aa r X i v : . [ m a t h . F A ] D ec Anti Lie-Trotter formula
Koenraad M.R. Audenaert , , ∗ and Fumio Hiai , † September 20, 2018 Department of Mathematics, Royal Holloway University of London,Egham TW20 0EX, United Kingdom Department of Physics and Astronomy, Ghent University,S9, Krijgslaan 281, B-9000 Ghent, Belgium Tohoku University (Emeritus),Hakusan 3-8-16-303, Abiko 270-1154, Japan
Abstract
Let A and B be positive semidefinite matrices. The limit of the expression Z p := ( A p/ B p A p/ ) /p as p tends to 0 is given by the well known Lie-Trotter-Kato formula. A similar formula holds for the limit of G p := ( A p B p ) /p as p tends to 0, where X Y is the geometric mean of X and Y . In this paper westudy the complementary limit of Z p and G p as p tends to ∞ , with the ultimategoal of finding an explicit formula, which we call the anti Lie-Trotter formula.We show that the limit of Z p exists and find an explicit formula in a special case.The limit of G p is shown for 2 × Primary 15A42, 15A16, 47A64
Key Words and Phrases:
Lie-Trotter-Kato product formula, Lie-Trotter formula,anti Lie-Trotter formula, positive semidefinite matrix, operator mean, geometricmean, log-majorization, antisymmetric tensor power, Grassmannian manifold ∗ E-mail: [email protected] † E-mail: [email protected] Introduction
When
H, K are lower bounded self-adjoint operators on a Hilbert space H and H + , K + are their positive parts, the sum of H and K can be given a precise meaning as a lowerbounded self-adjoint operator on the subspace H , which is defined as the closure ofdom H / ∩ dom K / . We denote this formal sum as H ˙+ K . Then the well-knownLie-Trotter-Kato product formula, as originally established in [18, 11] and refined bymany authors, expresses the convergencelim n →∞ ( e − tH/n e − tK/n ) n = e − t ( H ˙+ K ) P , t > , in the strong operator topology (uniformly in t ∈ [ a, b ] for any 0 < a < b ), where P is the orthogonal projection onto H . Although this formula is usually stated fordensely-defined H, K , the proof in [11] applies to the improper case (i.e.,
H, K are notdensely-defined) as well, under the convention that e − tH = 0 on (dom H ) ⊥ for t > e − tK .The Lie-Trotter-Kato formula can easily be modified to symmetric form and with acontinuous parameter as [8, Theorem 3.6]lim p ց ( e − ptH/ e − ptK e − ptH/ ) /p = e − t ( H ˙+ K ) P , t > . When restricted to matrices (and to t = 1) this can be rephrased aslim p ց ( A p/ B p A p/ ) /p = P exp(log A ˙+ log B ) , (1.1)where A and B are positive semidefinite matrices (written as A, B ≥ P is now the orthogonal projection onto the intersection of the supports of A, B andlog A ˙+ log B is defined as P (log A ) P + P (log B ) P .When σ is an operator mean [13] corresponding to an operator monotone function f on (0 , ∞ ) such that α := f ′ (1) is in (0 , p ց ( e − ptH σ e − ptK ) /p = e − t ((1 − α ) H ˙+ αK ) , t > , in the strong operator topology, for a bounded self-adjoint operator H and a lower-bounded self-adjoint operator K on H . Although it is not known whether the aboveformula holds even when both H, K are lower bounded (and unbounded), we can verifythat (1.1) has the operator mean versionlim p ց ( A p σ B p ) /p = P exp((1 − α ) log A ˙+ α log B ) , (1.2)for matrices A, B ≥
0. A proof of (1.2) is supplied in an appendix of this paper sinceit is not our main theme. 2n particular, let σ be the geometric mean A B (introduced first in [17] and furtherdiscussed in [13]), corresponding to the operator monotone function f ( x ) = x / (hence α = 1 / p ց ( A p B p ) /p = P exp(log A ˙+ log B ) , (1.3)which has the same right-hand side as (1.1).It turns out that the convergence of both (1.1) and (1.3) is monotone in the log-majorization order. For d × d matrices X, Y ≥
0, the log-majorization relation X ≺ (log) Y means that k Y i =1 λ i ( X ) ≤ k Y i =1 λ i ( Y ) , ≤ k ≤ d, with equality for k = d , where λ ( X ) ≥ · · · ≥ λ d ( X ) are the eigenvalues of X sorted indecreasing order and counting multiplicities. The Araki-Lieb-Thirring inequality canbe written in terms of log-majorization as( A p/ B p A p/ ) /p ≺ (log) ( A q/ B q A q/ ) /q if 0 < p < q, (1.4)for matrices A, B ≥
0, see [14, 3, 2]. One can also consider the complementary versionof (1.4) in terms of the geometric mean. Indeed, for
A, B ≥ A q B q ) /q ≺ (log) ( A p B p ) /p if 0 < p < q. (1.5)Hence, for matrices A, B ≥
0, we see that Z p := ( A p/ B p A p/ ) /p and G p := ( A p B p ) /p both tend to P exp(log A ˙+ log B ) as p ց
0, with the former decreasing (by (1.4)) andthe latter increasing (by (1.5)) in the log-majorization order.The main topic of this paper is the complementary question about what happensto the limits of Z p and G p as p tends to ∞ instead of 0. Although this seems a naturalmathematical problem, we have not been able to find an explicit statement of concernin the literature. It is obvious that if A and B are commuting then G p = AB = Z p ,independently of p >
0. However, if A and B are not commuting, then the limitbehavior of Z p and its eigenvalues as p → ∞ is of a rather complicated combinatorialnature, and that of G p seems even more complicated.The problem of finding an explicit formula, which we henceforth call the anti Lie-Trotter formula, also emerges from recent developments of new R´enyi relative entropiesrelevant to quantum information theory. Indeed, the recent paper [4] proposed togeneralize the R´enyi relative entropy as D α,z ( ρ k σ ) := 1 α − (cid:0) ρ α/ z σ (1 − α ) /z ρ α/ z (cid:1) z for density matrices ρ, σ with two real parameters α, z , and discussed the limit formulaswhen α, z converge to some special values. The limit case of D α,z ( ρ k σ ) as z → α fixed is exactly related to our anti Lie-Trotter problem.3he rest of the paper is organized as follows. In Section 2 we prove the existenceof the limit of Z p as p → ∞ when A, B are d × d positive semidefinite matrices. InSection 3 we analyze the case when the limit eigenvalue list of Z p becomes λ i ( A ) λ i ( B )(1 ≤ i ≤ d ), the maximal case in the log-majorization order. In Section 4 we extendthe existence of the limit of Z p to that of (cid:0) A p/ · · · A p/ m − A pm A p/ m − · · · A p/ (cid:1) /p with morethan two matrices. Finally in Section 5 we treat G p ; however we can prove the existenceof the limit of G p as p → ∞ only when A, B are 2 × ( A p/ B p A p/ ) /p as p → ∞ Let A and B be d × d positive semidefinite matrices having the eigenvalues a ≥ · · · ≥ a d ( ≥
0) and b ≥ · · · ≥ b d ( ≥ { v , . . . , v d } be an orthonormal set of eigenvectors of A such that Av i = a i v i for i = 1 , . . . , d , and { w , . . . , w d } an orthonormal set of eigenvectors of B in a similar way. Then A and B are diagonalized as A = V diag( a , . . . , a d ) V ∗ = d X i =1 a i v i v ∗ i , (2.1) B = W diag( b , . . . , b d ) W ∗ = d X i =1 b i w i w ∗ i . (2.2)For each p > Z p := ( A p/ B p A p/ ) /p , (2.3)whose eigenvalues are denoted as λ ( p ) ≥ λ ( p ) ≥ · · · ≥ λ d ( p ), again in decreasingorder and counting multiplicities. Lemma 2.1.
