Schur product of matrices and numerical radius (range) preserving maps. Dedicated to Professor Roger Horn on the occasion of his sixty fifth birthday.

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1 Schu poduct of matices and numeical adius (ange) peseving maps Chi-Kwong Li 1 and Edwad Poon 2 Dedicated to Pofesso Roge Hon on the occasion of his sixty fifth bithday. Abstact Let F (A) be the numeical ange o the numeical adius of a squae matix A. Denote by A B the Schu poduct of two matices A and B. Chaacteizations ae given fo mappings on squae matices satisfying F (A B) = F (φ(a) φ(b)) fo all matices A and B. Analogous esults ae obtained fo mappings on Hemitian matices Mathematics Subject Classification. 15A04, 15A18, 15A60 Key wods and phases. Numeical ange, numeical adius, Schu poduct. 1 Intoduction Let M n be the algeba of n n complex matices. Denote the numeical ange and numeical adius of A M n by W (A) = {x Ax : x C n, x x = 1} and (A) = max{ µ : µ W (A)}. Thee has been consideable inteest in studying the stuctue of maps peseving the numeical ange o adius. Suppose U M n is a unitay matix. Define the map φ on M n by A U AU o A U A t U. (1.1) Then φ is a C -isomophism on the C -algeba M n, and a Jodan isomophism on the Jodan algeba H n of Hemitian n n matices. Evidently, φ is bijective linea and peseves the numeical ange, i.e., W (φ(a)) = W (A) fo all A. Pellegini [14] (see also [11]) obtained an inteesting esult on numeical ange peseving maps on a geneal C -algeba, which implies that a linea map φ : M n M n peseving the numeical ange must be of this fom. One easily deduces that the conclusion is also valid fo linea maps φ defined on H n. In [4], it was shown that a multiplicative map φ : M n M n satisfies W (φ(a)) = W (A) fo all A if and only if φ has the fom A U AU fo some U M n. In [7], the authos eplaced the condition that φ is multiplicative and peseves the numeical ange on the sujective map φ : M n M n by the condition that W (AB) = W (φ(a)φ(b)) fo all A, B, and showed that such a map has the fom A ±U AU fo some unitay opeato U M n. They also showed that a sujective map φ : M n M n satisfies W (ABA) = W (φ(a)φ(b)φ(a)) fo all A, B M n if and only if φ has the fom A µu AU o A µu A t U fo some unitay opeato U M n and µ C with µ 3 = 1. Simila esults fo 1 Depatment of Mathematics, College of William & May, Williamsbug, VA ckli@math.wm.edu. Li is an honoay pofesso of the Heilongjiang Univesity and an honoay pofesso of the Univesity of Hong Kong. His eseach was suppoted by a USA NSF gant and a HK RCG gant. 2 Depatment of Mathematics, Emby-Riddle Aeonautical Univesity, Pescott, AZ edwad.poon@eau.edu. 1

2 mappings on H n wee also obtained. It is inteesting to note that all the esults mentioned above show that unde athe mild assumptions, a numeical ange peseving map φ on V = M n o H n must be a multiple of the standad map (1.1). Thee is also inteest in studying numeical adius peseving maps on matices o opeatos. In [9] (see also [2]), it is shown that linea peseves of the numeical adius on V = M n o H n have the fom A µu AU o A µu A t U fo some unitay U and scala µ with µ = 1. By the esult in [4], if φ : V V is a multiplicative peseve of the numeical adius, then φ has the fom A µu AU o A µu AU fo some unitay U M n and unit scala µ. By the esult in [1], if φ : V V satisfies (φ(a) φ(b)) = (A B) fo all A, B V, then φ has the fom A µu A τ U + R fo some unit scala µ, R V, and unitay U M n, whee A τ denotes A, A t, A, o A. In this pape, we conside the Schu poduct (also known as the Hadamad poduct) of matices defined by (a ij ) (b ij ) = (a ij b ij ), which is quite diffeent fom the othe types of binay poducts on V. One easily sees that mappings φ in the fom (1.1) will not always satisfy W (A B) = W (φ(a) φ(b)) fo all A, B V (1.2) unless the matix in (1.1) is caefully chosen, say, U is a pemutation matix. On the othe hand, if a pemutation matix P is given, and a diagonal unitay matix D A is assigned to each A V, then a mapping φ of the fom A D AP t AP D A o A D AP t A t P D A will satisfy (1.2). A moe obscue opeation is to choose a matix R V so that R R = ( x i x j ) with x 1 = = x n = 1 and define the map φ by A R A. Then φ(a) φ(b) = R R (A B) = D x(a B)D x, whee D x is the diagonal matix with diagonal enties x 1,..., x n, and hence φ satisfies (1.2). It tuns out that the composition of the maps descibed above will be the totality of maps satisfying (1.2); see Theoem 1.2. Of couse, a mapping φ satisfying (1.2) will also satisfy (A B) = (φ(a) φ(b)) fo all A, B V. (1.3) But thee may be moe admissible maps. Fo example, the mappings A A and A A also satisfy (1.3). Also, if a unit scala µ A is assigned to each A V, then the mapping A µ A A also satisfies (1.3). Moe geneally, wheneve A is pemutationally simila to a diect sum of squae 2

