On a Decomposition Lemma for Positive Semi-Definite Block-Matrices

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for all A, B M+ and all symmetric (or unitarily invariant) norms. The recent survey ∈ n [4] provides a good exposition on these classical norms.

2 Some Consequences

If we ﬁrst use a unitary congruence with 1 I I J = − √2 I I where I is the identity of Mn, we observe that A+B A X ∗ 2 + ReX J J = + ∗ X∗ B A B ReX ∗ 2 − where stands for unspeciﬁed entries and ReX =(X + X∗)/2. Thus Lemma 1.1 yields: ∗ M+ Corollary 2.1. For every matrix in 2n written in blocks of the same size, we have a decomposition

A+B A X 2 + ReX 0 ∗ 0 0 ∗ = U U + V + V X∗ B 0 0 0 A B ReX 2 − for some unitaries U, V M2 . ∈ n This is equivalent to Corollary 2.2 below by the obvious unitary congruence A X A iX . X∗ B ≃ iX∗ B − In Corollary 2.1, if A,B,X have real entries, i.e., if we are dealing with an operator on a real Hilbert space of dimension 2n, then U, V can be taken with real entries, H thus are isometries on . Do we have the same for Corollary 2.2 ? The answer might H be negative, but an explicit counter-example would be desirable. M+ Corollary 2.2. For every matrix in 2n written in blocks of same size, we have a decomposition

A+B A X 2 + ImX 0 ∗ 0 0 ∗ = U U + V + V X∗ B 0 0 0 A B ImX 2 − for some unitaries U, V M2 . ∈ n 2 Here ImX =(X X∗)/2i. The decomposition allows to obtain some norm estimates − depending on how the full matrix is far from a block-diagonal matrix. If Z Mn, ∗ 1/2 ∈ its absolute value is Z := (Z Z) . If A, B Mn are Hermitian, A B means | | ∈ 1 ≤ B A M+. Firstly, by noticing that ImX ImX = X X∗ , we have: − ∈ n ≤| | 2 | − | M+ Corollary 2.3. For every matrix in 2n written in blocks of same size, we have A X 1 A + B + X X∗ 0 0 0 U | − | U ∗ + V V ∗ X∗ B ≤ 2 0 0 0 A + B + X X∗ | − | for some unitaries U, V M2 . ∈ n We may then obtain estimates for the class of symmetric norms . Such a norm k·k on M satisﬁes UA = AU = A for all A M and all unitaries U M . Since a n k k k k k k ∈ n ∈ n symmetric norm on Mn+m induces a symmetric norm on Mn we may assume that our norms are deﬁned on all spaces M , n 1. n ≥ M+ Corollary 2.4. For every matrix in 2n written in blocks of same size and for all symmetric norms, we have A X p 2|p−1| (A + B)p + X X∗ p X∗ B ≤ {k k k| − | k}

for all p> 0. Proof. We ﬁrst show the case 0

Applying again (2.1) with f(t)= tp, S = A + B and T = X X∗ yields the result for | − | 0

(see [3, Section 2] for much stronger results). With A + B + X X∗ 0 0 0 S = U | − | U ∗, T = V V ∗ 0 0 0 A + B + X X∗ | − | 3 we get from Corollary 2.3 and (2.2)

A X p (A + B + X X∗ )p X∗ B ≤k | − | k

and another application of (2.2) with S = 2(A + B) and T =2 X X∗ completes the | − | proof. M+ Corollary 2.5. For any matrix in 2n written in blocks of same size such that the right upper block X is accretive, we have

A X A + B + ReX X∗ B ≤k k k k

for all symmetric norms.

Proof. By Corollary 2.1, for all Ky Fan k-norms , k =1,..., 2n, we have k·kk A+B A X 2 + ReX 0 0 0 + + . X∗ B ≤ 0 0 0 A B k k 2 k

Equivalently,

A X A + B ↓ A + B ↓ + ReX + X∗ B ≤ 2 2 k k k

where Z↓ stands for the diagonal matrix listing the eigenvalues of Z M+ in decreasing ∈ n order. By using the triangle inequality for and the fact that k·kk Z↓ + Z↓ = Z↓ + Z↓ k 1 kk k 2 kk k 1 2 kk + for all Z1,Z2 M we infer ∈ n A X (A + B)↓ + (ReX)↓ . X∗ B k ≤ k

Hence A X (A + B)↓ + (ReX)↓ X∗ B ≤

for all symmetric norms. The triangle inequality completes the proof Corollary 2.6. For any matrix in M+ written in blocks of same size such that 0 / 2n ∈ W (X), the numerical range the of right upper block X, we have

A X A + B + X X∗ B ≤k k k k

for all symmetric norms.

4 Proof. The condition 0 / W (X) means that zX is accretive for some complex number ∈ z in the unit circle. Making use of the unitary congruence

A X A zX X∗ B ≃ zX∗ B we obtain the result from Corollary 2.5. 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 Wint(X). In case of the usual operator norm , this can be restated with the numerical radius w(X): k·k∞ Corollary 2.7. For any matrix in M+ written in blocks of same size such that 0 / 2n ∈ Wint(X), the relative interior of the numerical range the of right upper block X, we have

A X A + B + w(X). X∗ B ≤k k∞ ∞

In case of the operator norm, we also infer from Corollary 2.1 the following result: M+ Corollary 2.8. For any matrix in 2n written in blocks of same size, we have

A X A + B +2w(X). X∗ B ≤k k∞ ∞

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 2.1. k k∞ Example 2.9. By letting

1 0 0 0 0 1 A = , B = , X = 0 0 0 1 0 0 we have an equality case in the previous corollary. This example also gives an equality case in Corollary 2.4 for the operator norm and any p 1. (For any 0

From Corollary 2.2 we also recapture in the next corollary the majorization result obtained in [5] for positive block-matrices whose off-diagonal blocks are Hermitian. Ex- ample 2.9 shows that the Hermitian requirement on the off-diagonal blocks is necessary. M+ Corollary 2.10. Given any matrix in 2n written in blocks of same size with Hermitian off-diagonal blocks, we have

A X A + B X B ≤k k

for all symmetric norms.

5 Letting X = 0 in the above corollary we get the basic inequality (1.1). The last two corollaries seem to be folklore. M+ Corollary 2.11. Given any matrix in 2n written in blocks of same size, we have A X A X 2 A B X∗ B ⊕ X∗ B ≤ k ⊕ k

for all symmetric norms.

Proof. This follows from (1.1) and the obvious unitary congruence

A X A X A X A X − X∗ B ⊕ X∗ B ≃ X∗ B ⊕ X∗ B −

Let , 1 p < , denote the usual Schatten p-norms. The previous corollary ||·||p ≤ ∞ entails the last one: M+ Corollary 2.12. Given any matrix in 2n written in blocks of same size, we have A X 21−1/p( A p + B p)1/p X∗ B ≤ k kp k kp p for all p [1, ). ∈ ∞ Note that if A = X = B we have an equality case in Corollary 2.12. Remark 2.13. Lemma 1.1 is still valid for compact operators on a Hilbert space, by taking U and V as partial isometries. A similar remark holds for the subadditivity inequality (2.1). Hence the symmetric norm inequalities in this paper may be extended to the setting of normed ideals of compact operators. The lack of counter-example suggests that the following could hold:

Conjecture 2.14. Corollary 2.10 is still true when the oﬀ-diagonal blocks are normal.

References

[1] J. S. Aujla and J.-C. Bourin, Eigenvalues inequalities for convex and log-convex functions, Linear Algebra Appl. 424 (2007), 25–35.

[2] J.-C. Bourin and F. Hiai, Norm and anti-norm inequalities for positive semi-deﬁnite matrices, Internat. J. Math. 63 (2011), 1121-1138.

[3] J.-C. Bourin and E.-Y. Lee, Unitary orbits of Hermitian operators with convex and concave functions, preprint (arXiv:1109.2384).

[4] F. Hiai, Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization (GSIS selected lectures), Interdisciplinary Information Sciences 16 (2010), 139-248.

6 [5] M. Lin and H. Wolkowicz, An eigenvalue majorization inequality for positive semideﬁnite block matrices, Linear Multilinear Algebra 2012, in press.

J.-C. Bourin, Laboratoire de mathématiques, Universitéde Franche-Comté, 25 000 Besancon, France. [email protected]

Eun-Young Lee Department of mathematics, Kyungpook National University, Daegu 702-701, Korea. eylee89@ knu.ac.kr

Minghua Lin Department of applied mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada. [email protected]