On a decomposition lemma for positive semi-definite block-matrices

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1 On a decomposition lemma for positive semi-definite bloc-matrices arxiv: v1 [math.fa] Feb 01 Jean-Christophe ourin, Eun-Young Lee, Minghua Lin January, 01 Abstract This short note, in part of expository nature, points out several new or recent consequences of a quite nice decomposition for positive semi-definite matrices. Keywords: Operator inequalities, positive semi-definite matrix, unitary orbit, symmetric norm. AMS subjects classification 010: 15A60, 47A30, 15A4. 1 A decomposition lemma Let M n denote the space of n n complex matrices, or operators on a finite dimensional Hilbert space, and let M + n be the positive (semi-definite) part. For positive blocmatrices, A X X M + n+m, with A M + n, M + m, we have a remarable decomposition lemma for elements in M + n+m noticed in [3]: Lemma 1.1. For every matrix in M + n+m written in blocs, we have the decomposition A X X = U A 0 V 0 for some unitaries U, V M n+m. The motivation for such a decomposition is various inequalities for convex or concave functions of positive operators partitioned in blocs. These results are extensions of some classical majorization, Rotfel d and Minowsi type inequalities. Lemma 1.1 actually implies a host of such inequalities as shown in the recent papers [] and [3] where a proof of Lemma 1.1 can be found too. Supported by ANR 011-S Research supported by asic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( ) 1

2 This note aims to give in the next section further consequences of the above decomposition lemma, including a simple proof of a recent majorization shown in [5]. Most of the corollaries below are rather straightforward consequences of Lemma 1.1, except Corollary.4 which also requires some more elaborated estimates. Corollary.10 is the majorization given in [5], a remarable extension of the basic and useful inequality A 0 A+ (1.1) 0 for all A, M + n and all symmetric (or unitarily invariant) norms. The recent survey [4] provides a good exposition on these classical norms. Some Consequences If we first use a unitary congruence with J = 1 [ I I I I where I is the identity of M n, we observe that [ A X A+ ] J X J +ReX = A+ ReX where stands for unspecified entries and ReX = (X+X )/. Thus Lemma 1.1 yields: Corollary.1. For every matrix in M + n written in blocs of the same size, we have a decomposition [ A X A+ ] +ReX 0 X = U 0 A+ V ReX for some unitaries U, V M n. This is equivalent to Corollary. below by the obvious unitary congruence A X A ix X ix. In Corollary.1, if A,,X have real entries, i.e., if we are dealing with an operator on a real Hilbert space H of dimension n, then U,V can be taen with real entries, thus are isometries on H. Do we have the same for Corollary.? The answer might be negative, but an explicit counter-example would be desirable. Corollary.. For every matrix in M + n written in blocs of same size, we have a decomposition [ A X A+ ] +ImX 0 X = U 0 A+ V ImX for some unitaries U, V M n. ]

3 Here ImX = (X X )/i. The decomposition allows to obtain some norm estimates depending on how the full matrix is far from a bloc-diagonal matrix. If Z M n, its absolute value is Z := (Z Z) 1/. If A, M n are Hermitian, A means A M + n. Firstly, by noticing that ImX ImX = 1 X X, we have: Corollary.3. For every matrix in M + n written in blocs of same size, we have A X X 1 { [ } A+ + X X U 0 ]V 0 A+ + X X for some unitaries U, V M n. We may then obtain estimates for the class of symmetric norms. Such a norm on M n satisfies UA = AU = A for all A M n and all unitaries U M n. Since a symmetric norm on M n+m induces a symmetric norm on M n we may assume that our norms are defined on all spaces M n, n 1. Corollary.4. For every matrix in M + n written in blocs of same size and for all symmetric norms, we have p A X X p 1 { (A+) p + X X p } for all p > 0. Proof. We first show the case 0 < p < 1. From [1] (for a proof see also, [3, Section 3]) it is nown that: If S,T M + n and if f : [0, ) [0, ) is concave, then, for some unitary U,V M + n, f(s +T) Uf(S)f(T)V. (.1) Applying (.1) to f(t) = t p and the RHS of Corollary.3 with S = 1 A+ + X X U 0 U, T = 1 V 0 A+ + X X we obtain p A X X 1 p { (A+ + X X ) p } Applying again (.1) with f(t) = t p, S = A+ and T = X X yields the result for 0 < p < 1. To get the inequality for p 1, it suffices to use in the RHS of Corollary.3 the elementary inequality, for S,T M + n, ( ) p S +T Sp + T p (.) (see [3, Section ] for much stronger results). With A+ + X X S = U 0 U, T = V 3 V 0 A+ + X X V