For every i = 1 , . . . , d the limit λ i := lim p →∞ λ i ( p ) (2.4) exists, and a b ≥ λ · · · ≥ λ d ≥ a d b d .Proof. Since ( a b ) p I ≥ A p/ B p A p/ ≥ ( a d b d ) p I , we have a b ≥ λ i ( p ) ≥ a d b d for all i = 1 , . . . , d and all p >
0. By the Araki-Lieb-Thirring inequality [3] (or the log-majorization [2]), for every k = 1 , . . . , d we have k Y i =1 λ i ( p ) ≤ k Y i =1 λ i ( q ) if 0 < p < q. (2.5)4herefore, the limit η k of Q ki =1 λ i ( p ) as p → ∞ exists for any k = 1 , . . . , d so that η ≥ · · · ≥ η d ≥
0. Let m (0 ≤ m ≤ d ) be the largest k such that η k > m := 0if η = 0). When 1 ≤ k ≤ m , we have λ k ( p ) → η k /η k − (where η := 1) as p → ∞ .When m < d , λ m +1 ( p ) → η m +1 /η m = 0 as p → ∞ . Hence λ k ( p ) → k > m .Therefore, the limit of λ i ( p ) as p → ∞ exists for any i = 1 , . . . , d . The latter assertionis clear now. Lemma 2.2.
The first eigenvalue in (2.4) is given by λ = max { a i b j : ( V ∗ W ) ij = 0 } , where ( V ∗ W ) ij denotes the ( i, j ) entry of V ∗ W .Proof. Write V ∗ W = [ u ij ]. We observe that (cid:0) V ∗ A p/ B p A p/ V (cid:1) ij = d X k =1 u ik u jk a p/ i a p/ j b pk . In particular, (cid:0) V ∗ A p/ B p A p/ V (cid:1) ii = d X k =1 | u ik | a pi b pk and hence we have λ ( p ) p ≤ Tr A p/ B p A p/ = d X i =1 d X k =1 | u ik | a pi b pk ≤ d max { a pi b pk : u ik = 0 } , where Tr is the usual trace functional on d × d matrices. Therefore, λ ( p ) ≤ d /p max { a i b k : u ik = 0 } . (2.6)On the other hand, we have dλ ( p ) p ≥ Tr A p/ B p A p/ ≥ min {| u ik | : u ik = 0 } max { a pi b pk : u ik = 0 } so that λ ( p ) ≥ (cid:18) min {| u ik | : u ik = 0 } d (cid:19) /p max { a i b k : u ik = 0 } . (2.7)Estimates (2.6) and (2.7) give the desired expression immediately. In fact, they provethe existence of the limit in (2.4) as well apart from Lemma 2.1.In what follows, for each k = 1 , . . . , d we write I d ( k ) for the set of all subsets I of { , . . . , d } with | I | = k . For I, J ∈ I d ( k ) we denote by ( V ∗ W ) I,J the k × k submatrix of V ∗ W corresponding to rows in I and columns in J ; hence det( V ∗ W ) I,J denotes the corresponding minor of V ∗ W . We also write a I := Q i ∈ I a i and b I := Q i ∈ I b i . Since det( V ∗ W ) = 0, note that for any k = 1 , . . . , d and any I ∈ I d ( k )we have det( V ∗ W ) I,J = 0 for some J ∈ I d ( k ), and that for any J ∈ I d ( k ) we havedet( V ∗ W ) I,J = 0 for some I ∈ I d ( k ). 5 emma 2.3. For every k = 1 , . . . , d , λ λ · · · λ k = max { a I b J : I, J ∈ I d ( k ) , det( V ∗ W ) I,J = 0 } . (2.8) Proof.
For each k = 1 , . . . , d the antisymmetric tensor powers A ∧ k and B ∧ k (see [5])are given in the form of diagonalizations as A ∧ k = V ∧ k diag( a I ) I ∈I d ( k ) V ∧ k , B ∧ k = W ∧ k diag( b I ) I ∈I d ( k ) W ∧ k , and the corresponding representation of the (cid:0) nk (cid:1) × (cid:0) nk (cid:1) unitary matrix V ∗∧ k W ∧ k is givenby ( V ∗∧ k W ∧ k ) I,J = det( V ∗ W ) I,J , I, J ∈ I d ( k ) . Note that the largest eigenvalue of (cid:0) ( A ∧ k ) p/ ( B ∧ k ) p ( A ∧ k ) p/ (cid:1) /p = (cid:0) ( A p/ B p A p/ ) /p (cid:1) ∧ k is λ ( p ) λ ( p ) · · · λ k ( p ), whose limit as p → ∞ is λ λ · · · λ k by Lemma 2.1. ApplyLemma 2.2 to A ∧ k and B ∧ k to obtain expression (2.8).Let H be a d -dimensional Hilbert space (say, C d ), k be an integer with 1 ≤ k ≤ d ,and H ∧ k be the k -fold antisymmetric tensor of H . We write x ∧ · · · ∧ x k ( ∈ H ∧ k ) forthe antisymmetric tensor of x , . . . , x k ∈ H (see [5]). The next lemma says that theGrassmannian manifold G ( k, d ) is realized in the projective space of H ∧ k . Althoughthe lemma might be known to specialists, we cannot find a precise explanation in theliterature. So, for the convenience of the reader, we will present its sketchy proof inAppendix A based on [7]. Lemma 2.4.
There are constants α, β > (depending on only d and k ) such that α k P − Q k ≤ inf θ ∈ R k u ∧ · · · ∧ u k − e √− θ v ∧ · · · ∧ v k k ≤ β k P − Q k for all orthonormal sets { u , . . . , u k } and { v , . . . , v k } and the respective orthogonalprojections P and Q onto span { u , . . . , u k } and span { v , . . . , v k } , where k P − Q k is theoperator norm of P − Q and k · k inside infimum is the norm on H ∧ k . The main result of the paper is the next theorem showing the existence of limit forthe anti version of (1.1).
Theorem 2.5.
