A matrix subadditivity inequality for symmetric norms

Well-known subadditivity results for positive operators (of Brown-Kosaki and Rotfeld/Ando-Zhan types) are extended to Hermitian and normal ones. Applications to Cartesian decomposition and block-matrices are given.


Some recent results for positive operators
Several nice inequalities for concave functions of operators have been recently established in a serie of papers [5], [8], [7] and [6]. Most of these results are matrix versions of the obvious inequality f (a + b) ≤ f (a) + f (b) (1) for non-negative concave functions f on [0, ∞) and scalars a, b ≥ 0. By matrix version we mean suitable extension where scalars are replaced by n-by-n matrices, i.e., operators on an n-dimensional Hilbert space H. For instance, we have [8]: As usual, capital letters A, B, . . . stand for operators, A ≥ 0 refers to positive semidefinite, and a symmetric norm (or unitarily invariant) satisfies A = U AV for all A and all unitaries U, V . Thus, up to symmetric norms, the basic inequality (1) still holds on the cone of positive operator. This subadditivity result for norms can not be extended to the determinant, even in the case of an operator concave function such as f (t) = √ t. The most elementary case in the above theorem is for the trace norm. Then, the result can be restated as a famous trace inequality [11]: Rotfel'd Inequality. Let f be a concave function on [0, ∞) such that f (0) ≥ 0. Then, for all A, B ≥ 0, In the matrix setting, the concavity assumption is quite crucial as shown in the following simple remark [13]: holds for all two-by-two positive matrices A, B, then f is concave.
To prove this statement, take for s, t > 0, and observe that the trace inequality means that f is concave. Theorem 1.1 closed a list of papers of several authors including Ando-Zhan [1], and Kosem [10]. However, It remained natural to ask wether this result could be extended to the set of all Hermitian, or even all normal operators. We noticed a partial answer in [6]: Here the e-convexity property of f means that f (e t ) is convex on (−∞, ∞). In particular, the theorem holds for the power functions f (t) = t p , 1 ≥ p ≥ 0. This result for normal operators entails several estimates for block matrices. A special case involving an operator partitioned in four normal blocks A, B, C, D of same size is for all symmetric norms and 0 ≤ p ≤ 1. These estimates, comparing an operator on H⊕H with a related operator on H, differ from the usual ones in the literature where the norm of the full matrix is evaluated with the norms of its blocks, for instance, [9] and [3]. In the subsequent sections we solve the conjectures in [6] by showing that the assumption of e-convexity is not necessary in Theorem 1.2, in its application to block-matrices and in some related inequalities. The proof of Theorem 1.2 given in [7] reduced to the positive case by using the fact that for any normal A, B and any non-negative e-convex functions f (t), we have for all symmetric norms. This is no longer true if the e-convexity assumption is dropped. In fact, one can easily find two-by-two positive semi-definite matrices A, B and a non-negative concave function f (t) on [0, ∞) such that for all symmetric norms which are not a multiple scalar of the usual operator norm.
For instance, take f (t) = min{t, √ 2/2} and The main point of the forthcoming proof is to overcome this difficulty. This proof can be adapted in order to obtain a version for normal operators of the following companion result to Theorem 1.1:

Subadditivity results for normal operators
We have the following norm inequalities: Then, for all symmetric norms,

This is a matrix version of the obvious inequality
. But the left inequality can not be extended to matrices. Indeed it is easy to find two-by-two matrices Z = A + iB -a simple example is given with A, B defined in (2) -with the eigenvalue relation Thus, there are some non-negative concave functions like for all symmetric norms which are not a multiple scalar of the usual operator norm. Let A, B be general operators. Applying Theorem 2.1 to the Hermitian operators so that, letting B = A * yields: Note that equality occurs in Corollary 2.3 whenever f (0) = 0 and where X is arbitrary. Note also that it may happen that for some concave functions and some symmetric norms, for instance when and the norm is the sum of the two largest singular values. At the end of this section, we will see some application of Theorem 2.1 to partitioned operators. Now, we turn to the proof of Theorem 2.1. We start by recalling the Ky Fan Principle. The Ky Fan k-norms of A, k = 1, 2, . . . , n are defined as the sum of its k largest singular values, Let A, B such that A (k) ≤ B (k) for all k = 1, 2, . . . , n. Then, the vector of the singular values of A lies in the convex hull of the permuted singular values of B multiplied by ±1, This can be proved by using the Hyperplan separation process to reach a contradiction, see [12] for details and [2] for alternative proofs. From this convexity statement follows a useful fact: Ky Fan Principle. Suppose that A (k) ≤ B (k) for all Ky-Fan k-norms. Then, we have A ≤ B for all symmetric norms.
We also need two elementary, well-known lemmas. For A, B ≥ 0 it is sometimes convenient to write A ≺ w B to mean that A ≤ B for all symmetric norms.
Combining this with