4 we get from Corollary.3 and (.) p A X X (A+ + X X ) p and another application of (.) with S = (A+) and T = X X completes the proof. Corollary.5. For any matrix in M + n written in blocs of same size such that the right upper bloc X is accretive, we have A X X A+ + ReX Proof. y Corollary.1, for all Ky Fan -norms, = 1,...,n, we have [ A+ X ] +ReX 0 + ]. Equivalently, X ] 0 A+ ( A+ ( ) A+ +ReX) + where Z stands for the diagonal matrix listing the eigenvalues of Z M + n in decreasing order. y using the triangle inequality for and the fact that for all Z 1,Z M + n Hence we infer X ] Z 1 + Z = Z 1 +Z (A+) +(ReX). A X X (A+) +(ReX) The triangle inequality completes the proof Corollary.6. For any matrix in M + n written in blocs of same size such that 0 / W(X), the numerical range the of right upper bloc X, we have A X X A+ + X 4

5 Proof. The condition 0 / W(X) means that zx is accretive for some complex number z in the unit circle. Maing use of the unitary congruence A X A zx X zx we obtain the result from Corollary.5. The condition 0 / W(X) in the previous corollary can obviously be relaxed to 0 does not belong to the relative interior of X, denoted by W int (X). In case of the usual operator norm, this can be restated with the numerical radius w(x): Corollary.7. For any matrix in M + n written in blocs of same size such that 0 / W int (X), the relative interior of the numerical range the of right upper bloc X, we have X ] A+ +w(x). In case of the operator norm, we also infer from Corollary.1 the following result: Corollary.8. For any matrix in M + n written in blocs of same size, we have X ] A+ +w(x). Once again, the proof follows by replacing X by zx where z is a scalar in the unit circle such that w(x) = w(zx) = Re(zX) and then by applying Corollary.1. Example.9. y letting A = 1 0, =, X = we have an equality case in the previous corollary. This example also gives an equality case in Corollary.4 for the operator norm and any p 1. (For any 0 < p < 1 and for the trace norm, equality occurs in Corollary.4 with A = and X = 0.) From Corollary. we also recapture in the next corollary the majorization result obtained in [5] for positive bloc-matrices whose off-diagonal blocs are Hermitian. Example.9 shows that the Hermitian requirement on the off-diagonal blocs is necessary. Corollary.10. Given any matrix in M + n written in blocs of same size with Hermitian off-diagonal blocs, we have A X A+ X 5

6 Letting X = 0 in the above corollary we get the basic inequality (1.1). The last two corollaries seem to be follore. Corollary.11. Given any matrix in M + n written in blocs of same size, we have A X A X X X A Proof. This follows from (1.1) and the obvious unitary congruence A X A X A X A X X X X X Let p, 1 p <, denote the usual Schatten p-norms. The previous corollary entails the last one: Corollary.1. Given any matrix in M + n written in blocs of same size, we have X ] 1 1/p ( A p p + p p )1/p for all p [1, ). p Note that if A = X = we have an equality case in Corollary.1. Remar.13. Lemma 1.1 is still valid for compact operators on a Hilbert space, by taing U and V as partial isometries. A similar remar holds for the subadditivity inequality (.1). Hence the symmetric norm inequalities in this paper may be extended to the setting of normed ideals of compact operators. The lac of counter-example suggests that the following could hold: Conjecture.14. Corollary.10 is still true when the off-diagonal blocs are normal. References [1] J. S. Aujla and J.-C. ourin, Eigenvalues inequalities for convex and log-convex functions, Linear Algebra Appl. 44 (007), [] J.-C. ourin and F. Hiai, Norm and anti-norm inequalities for positive semi-definite matrices, Internat. J. Math. 63 (011), [3] J.-C. ourin and E.-Y. Lee, Unitary orbits of Hermitian operators with convex and concave functions, preprint (arxiv: ). [4] F. Hiai, Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization (GSIS selected lectures), Interdisciplinary Information Sciences 16 (010),

7 [5] M. Lin and H. Wolowicz, An eigenvalue majorization inequality for positive semidefinite bloc matrices, Linear Multilinear Algebra 01, in press. J.-C. ourin, Laboratoire de mathématiques, Université de Franche-Comté, esancon, France. jcbourin@univ-fcomte.fr Eun-Young Lee Department of mathematics, Kyungpoo National University, Daegu , Korea. eylee89@ nu.ac.r Minghua Lin Department of applied mathematics, University of Waterloo, Waterloo, ON, NL 3G1, Canada. mlin87@ymail.com 7