For every d × d positive semidefinite matrices A and B the matrix Z p in (2.3) converges as p → ∞ to a positive semidefinite matrix.Proof. By replacing A and B with V AV ∗ and V BV ∗ , respectively, we may assumethat V = I and so A = diag( a , . . . , a d ) , B = W diag( b , . . . , b d ) W ∗ . { u ( p ) , . . . , u d ( p ) } of C d for which we have Z p u i ( p ) = λ i ( p ) u i ( p ) for 1 ≤ i ≤ d . Let λ i be given in Lemma 2.1, and assume that 1 ≤ k < d and λ ≥ · · · ≥ λ k > λ k +1 . Moreover, let λ ( Z ∧ kp ) ≥ λ ( Z ∧ kp ) ≥ . . . be the eigenvaluesof Z ∧ kp in decreasing order. We note thatlim p →∞ λ ( Z ∧ kp ) = lim p →∞ λ ( p ) · · · λ k − ( p ) λ k ( p )= λ . . . λ k − λ k > λ · · · λ k − λ k +1 = lim p →∞ λ ( Z ∧ kp ) . (2.9)Hence it follows that λ ( Z ∧ kp ) is a simple eigenvalue of Z ∧ kp for every p sufficiently large.Letting w I,J := det W I,J for
I, J ∈ I d ( k ) we compute( Z ∧ kp ) p = ( A ∧ k ) p/ W ∧ k ((diag( b , . . . , b d )) ∧ k ) p ( W ∧ k ) ∗ ( A ∧ k ) p/ = diag( a p/ I ) I (cid:2) w I,J (cid:3)
I,J diag( b pI ) I (cid:2) w J,I (cid:3)
I,J diag( a p/ I ) I = X K ∈I d ( k ) w I,K w J,K a p/ I a p/ J b pK I,J = η pk X K ∈I d ( k ) w I,K w J,K a / I a / J b K η k ! p I,J , where η k := λ λ · · · λ k > η k = max { a I b K : I, K ∈ I d ( k ) , w I,K = 0 } due to Lemma 2.3. We now define∆ k := (cid:8) ( I, K ) ∈ I d ( k ) : w I,K = 0 and a I b K = η k (cid:9) . Then we have (cid:18) Z ∧ kp η k (cid:19) p = X K ∈I d ( k ) w I,K w J,K a / I a / J b K η k ! p I,J −→ Q := X K ∈I d ( k ) w I,K w J,K δ I,J,K I,J , where δ I,J,K := ( I, K ) , ( J, K ) ∈ ∆ k , . Since Q I,I ≥ | w I,K | > I, K ) ∈ ∆ k , note that Q = 0. Furthermore, sincethe eigenvalue λ ( Z ∧ kp ) is simple (if p large), it follows from (2.9) that the limit Q of7 Z ∧ kp /η k (cid:1) p must be a rank one projection ψψ ∗ up to a positive scalar multiple, where ψ is a unit vector in ( C d ) ∧ k . Since the unit eigenvector u ( p ) ∧ · · · ∧ u k ( p ) of Z ∧ kp corresponding to the largest (simple) eigenvalue coincides with that of (cid:0) Z ∧ kp /η k (cid:1) p , weconclude that u ( p ) ∧· · ·∧ u k ( p ) converges ψ up to a scalar multiple e √− θ . Therefore, byLemma 2.4 the orthogonal projection onto span { u ( p ) , . . . , u k ( p ) } converges as p → ∞ .Assume now that λ = · · · = λ k > λ k +1 = · · · = λ k > · · · > λ k s − +1 = · · · = λ k s ( k s = d ) . From the fact proved above, the orthogonal projection onto span { u ( p ) , . . . , u k r ( p ) } converges for any r = 1 , . . . , s −
1, and this is trivial for r = s . Therefore, the orthogonalprojection onto span { u k r − +1 ( p ) , . . . , u k r ( p ) } converges to a projection P r for any r =1 , . . . , s , and thus Z p converges to P sr =1 λ k r P r .For 1 ≤ k ≤ d define η k by the right-hand side of (2.8). Then Lemma 2.3 (see alsothe proof of Lemma 2.1) implies that, for k = 1 , . . . , d , λ k = η k η k − if η k > η := 1), and λ k = 0 if η k = 0. So one can effectively compute the eigenvaluesof Z := lim p →∞ Z p ; however, it does not seem that there is a simple algebraic methodto compute the limit matrix Z . Let A and B be d × d positive semidefinite matrices with diagonalizations (2.1) and(2.2). For each d × d matrix X we write s ( X ) ≥ s ( X ) ≥ · · · ≥ s d ( X ) for the singularvalues of X in decreasing order with multiplicities. For each p > k = 1 , . . . , d ,since Q ki =1 λ i ( p ) = (cid:0)Q ki =1 s i ( A p/ B p/ ) (cid:1) /p , by the majorization results of Gel’fand andNaimark and of Horn (see, e.g., [15, 5, 9]), we have k Y j =1 a i j b n +1 − i j ≤ k Y j =1 λ j ( p ) ≤ k Y j =1 a j b j for any choice of 1 ≤ i < i < · · · < i k ≤ d , and for k = d d Y i =1 λ i ( p ) = det A · det B = d Y i =1 a i b i . That is, for any p > a i b n +1 − i ) di =1 ≺ (log) ( λ i ( p )) di =1 ≺ (log) ( a i b i ) di =1 (3.1)8ith the notation of log-majorization, see [2]. Letting p → ∞ gives( a i b n +1 − i ) di =1 ≺ (log) ( λ i ) di =1 ≺ (log) ( a i b i ) di =1 (3.2)for the eigenvalues λ ≥ · · · ≥ λ d of Z = lim p →∞ Z p . In general, we have nothingto say about the position of ( λ i ) di =1 in (3.2). For instance, when V ∗ W becomes thepermutation matrix corresponding to a permutation ( j , . . . , j d ) of (1 , . . . , d ), we have Z p = V diag( a b j , . . . , a d b j d ) V ∗ independently of p > λ i ) = ( a i b j i ).In this section we clarify the case when ( λ i ) di =1 is equal to ( a i b i ) di =1 , the maximal casein the log-majorization order in (3.2). To do this, let 0 = i < i < · · · < i l − < i l = d and 0 = j < j < · · · < j m − < j m = d be taken so that a = · · · = a i > a i +1 = · · · = a i > · · · > a i l − +1 = · · · = a i l ,b = · · · = b j > b j +1 = · · · = b j > · · · > b j m − +1 = · · · = b j m . Theorem 3.1.
In the above situation the following conditions are equivalent:(i) λ i = a i b i for all i = 1 , . . . , d ;(ii) for every k = 1 , . . . , d so that i r − < k ≤ i r and j s − < k ≤ j s , there are I k , J k ∈ I d ( k ) such that { , . . . , i r − } ⊂ I k ⊂ { , . . . , i r } , { , . . . , j s − } ⊂ J k ⊂ { , . . . , j s } , det( V ∗ W ) I k ,J k = 0; (iii) the property in (ii) holds for every k ∈ { i , . . . , i l − , j , . . . , j m − } .Proof. (i) ⇔ (ii). By Lemma 2.3 condition (ii) means that k Y i =1 λ i = k Y i =1 a i b i , k = 1 , . . . , d. It follows (see the proof of Lemma 2.1) that this is equivalent to (i).(ii) ⇒ (iii) is trivial.(iii) ⇒ (i). By Lemma 2.3 again condition (iii) means that h Y i =1 λ i = h Y i =1 a i b i for all h ∈ { i , . . . , i l − , j , . . . , j m − } . (3.3)This holds also for h = d thanks to (3.2). We need to prove that Q ki =1 λ i = Q ki =1 a i b i for all k = 1 , . . . , d . Now, let i r − < k ≤ i r and j s − < k ≤ j s as in condition (ii). If k = i r or k = j s , then the conclusion has already been stated in (3.3). So assume that9 r − < k < i r and j s − < k < j s . Set h := max { i r − , j s − } and h := min { i r , j s } sothat h < k < h . By (3.3) for h = h , h we have h Y i =1 λ i = h Y i =1 a i b i > , h Y i =1 λ i = h Y i =1 a i b i . Since a i = a h and b i = b h for h < i ≤ h , we have Q h i = h +1 λ i = ( a h b h ) h − h . By(3.2) we furthermore have Q h +1 i =1 λ i ≤ Q h +1 i =1 a i b i and hence a h b h ≥ λ h +1 ≥ λ h +2 ≥ · · · ≥ λ h . Therefore, λ i = a h b h for all i with h + 1 < i ≤ h , from which Q ki =1 λ i = Q ki =1 a i b i follows for h < k < h . Proposition 3.2.
Assume that the equivalent conditions of Theorem 3.1 hold. Then,for each r = 1 , . . . , l , the spectral projection of Z corresponding to the set of eigen-values { a i r − +1 b i r − +1 , . . . , a i r b i r } is equal to the spectral projection P i r i = i r − +1 v i v ∗ i of A corresponding to a i r . Hence Z is of the form Z = d X i =1 a i b i u i u ∗ i for some orthonormal set { u , . . . , u d } such that P i r i = i r − +1 u i u ∗ i = P i r i = i r − +1 v i v ∗ i for r = 1 , . . . , l .Proof. In addition to Theorem 2.5 we may prove that, for each k ∈ { i , . . . , i l − } , thespectral projection of Z p corresponding to { λ ( p ) , . . . , λ k ( p ) } converges to P ki =1 v i v ∗ i .Assume that k = i r with 1 ≤ r ≤ l −
1. When j s − < k < j s , by condition (iii)of Theorem 3.1 we have det( V ∗ W ) { ,...,k } , { ,...,j s − ,j ′ s ,...,j ′ k } = 0 for some { j ′ s , . . . , j ′ k } ⊂{ j s − + 1 , . . . , j s } . By exchanging w j ′ s , . . . , w j ′ k with w j s − +1 , . . . , w k we may assumethat det( V ∗ W ) { ,...,k } , { ,...,k } = 0. Furthermore, by replacing A and B with V AV ∗ and V BV ∗ , respectively, we may assume that V = I . So we end up assuming that A = diag( a , . . . , a d ) , B = W diag( b , . . . , b d ) W ∗ , and det W (1 , . . . , k ) = 0, where W (1 , . . . , k ) denotes the principal k × k submatrix ofthe top-left corner. Let { e , . . . , e d } be the standard basis of C d . By Theorem 3.1 wehave lim p →∞ λ ( Z ∧ kp ) = k Y i =1 a i b i > k − Y i =1 a i b i · a k +1 b k +1 = lim p →∞ λ ( Z ∧ kp )so that the largest eigenvalue of Z ∧ kp is simple for every sufficiently large p . Let { u ( p ) , . . . , u d ( p ) } be an orthonormal basis of C d for which Z p u i ( p ) = λ i ( p ) u i ( p ) for1 ≤ i ≤ d . Then u ( p ) ∧ · · · ∧ u k ( p ) is the unit eigenvector of Z ∧ kp corresponding to the10igenvalue λ ( Z ∧ kp ). We now show that u ( p ) ∧ · · · ∧ u k ( p ) converges to e ∧ · · · ∧ e k in( C d ) ∧ k . We observe that( A ∧ k ) p/ = diag (cid:0) a p/ I (cid:1) I = a p/ { ,...,k } diag (cid:18) , α p/ , . . . , α p/ ( dk ) (cid:19) with respect to the basis (cid:8) e i ∧ · · · ∧ e i k : I = { i , . . . , i k } ∈ I d ( k ) (cid:9) , where the firstdiagonal entry 1 corresponds to e ∧ · · · ∧ e k and 0 ≤ α h < ≤ h ≤ (cid:0) dk (cid:1) . Similarly, (cid:0) (diag( b , . . . , b d )) ∧ k (cid:1) p = b p { ,...,k } diag (cid:18) , β p , . . . , β p ( dk ) (cid:19) , where 0 ≤ β h ≤ ≤ h ≤ (cid:0) dk (cid:1) . Moreover, W ∧ k is given as W ∧ k = (cid:2) w I,J (cid:3)
I,J = w · · · w ( dk )... . . . ... w ( dk ) · · · w ( dk )( dk ) , where w I,J = det W I,J and so w = det W (1 , . . . , k ) = 0. As in the proof of Theorem2.5 we now compute( Z ∧ kp ) p = ( A ∧ k ) p/ W ∧ k (cid:0) (diag( b , . . . , b d )) ∧ k (cid:1) p ( W ∧ k ) ∗ ( A ∧ k ) p/ = (cid:0) a { ,...,k } b { ,...,k } (cid:1) p ( dk ) X h =1 w ih w jh α p/ i α p/ j β ph ( dk ) i,j =1 , where α = β = 1. As p → ∞ we have ( dk ) X h =1 w ih w jh α p/ i α p/ j β ph −→ diag X h : β h =1 | w h | , , . . . , ! Since the unit eigenvector of Z ∧ kp corresponding to the largest eigenvalue coincideswith that of (cid:20)P ( dk ) h =1 w ih w jh α p/ i α p/ j β ph (cid:21) , it follows that u ( p ) ∧ · · · ∧ u k ( p ) converges to e ∧ · · · ∧ e k up to a scalar multiple e √− θ , θ ∈ R . By Lemma 2.4 this implies thedesired assertion. Corollary 3.3.
If the eigenvalues a , . . . , a d of A are all distinct and the conditions ofTheorem 3.1 hold, then lim p →∞ ( A p/ B p A p/ ) /p = V diag( a b , a b , . . . , a d b d ) V ∗ . In particular, when the eigenvalues of A are all distinct and so are those of B , theconditions of Theorem 3.1 means that all the leading principal minors of V ∗ W arenon-zero. 11 Extension to more than two matrices
Let A , . . . , A m be d × d positive semidefinite matrices with diagonalizations A l = V l D l V ∗ l , D l = diag (cid:0) a ( l )1 , . . . , a ( l ) d (cid:1) , ≤ l ≤ m. For each p > Z p := (cid:0) A p/ A p/ · · · A p/ m − A pm A p/ m − · · · A p/ A p/ (cid:1) /p , = V (cid:0) D p/ W · · · D p/ m − W m − D pm W ∗ m − D p/ m − · · · W ∗ D p/ (cid:1) /p V ∗ , where W l := V ∗ l V l +1 = h w ( l ) ij i di,j =1 , ≤ l ≤ m − . The eigenvalues of Z p are denoted as λ ( p ) ≥ λ ( p ) ≥ · · · ≥ λ d ( p ) in decreasing order.Although the log-majorization in (2.5) is no longer available in the present situation,we can extend Lemma 2.2 as follows. Lemma 4.1.
The limit λ := lim p →∞ λ ( p ) exists and λ = max (cid:8) a (1) i a (2) i · · · a ( m ) i m : w ( i , i , . . . , i m ) = 0 (cid:9) , (4.1) where w ( i , i , . . . , i m ):= Xn w (1) i j w (2) j j · · · w ( m ) j m − i m : 1 ≤ j , . . . , j m − ≤ d, a (2) j · · · a ( m − j m − = a (2) i · · · a ( m − i m − o . Moreover, a (1)1 · · · a ( m )1 ≥ λ ≥ a (1) d · · · a ( m ) d .Proof. We notice that (cid:2) V ∗ Z pp V (cid:3) ii = (cid:2) D p/ W · · · D p/ m − W m − D pm W ∗ m − D p/ m − · · · W ∗ D p/ (cid:3) ii = X i ,...,i m − ,k,j m − ,...,j (cid:0) a (1) i (cid:1) p/ w (1) ii (cid:0) a (2) i (cid:1) p/ · · · w ( m − i m − i m − (cid:0) a ( m − i m − (cid:1) p/ × w ( m − i m − k (cid:0) a ( m ) k (cid:1) p w ( m − j m − k (cid:0) a ( m − j m − (cid:1) p/ w ( m − j m − j m − · · · (cid:0) a (2) j (cid:1) p/ w (1) ij (cid:0) a (1) i (cid:1) p/ = X k X i ,...,i m − w (1) ii w (2) i i · · · w ( m − i m − k (cid:0) a (1) i a (2) i · · · a ( m − i m − a ( m ) k (cid:1) p/ × X j ,...,j m − w (1) ij w (2) j j · · · w ( m − j m − k (cid:0) a (1) i a (2) j · · · a ( m − j m − a ( m ) k (cid:1) p/ = X k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ,...,j m − w (1) ij w (2) j j · · · w ( m − j m − k (cid:0) a (1) i a (2) j · · · a ( m − j m − a ( m ) k (cid:1) p/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . η be the right-hand side of (4.1). From the above expression we have λ ( p ) p ≤ Tr V ∗ Z pp V = X i,k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ,...,j m − w (1) ij w (2) j j · · · w ( m − j m − k (cid:0) a (1) i a (2) j · · · a ( m − j m − a ( m ) k (cid:1) p/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ M η p , where M > p . Therefore, lim sup p →∞ λ ( p ) ≤ η . Onthe other hand, let ( i, i , . . . , i m − , k ) be such that a (1) i a (2) i · · · a ( m − i m − a ( m ) k = η , and let δ := | w ( i, i , . . . , i m − , k ) | >
0. Then we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ,...,j m − w (1) ij w (2) j j · · · w ( m − j m − k (cid:0) a (1) i a (2) j · · · a ( m − j m − a ( m ) k (cid:1) p/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δη p/ − M ′ α p/ for some constants M ′ > α > α < η . Therefore, for sufficiently large p we have δη p/ − M ′ α p/ > dλ ( p ) p ≥ Tr V ∗ Z pp V ≥ (cid:0) δη p/ − M ′ α p/ (cid:1) = δ η p (cid:18) − M ′ δ (cid:18) αη (cid:19) p/ (cid:19) so that lim inf p →∞ λ ( p ) ≥ η . The latter assertion is obvious. Lemma 4.2.
For every i = 1 , . . . , d the limit λ i := lim p →∞ λ i ( p ) exists.Proof. For every k = 1 , . . . , d apply Lemma 4.1 to A ∧ k , . . . , A ∧ km to see thatlim p →∞ λ ( p ) λ ( p ) · · · λ k ( p )exists. Hence, the limit lim p →∞ λ i ( p ) exists for i = 1 , . . . , d as in the proof of Lemma2.1. Theorem 4.3.
For every d × d positive semidefinite matrices A , . . . , A m the matrix Z p = (cid:0) A p/ A p/ · · · A p/ m − A pm A p/ m − · · · A p/ A p/ (cid:1) /p converges as p → ∞ .Proof. The proof is similar to that of Theorem 2.5. Choose an orthogonal basis { u ( p ) , . . . , u d ( p ) } of C d such that Z p u i ( p ) = λ i ( p ) u i ( p ) for 1 ≤ i ≤ d . Let k (1 ≤ k < d )be such that λ ≥ · · · ≥ λ k > λ k +1 . Since (2.9) holds in the present case too, λ ( Z ∧ kp )is a simple eigenvalue of Z ∧ kp for every p sufficiently large. For I, J ∈ I d ( k ) we write w ( l ) I,J := det W ( l ) I,J for 1 ≤ l ≤ m − a ( l ) I := Q i ∈ I a ( l ) i for 1 ≤ l ≤ m . We have (cid:2) V ∗∧ k ( Z ∧ kp ) p V ∧ k (cid:3) I,J X K ∈I d ( k ) X I ,...,I m − w (1) I,I w (2) I ,I · · · w ( m − I m − ,K (cid:0) a (1) I a (2) I · · · a ( m − I m − a ( m ) K (cid:1) p/ × X J ,...,J m − w (1) J,J w (2) J ,J · · · w ( m − J m − ,K (cid:0) a (1) J a (2) J · · · a ( m − J m − a ( m ) K (cid:1) p/ = η pk X K ∈I d ( k ) X I ,...,I m − w (1) I,I w (2) I ,I · · · w ( m − I m − ,K a (1) I a (2) I · · · a ( m − I m − a ( m ) K η k ! p/ × X J ,...,J m − w (1) J,J w (2) J ,J · · · w ( m − J m − ,K a (1) J a (2) J · · · a ( m − J m − a ( m ) K η k ! p/ , where η k := λ λ · · · λ k = max (cid:8) a (1) I a (2) I · · · a ( m − I m − a ( m ) I m : w k ( I , I , . . . , I m − , I m ) = 0 (cid:9) and w k ( I , I , . . . , I m − , I m ):= Xn w (1) I J w (2) J J · · · w ( m − J m − I m : J , . . . , J m − ∈ I d ( k ) , a (2) J · · · a ( m − J m − = a (2) I · · · a ( m − I m − o . We see that V ∗∧ k (cid:18) Z ∧ kp η k (cid:19) p V ∧ k −→ Q := " X K ∈I d ( k ) v k ( I, K ) v k ( J, K ) I,J as p → ∞ , where v k ( I, K ) := w k ( I, I , . . . , I m − , K )if w k ( I, I , . . . , I m − , K ) = 0 and a (1) I a (2) I · · · a ( m − I m − a ( m ) K = η k for some I , . . . , I m − ∈I d ( k ), and otherwise v k ( I, K ) := 0. Since Q I,I ≥ | v k ( I, K ) | > I, K ∈ I d ( k ),note that Q = 0. The remaining proof is the same as in that of Theorem 2.5. ( A p B p ) /p as p → ∞ Another problem, seemingly more interesting, is to know what is shown on the con-vergence ( A p σ B p ) /p as p → ∞ , the anti-version of (1.2) (or Theorem B.1). Forexample, when σ = ▽ , the arithmetic mean, the increasing limit of ( A p ▽ B p ) /p = (cid:0) ( A p + B p ) / (cid:1) /p as p → ∞ exists and A ∨ B := lim p →∞ ( A − p ▽ B − p ) − /p = lim p →∞ ( A p + B p ) /p (5.1)is the supremum of A, B with respect to some spectral order among Hermitian matrices,see [12] and [1, Lemma 6.5]. When σ = !, the harmonic mean, we have the infimumcounterpart A ∧ B := lim p →∞ ( A p ! B p ) /p , the decreasing limit.14n this section we are interested in the case where σ = p > d × d positive semidefinite matrices A, B with the diagonalizations in(2.1) and (2.2) we define G p := ( A p B p ) /p , (5.2)which is given as (cid:0) A p/ ( A − p/ B p A − p/ ) / A p/ (cid:1) /p if A >
0. The eigenvalues of G p aredenoted as λ ( G p ) ≥ · · · ≥ λ d ( G p ) in decreasing order. Proposition 5.1.
For every i = 1 , . . . , d the limit b λ i := lim p →∞ λ i ( G p ) exists, and a b ≥ b λ ≥ · · · ≥ b λ d ≥ a d b d . Furthermore, ( a i b d +1 − i ) di =1 ≺ (log) (cid:0)b λ i (cid:1) di =1 ≺ (log) ( a i b i ) di =1 . (5.3) Proof.