Proof. Note that
where ≃ means unitarily congruent. Combining with gives the lemma. 2 Proof of Theorem 2.1. It suffices to prove the result when A and B are Hermitian. The general case then follows by replacing A, B bỹ and by using normality of A and B. Therefore assume that A, B are Hermitian with decomposition in positive and negative parts, Let g(t) = f (t) − f (0) and note that, for each Ky Fan k-norm, . Hence, it suffices to prove the result for g(t), or equivalently when f (0) = 0. This assumption implies Now, given two positive n-by-n matrices X and Y with direct sum for all j = 1, 2, . . . , n. Indeed, for some subspace S ⊂ H we have for all j = 1, 2, . . . , n. Replacing in (4) X by A + + B + and Y by A − + B − we then get for all j = 1, 2, . . . , n. Since f is non-decreasing, it follows for all j = 1, 2, . . . , n, so that for all symmetric norms. By Theorem 1.1 combined with Lemma 1, followed by application of Lemma 2, we then obtain and making use of relations (3) ends the proof. 2 Let us now give some application for Block-matrices. The most obvious one is for a Hermitian matrix A B B * C partitioned in four blocks of same size. Then by using Theorem 2.1 for the decomposition in two Hermitian and then using Lemma 1, we have for We may then obtain results for some matrices partitioned in m 2 blocks of same size.
Proof. We prove this corollary via Theorem 2.1. for a partition in four blocks The proof for a partition in m 2 blocks is similar by using the version of Theorem 2.1 for m operators. LetÃ = 0 A A * 0 and note that where the symbol ≃ stands for unitarily equivalent. On the other hand are Hermitian. Therefore, Theorem 2.1 yields, for all symmetric norms; that is, using the shorthand symbol ≺ w , Gathering the two first lines, and the two last ones, we have via Lemmas 2 and 1 By using (6) we then obtain, using normality of S, T, R, Q, which is equivalent to inequalities for symmetric norms.  Proof. Consider the polar decompositions A = |A * |U and B = V |B|, note that and apply Theorem 2.1. 2 The assumption in Corollary 2.4 requiring normality of each block is rather special. The next corollary generalizes (5) and meets the simple requirement that the full matrix is Hermitian.
Proof. The proof of Corollary 2.4 actually shows that for a general block-matrix A = (A i, j ) partitioned in blocks of same size, we have for all non-negative concave function f . Assuming A Hermitian, we have A * i, j = A j, i and Corollary 2.6 follows. Indeed, we can derive Theorem 3.1 from Theorem 1.3 in a quite similar way of the one we derive Theorem 2.1 from Theorem 1.1. The proof of Theorem 3.1 starts by noticing that we can assume that A is Hermitian. Then, using the decomposition in positive and negative parts A = A + − A − we have, as in the proof of Theorem 2.1, and we may proceed as previously.
When we deal with the trace norm, the fact that f is positive on the whole halfline is not essential, as in Rotfel'd inequality. Hence we have the following corollary, extending to normal operators a result from [4]. Corollary 3.2 follows from Theorem 3.1 by approaching f (t) with g(t) + at for some scalar a and some non-negative concave function g(t). Theorem 2.1 and 3.1 can be combined in a unique statement, extending the main result in [7]: be expansive and let f be a non-negative concave function on [0, ∞). Then, for all symmetric norms, It would be elegant and interesting to state this theorem in the more general framework of positive linear maps Φ between matrix algebras. This leads to the problem of characterizing the positive linear maps Φ such that for all normal operators N , all non-negative concave functions and all symmetric norms. Some furher questions are considered in [6]. For sake of completeness, we mention that when f (t) is a non-negative convex function vanishing at 0, then inequalities of Theorem 1.1-1.3 are reversed. For instance we have [7] Theorem 3.4. Let {A i } m i=1 be positive and let {Z i } m i=1 be expansive. Then, for all symmetric norms and all p > 1, If Z i = I for all i, it is a famous result of Ando-Zhan [1] and of Bhatia-Kittaneh [3] in case of integer exponents. The very special case Tr (A p 1 + A p 2 ) ≤ Tr (A 1 + A 2 ) p is Mc-Carthy's inequality [13, p. 20]. Note that the positivity assumption in Theorem 3.4 can not be replaced by a normality one.
When we consider contractive congruences and positive operators, then there exist several Jensen type inequalities, not only for norms but also for eigenvalues (cf. [4] [5]). The proof are much simpler than in the expansive case, where some unexpected counterexamples may occur (see discussion and counterexamples in [4] [5]). We give an example of such results: