OF SELFADJOINT INVERTIBLE ELEMENTS IN A C -ALGEBRA. Integral Equations Operator Theory 16 (1993), no. 3, Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38

41 THANKS Corach, Gustavo; Porta, Horacio; Recht, Lázaro. CONVEXITY OF THE GEODESIC DISTANCE ON SPACES OF POSITIVE OPERATORS. Illinois J. Math. 38 (1994), no. 1, Grossman, Nathaniel. HILBERT MANIFOLDS WITHOUT EPICONJUGATE POINTS. Proc. Amer. Math. Soc Hadamard, Jacques. SUR LA FORME DES LIGNES GÉODÉSIQUES Á L INFINI ET SUR LES GÉODÉSIQUES DES SURFACES RÉGLÉES DU SECOND ORDRE. (French) Bull. Soc. Math. France 26 (1898), S. Helgason. DIFFERENTIAL GEOMETRY, LIE GROUPS, AND SYMMETRIC SPACES. Pure and Applied Mathematics, 80. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38

42 THANKS F. Hiai, H. Kosaki. COMPARISON OF VARIOUS MEANS FOR OPERATORS. J. Funct. Anal. 163 (1999), no. 2, F. Kittaneh. ON SOMEOPERATOR INEQUALITIES. Linear Algebra Appl. 208/209 (1994), Lawson, Jimmie; Lim, Yongdo. METRIC CONVEXITY OF SYMMETRIC CONES. Osaka J. Math. 44 (2007), no. 4, Larotonda, Gabriel. NONPOSITIVE CURVATURE: AGEOMETRICALAPPROACHTOHILBERT SCHMIDT OPERATORS. Differential Geom. Appl. 25 (2007), no. 6, Larotonda, Gabriel. NORM INEQUALITIES IN OPERATOR IDEALS. J. Funct. Anal. 255 (2008), no. 11, Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38

43 THANKS McAlpin, John Harris. INFINITE DIMENSIONAL MANIFOLDS AND MORSE THEORY. Thesis (Ph.D.) Columbia University. ProQuest LLC, Ann Arbor, MI, M. Migloli. DECOMPOSITIONS AND COMPLEXIFICATIONS OF HOMOGENEOUS SPACES. Preprint (2013), arxiv: Milnor, John. MORSE THEORY. BASED ON LECTURE NOTES BY M. SPIVAK AND R. WELLS. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J Mostow, George. SOME NEW DECOMPOSITION THEOREMS FOR SEMI-SIMPLE GROUPS. Mem. Amer. Math. Soc. 1955, (1955), no. 14, Neeb, Karl-Hermann. A CARTAN-HADAMARD THEOREM FOR BANACH-FINSLER MANIFOLDS. Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38

44 THANKS Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000). Geom. Dedicata 95 (2002), Porta, Horacio; Recht, Lázaro. CONDITIONAL EXPECTATIONS AND OPERATOR DECOMPOSITIONS. Ann. Global Anal. Geom. 12 (1994), no. 4, Gabriel Larotonda (CONICET-UNGS) ICTP-2013 Matrix Geometries 38 / 38