Since ( a b ) p/ I ≥ A p B p ≥ ( a d b d ) p/ I , we have a b ≥ λ i ( G p ) ≥ a d b d for all i = 1 , . . . , d and p >
0. By the log-majorization result in [2, Theorem 2.1], for every k = 1 , . . . , d we have k Y i =1 λ i ( G p ) ≥ k Y i =1 λ i ( G q ) if 0 < p < q. (5.4)This implies that the limit of Q ki =1 λ i ( G p ) as p → ∞ exists for every k = 1 , . . . , d , andhence the limit λ i ( G p ) exists for i = 1 , . . . , d as in the proof of Lemma 2.1.To prove the latter assertion, it suffices to show that( a i b d +1 − i ) di =1 ≺ (log) ( λ i ( G )) di =1 ≺ (log) ( a i b i ) di =1 (5.5)for G = ( A B ) . Indeed, applying this to A p and B p we have( a i b d +1 − i ) di =1 ≺ (log) ( λ i ( G p )) di =1 ≺ (log) ( a i b i ) di =1 so that (5.3) follows by letting p → ∞ . To prove (5.5), we may by continuity assumethat A >
0. By [2, Corollary 2.3] and (3.1) we have( λ i ( G )) di =1 ≺ (log) (cid:0) λ i ( A / BA / ) (cid:1) di =1 ≺ (log) ( a i b i ) di =1 . Since G / A − G / = B , there exists a unitary matrix V such that A − / G A − / = V BV ∗ and hence G = A / V BV ∗ A / . Since λ i ( V BV ∗ ) = b i , by the majorization ofGel’fand and Naimark we have( a i b d +1 − i ) di =1 ≺ (log) ( λ i ( G )) di =1 , proving (5.5) 15n view of (2.5) and (5.4) we may consider G p as the complementary counterpart of Z p in some sense; yet it is also worth noting that G p is symmetric in A and B while Z p is not. Our ultimate goal is to prove the existence of the limit of G p in (5.2) as p → ∞ similarly to Theorem 2.5 and to clarify, similarly to Theorem 3.1, the minimalcase when (cid:0)b λ i (cid:1) di =1 is equal to the decreasing rearrangement of ( a i b d +1 − i ) di =1 . However,the problem seems much more difficult, and we can currently settle the special case of2 × Proposition 5.2.
Let A and B be × positive semidefinite matrices with the diag-onalizations (2.1) and (2.2) with d = 2 . Then G p in (5.2) converges as p → ∞ to apositive semidefinite matrix whose eigenvalues are (cid:0)b λ , b λ (cid:1) = ( ( a b , a b ) if ( V ∗ W ) = 0 , (max { a b , a b } , min { a b , a b } ) if ( V ∗ W ) = 0 . Proof.
Since G p = V (cid:0) (diag( a , a )) p V ∗ W diag( b , b ) V ∗ W ) p (cid:1) /p V ∗ , we may assume without loss of generality that V = I (then V ∗ W = W ).First, when W = 0 (hence W is diagonal), we have for every p > G p = diag( a b , a b ) . Next, when W = 0 (hence W = (cid:20) w w (cid:21) with | w | = | w | = 1), we have for every p > G p = diag( a b , a b ) . In the rest it suffices to consider the case where W = (cid:20) w w w w (cid:21) with w ij = 0 forall i, j = 1 ,
2. First, assume that det A = det B = 1 so that a a = b b = 1. Forevery p >
0, since det A p = det B p = 1, it is known [16, Proposition 3.11] (also [6,Proposition 4.1.12]) that A p B p = A p + B p p det( A p + B p )so that G p = ( A p + B p ) /p (cid:0) det( A p + B p ) (cid:1) /p . Compute A p + B p = (cid:20) a p + | w | b p + | w | b p w w b p + w w b p w w b p + w w b p a p + | w | b p + | w | b p (cid:21) (5.6)and det( A p + B p ) = 1 + | w | ( a b ) p + | w | ( a b ) p + | w | ( a b ) p + | w | ( a b ) p | w w − w w | . (5.7)Hence we havelim p →∞ (cid:0) det( A p + B p ) (cid:1) /p = a b , lim p →∞ (cid:0) Tr ( A p + B p ) (cid:1) /p = max { a , b } . Therefore, thanks to (5.1) we havelim p →∞ G p = ( A ∨ B ) a b . Since 12 Tr ( A p B p ) ≤ (cid:0) λ ( G p ) (cid:1) p/ ≤ Tr ( A p B p ) , we obtain b λ = lim p →∞ (cid:0) Tr ( A p B p ) (cid:1) /p = lim p →∞ (cid:0) Tr ( A p + B p ) (cid:1) /p (cid:0) det( A p + B p ) (cid:1) /p = max { a , b } a b = max (cid:26) a b , b a (cid:27) = max { a b , a b } . Furthermore, b λ = min { a b , a b } follows since b λ b λ = 1.For general A, B > α := √ det A and β := √ det B . Since G p = αβ (cid:0) ( α − A ) p β − B ) p (cid:1) /p , we see from the above case that G p converges as p → ∞ and b λ = αβ max { ( α − a )( β − b ) , ( α − a )( β − b ) } = max { a b , a b } , and similarly for b λ .The remaining is the case when a and/or b = 0. We may assume that a , b > A = 0 or B = 0 is trivial. When a = b = 0, since a − A and b − B are non-commuting rank one projections, we have G p = 0 for all p > a = 0 and B >
0. Then we may assume that a = 1 anddet B = 1. For ε > A ε := diag(1 , ε ). Since det( ε − A ε ) = 1, we have A pε B p = ε p/ (cid:0) ( ε − A ε ) p B p (cid:1) = ε p/ ( ε − A ε ) p + B p q det (cid:0) ( ε − A ε ) p + B p (cid:1) . By use of (5.6) and (5.7) with a = ε − and a = ε we compute A p B p = lim ε ց A pε B p = (cid:0) | w | b p + | w | b p (cid:1) − / diag(1 , p →∞ G p = diag( b − ,
0) = diag( b , , which is the desired assertion in this final situation.17 Proof of Lemma 2.4
We may assume that H = C d by fixing an orthonormal basis of H . Let G ( k, d )denote the Grassmannian manifold consisting of k -dimensional subspaces of H . Let O k,d denote the set of all u = ( u , . . . , u k ) ∈ H k such that u , . . . , u k are orthonormalin H . Consider O k,d as a metric space with the metric d ( u, v ) := k X i =1 k u i − v i k ! / , u = ( u , . . . , u k ) , v = ( v , . . . , v k ) ∈ H k . Moreover, let e H k,d be the set of projectivised vectors u = u ∧ · · · ∧ u k in H ∧ k of norm1, i.e., the quotient space of H k,d := { u ∈ H ∧ k : u = u ∧ · · · ∧ u k , k u k = 1 } under theequivalent relation u ∼ v on H k,d defined as u = e iθ v for some θ ∈ R . We then havethe commutative diagram: O k,d ✲ π G ( k, d ) e H k,d ❅❅❅❅❅❘ e π ❄ φ where π and e π are surjective maps defined for u = ( u , . . . , u k ) ∈ O k,d as π ( u ) := span { u , . . . , u k } , e π ( u ) := [ u ∧ · · · ∧ u k ] , the equivalence class of u ∧ · · · ∧ u k , and φ is the canonical representation of G ( k, d ) by the k th antisymmetric tensors (orthe k th exterior products).As shown in [7], the standard Grassmannian topology on G ( k, d ) is the final topology(the quotient topology) from the map π and it coincides with the topology induced bythe gap metric: d gap ( U , V ) := k P U − P V k for k -dimensional subspaces U , V of H and the orthogonal projections P U , P V ontothem. On the other hand, consider the quotient topology on e H k,d induced from thenorm on H k,d ⊂ H ∧ k , which is determined by the metric e d ( e π ( u ) , e π ( v )) := inf θ ∈ R k u ∧ · · · ∧ u k − e √− θ v ∧ · · · ∧ v k k , u, v ∈ O k,d . It is easy to prove that e π : ( O k,d , d ) → ( e H k,d , e d ) is continuous. Since ( O k,d , d ) iscompact, it thus follows that the final topology on e H k,d from the map e π coincides withthe e d -topology.It is clear from the above commutative diagram that the final topology on G ( k, d )from π is homeomorphic via φ to that on e H k,d from e π . Hence φ is a homeomorphism18rom ( G ( k, d ) , d gap ) onto ( e H k,d , e d ). From the homogeneity of ( G ( k, d ) , d gap ) and ( e H k,d , e d )under the unitary transformations there exist constant α, β > k, d ) such that α k P π ( u ) − P π ( v ) k ≤ e d ( e π ( u ) , e π ( v )) ≤ β k P π ( u ) − P π ( v ) k , u, v ∈ O k,d , which is the desired inequality. B Proof of (1.2)
This appendix is aimed to supply the proof of (1.2) for matrices
A, B ≥
0. Through-out the appendix let
A, B be d × d positive semidefinite matrices with the supportprojections A , B . We define log A in the generalized sense aslog A := (log A ) A , i.e., log A is defined by the usual functional calculus on the range of A and it is zero onthe range of A ⊥ = I − A , and similarly log B := (log B ) B . We write P := A ∧ B and log A ˙+ log B := P (log A ) P + P (log B ) P . Note [10, Section 4] that P exp(log A ˙+ log B ) = lim ε ց exp(log( A + εA ⊥ ) + log( B + εB ⊥ ))= lim ε ց exp(log( A + εI ) + log( B + εI )) . (B.1)Now, let σ be an operator mean with the representing operator monotone function f on (0 , ∞ ), and let α := f ′ (1). Note that 0 ≤ α ≤ α = 0 (resp., α = 1) then A σ B = A (resp., A σB = B ) so that ( A p σ B p ) /p = A (resp., ( A p σ B p ) /p = B ) forall A, B ≥ p >
0. So in the rest we assume that 0 < α < Theorem B.1.
With the above assumptions, for every
A, B ≥ , lim p ց ( A p σ B p ) /p = P exp((1 − α ) log A ˙+ α log B ) . (B.2)From (B.1) we may writelim p ց ( A p σ B p ) /p = lim ε ց exp((1 − α ) log( A + εI ) + α log( B + εI ))= lim ε ց lim p ց (( A + εI ) p σ ( B + εI ) p ) /p . The next lemma is essential to prove the theorem. The proof of the lemma is aslight modification of that of [10, Lemma 4.1].19 emma B.2.
For each p ∈ (0 , p ) with some p > , a Hermitian matrix Z ( p ) is givenin the × block form as Z ( p ) = (cid:20) Z ( p ) Z ( p ) Z ∗ ( p ) Z ( p ) (cid:21) , where Z ( p ) is k × k , Z ( p ) is l × l and Z ( p ) is k × l . Assume:(a) Z ( p ) → Z and Z ( p ) → Z as p ց ,(b) there is a δ > such that pZ ( p ) ≤ − δI l for all p ∈ (0 , p ) .Then e Z ( p ) −→ (cid:20) e Z
00 0 (cid:21) as p ց . Proof.
We list the eigenvalues of Z ( p ) in decreasing order (with multiplicities) as λ ( p ) ≥ · · · ≥ λ k ( p ) ≥ λ k +1 ( p ) ≥ · · · ≥ λ m ( p )together with the corresponding orthonormal eigenvectors u ( p ) , . . . , u k ( p ) , u k +1 ( p ) , . . . , u m ( p ) , where m := k + l . Then e Z ( p ) = m X i =1 e λ i ( p ) u i ( p ) u i ( p ) ∗ . (B.3)Furthermore, let µ ( p ) ≥ · · · ≥ µ k ( p ) be the eigenvalues of Z ( p ) and µ ≥ · · · ≥ µ k be the eigenvalues of Z Then µ i ( p ) → µ i as p ց r X i =1 µ i ( p ) ≤ r X i =1 λ i ( p ) , ≤ r ≤ k. (B.4)Since pZ ( p ) ≤ (cid:20) pZ ( p ) pZ ( p ) pZ ∗ ( p ) − δI l (cid:21) −→ (cid:20) − δI l (cid:21) as p ց k < i ≤ m , pλ i ( p ) < − δ/ p > p ց λ i ( p ) = −∞ , k < i ≤ m. (B.5)Hence, it suffices to prove that for any sequence ( p > ) p n ց { p ′ n } of { p n } such that we have for 1 ≤ i ≤ kλ i ( p ′ n ) −→ µ i as n → ∞ , (B.6) u i ( p ′ n ) −→ v i ⊕ ∈ C k ⊕ C l as n → ∞ , (B.7)20 v i = µ i v i . (B.8)Indeed, it then follows that v , . . . , v k are orthonormal vectors in C k , so from (B.3) and(B.5) we obtain lim n →∞ e Z ( p ′ n ) = k X i =1 e µ i v i v ∗ i ⊕ e Z ⊕ . Now, replacing { p n } with a subsequence, we may assume that u i ( p n ) itself convergesto some u i ∈ C m for 1 ≤ i ≤ k . Writing u i ( p n ) = v ( n ) i ⊕ w ( n ) i in C k ⊕ C l , we have λ ( p n ) = (cid:10) v ( n ) i ⊕ w ( n ) i , Z ( p n )( v ( n ) i ⊕ w ( n ) i ) (cid:11) = (cid:10) v ( n ) i , Z ( p ) v ( n ) i (cid:11) + 2Re (cid:10) v ( n ) i , Z ( p n ) w ( n ) i (cid:11) + (cid:10) w ( n ) i , Z ( p n ) w ( n ) i (cid:11) ≤ (cid:10) v ( n ) i , Z ( p n ) v ( n ) i (cid:11) + 2Re (cid:10) v ( n ) i , Z ( p n ) w ( n ) i (cid:11) − δp n (cid:13)(cid:13) w ( n ) i (cid:13)(cid:13) (B.9)due to assumption (b). For i = 1, since µ ( p n ) ≤ λ ( p n ) by (B.4) for r = 1, it followsfrom (B.9) that p n µ ( p n ) ≤ p n k Z ( p n ) k + 2 p n k Z ( p n ) k − δ (cid:13)(cid:13) w ( n )1 (cid:13)(cid:13) , where k Z ( p n ) k and k Z ( p n ) k are the operator norms. As n → ∞ ( p n ց w ( n )1 → u ( p n ) → u = v ⊕ C k ⊕ C l . From (B.9)again we furthermore havelim sup n →∞ λ ( p n ) ≤ h v , Z v i ≤ µ ≤ lim inf n →∞ λ ( p n )since µ ( p n ) ≤ λ ( p n ) and µ ( p n ) → µ . Therefore, λ ( p n ) → h v , Z v i = µ andhence Z v = µ v . Next, when k ≥ i = 2, since λ ( p n ) is bounded below by(B.4) for r = 2, it follows as above that w ( n )2 → u ( p n ) → u = v ⊕ n →∞ λ ( p n ) ≤ h v , Z v i ≤ µ ≤ lim inf n →∞ λ ( p n )so that λ ( p n ) → h v Z v i = µ and Z v = µ v , since µ is the largest eigenvalueof Z restricted to { v } ⊥ ∩ C k . Repeating this argument we obtain (B.6)–(B.8) for1 ≤ i ≤ k .Note that the lemma and its proof hold true even when the assumption Z ( p ) → Z in (b) is slightly relaxed into p / Z ( p ) → p ց
0. (For this, from (B.9) note that p − / n w ( n ) i → Z ( p n ) w ( n ) i → Proof of Theorem B.1.
Let us divide the proof into two steps. In the proof below wedenote by ▽ α and ! α the weighted arithmetic and harmonic operator means having therepresenting functions (1 − α ) + αx and x/ ((1 − α ) x + α ), respectively. Note that A ! α B ≤ A σ B ≤ A ▽ α B, A, B ≥ . tep 1. First, we prove the theorem in the case where
P σ Q = P ∧ Q for all orthogonalprojections P, Q (this is the case, for instance, when σ is the weighted harmonic opera-tor mean ! α , see [13, Theorem 3.7]). Let H be the range of P (= A ! α B = A σ B ).From the operator monotonicity of log x ( x >
0) it follows that, for every p > p log( A p ! α B p ) (cid:12)(cid:12) H ≤ p log( A p σ B p ) (cid:12)(cid:12) H ≤ p log (cid:0) P ( A p ▽ α B p ) P (cid:1)(cid:12)(cid:12) H . (B.10)For every ε > A + εA ⊥ ) p ! α ( B + εB ⊥ ) p = (cid:0) ( A + εA ⊥ ) − p ▽ α ( B + εB ⊥ ) − p (cid:1) − = (cid:0) A − p ▽ α B − p + ε − p ( A ⊥ ▽ α B ⊥ ) (cid:1) − , where A − p = ( A − ) p and B − p = ( B − ) p are taken as the generalized inverses. There-fore, P (cid:0) ( A + εA ⊥ ) p ! α ( B + εB ⊥ ) p (cid:1) P ≥ (cid:0) P (cid:0) A − p ▽ α B − p + ε − p ( A ⊥ ▽ α B ⊥ ) (cid:1) P (cid:1) − = (cid:0) P ( A − p ▽ α B − p ) P (cid:1) − , (B.11)since the support projection of A ⊥ + B ⊥ is A ⊥ ∨ B ⊥ = P ⊥ . In the above, ( − ) − isthe generalized inverse (with support H ) and the inequality follows from the operatorconvexity of x − ( x > ε ց A p ! α B p = P ( A p ! α B p ) P ≥ (cid:0) P ( A − p ▽ α B − p ) P (cid:1) − so that 1 p log( A p ! α B p ) (cid:12)(cid:12) H ≥ − p log (cid:0) P ( A − p ▽ α B − p ) P (cid:1)(cid:12)(cid:12) H . (B.12)Combining (B.10) and (B.12) yields − p log (cid:0) P ( A − p ▽ α B − p ) P (cid:1)(cid:12)(cid:12) H ≤ p log( A p σ B p ) (cid:12)(cid:12) H ≤ p log (cid:0) P ( A p ▽ α B p ) P (cid:1)(cid:12)(cid:12) H . (B.13)Since A − p = A − p log A + o ( p ) , B − p = B − p log B + o ( p )as p ց
0, we have A − p ▽ α B − p = A ▽ α B − p ((log A ) ▽ α (log B )) + o ( p )so that P ( A − p ▽ α B − p ) P = P − p ((1 − α ) log A ˙+ α log B ) + o ( p ) . Therefore, − p log (cid:0) P ( A − p ▽ B − p ) P (cid:1)(cid:12)(cid:12) H = ((1 − α ) log A ˙+ α log B ) (cid:12)(cid:12) H + o (1) . (B.14)22imilarly,1 p log (cid:0) P ( A p ▽ B p ) P (cid:1)(cid:12)(cid:12) H = ((1 − α ) log A ˙+ α log B ) (cid:12)(cid:12) H + o (1) . (B.15)From (B.13)–(B.15) we obtainlim p ց p log( A p σ B p ) (cid:12)(cid:12) H = ((1 − α ) log A ˙+ α log B ) (cid:12)(cid:12) H , which yields the required limit formula. Step 2.
For a general operator mean σ the integral representation theorem [13, Theorem4.4] says that there are 0 ≤ θ ≤
1, 0 ≤ β ≤ τ such that σ = θ ▽ β + (1 − θ ) τ and P τ Q = P ∧ Q for all orthogonal projections P, Q . Moreover, τ has the representingoperator monotone function g on (0 , ∞ ) for which γ := g ′ (1) ∈ (0 ,
1) and α = θβ + (1 − θ ) γ. We may assume that 0 < θ ≤ θ = 0 was shown in Step 1. Moreover,when θ = 1, we have β = α ∈ (0 , < θ ≤ < β <
1. Let
A, B ≥ A σ B = θA ▽ β B +(1 − θ )( A ∧ B )has the support projection A ∨ B . Let H , H and H denote the ranges of A ∨ B , P = A ∧ B and A ∨ B − P , respectively, so that H = H ⊕ H . Note that thesupport of A p σ B p for any p > H . We will describe p log( A p σ B p ) (cid:12)(cid:12) H in the 2 × H = H ⊕ H . Let Z := ((1 − γ ) log A ˙+ γ log B ) | H . It follows from Step 1 that lim p ց ( A p τ B p ) /p = P e Z P and hence A p τ B p = P (cid:0) e Z + o (1) (cid:1) p P = P (cid:0) I H + p log (cid:0) e Z + o (1) (cid:1) + o ( p ) (cid:1) P = P (cid:0) I H + pZ + o ( p ) (cid:1) P = P + p ((1 − γ ) log A ˙+ γ log B ) + o ( p ) . In the above, the third equality follows since log (cid:0) e Z + o (1) (cid:1) = Z + o (1). On the otherhand, we have A p ▽ β B p = A ▽ β B + p ((log A ) ▽ β (log B )) + o ( p ) . Therefore, we have A p σ B p = θ ( A ▽ β B ) + (1 − θ ) P pθ ((log A ) ▽ β (log B )) + p (1 − θ )((1 − γ ) log A ˙+ γ log B ) + o ( p ) . Setting C := (cid:0) θ ( A ▽ β B ) + (1 − θ ) P (cid:1)(cid:12)(cid:12) H ,H := (cid:0) θ ((log A ) ▽ β (log B )) + (1 − θ )((1 − γ ) log A ˙+ γ log B ) (cid:1)(cid:12)(cid:12) H , we write 1 p log( A p σ B p ) (cid:12)(cid:12) H = 1 p log( C + pH + o ( p )) , (B.16)which C is a positive definite contraction on H and H is a Hermitian operator on H . Note that the eigenspace of C for the eigenvalue 1 is H . Hence, with a basisconsisting of orthonormal eigenvectors for C we may assume that C is diagonal so that C = diag( c , . . . , c m ) with c = · · · = c k = 1 > c k +1 ≥ · · · ≥ c m > m = dim H and k = dim H .Applying the Taylor formula (see, e.g., [9, Theorem 2.3.1] to log( C + pH + o ( p )) wehave log( C + pH + o ( p )) = log C + pD log( C )( H ) + o ( p ) , (B.17)where D log( C ) denotes the Fr´echet derivative of the matrix functional calculus bylog x at C . The Daleckii and Krein’s derivative formula (see, e.g., [9, Theorem 2.3.1])says that D log( C )( H ) = " log c i − log c j c i − c j mi,j =1 ◦ H, (B.18)where ◦ denotes the Schur (or Hadamard) product and (log c i − log c j ) / ( c i − c j ) isunderstood as 1 /c i when c i = c j . We write D log( C )( H ) in the 2 × H ⊕ H as (cid:20) Z Z Z ∗ Z (cid:21) where Z := P HP | H . By (B.16)–(B.18) we can write1 p log( A p σ B p ) = 1 p log C + D log( C )( H ) + o (1) = (cid:20) Z ( p ) Z ( p ) Z ∗ ( p ) Z ( p ) (cid:21) , where Z ( p ) = Z + o (1) , Z ( p ) = Z + o (1) ,Z ( p ) = 1 p diag(log c k +1 , . . . , log c m ) + Z + o (1) . This 2 × Z ( p ) := p log( A p σ B p ) (cid:12)(cid:12) H satisfies assumptions (a) and (b)of Lemma B.2 for p ∈ (0 , p ) with a sufficiently small p >
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