3 matices of smalle sizes, say, A 1 A k, one can take a pai of diagonal unitay matices D A, E A so that D A E A V and D A E A A = AD A E A (equivalently, D A AE A is pemutationally simila to µ 1 A 1 µ k A k fo some unit scalas µ 1,..., µ k ) and define φ(a) = D A AE A. Since A B will be pemutationally simila to a matix of the fom (A 1 B 11 ) (A k B kk ), and (X 1 X k ) = max{(x 1 ),..., (X k )}, we see that mappings constucted as above also satisfy (1.3). We will show that these ae the only additional maps needed to geneate (by compositions) all of the maps satisfying (1.3). Specifically, we have the following theoems (whee n 2 to avoid tivialities). Theoem 1.1. Let V = M n o H n, and let φ : V V. Then (A B) = (φ(a) φ(b)) fo all A, B V if and only if thee is a fixed pemutation matix P, a matix R V such that R R = ( x i x j ) with x 1 = = x n = 1, and a mapping A (D A, E A ) assigning each A V to a pai of diagonal unitay matices D A, E A satisfying D A E A V and D A E A A = AD A E A such that φ has the fom X R (P t D X X τ E X P ) fo all X V, whee X τ denotes X, X, X t, o X. (Of couse, X = X and X = X t if V = H n.) We note again that the condition on D A and E A simply means that if Q is a pemutation matix such that Q t AQ = A 1 A m, then D A and E A ae chosen such that Q t D A E A Q = λ 1 I λ m I accodingly. Theoem 1.2. Let V = M n o H n, and let φ : M n M n. Then W (A B) = W (φ(a) φ(b)) fo all A, B V if and only if thee is a fixed pemutation matix P, a matix R V such that R R = ( x i x j ) with x 1 = = x n = 1, and a mapping A D A fom V to the goup of diagonal unitay matices such that φ has the fom X R (P t D XXD X P ) o X R (P t D XX t D X P ). The sufficiencies of the theoems ae clea by ou discussion befoe the statements. We will pove the necessities in the next two sections. In ou discussion, v denotes the vecto obtained fom v C n by eplacing each enty by its absolute value; A has a simila meaning fo A M n. A vecto o a matix is said to be unimodula if all enties have moduli one. The matix in M n whose evey enty is one is denoted by J. We say that a vecto o a matix has suppot at cetain enties if all othe enties of the vecto o matix equal zeo. A matix A is decomposable if it is pemutationally simila to a diect sum of squae matices of smalle sizes; othewise, A is indecomposable. The Schu-invese of A is denoted by A ( 1), and is defined by (A ( 1) ) ij = A 1 ij if A ij 0, and (A ( 1) ) ij = 0 if A ij = 0. Denote by {E 11, E 12,..., E nn } the standad basis fo M n. 3

4 2 Poofs fo complex matices 2.1 Auxiliay esults Lemma 2.1. Suppose S M n has n 2 nonzeo elements such that X Y = 0 fo any X Y S. Then thee ae nonzeo scalas µ ij C such that S = {µ ij E ij : 1 i, j n}. Lemma 2.2. Let A be a nonnegative matix such that A+A t is ieducible. Let U be a unimodula matix (i.e., U ij = 1 fo all i, j). If (A) = (A U) then thee exist some unit scala µ and unimodula vecto w such that A U = A (µww ). Poof. Let x R n be the unique positive unit eigenvecto of (A + A t )/2, so (A) = x t Ax. Wite à = A U. Let v C n be a unit vecto such that (Ã) = v Ãv. Let D be a diagonal unitay such that D v = v. Then (Ã) = v t D ÃD v v t D ÃD v x t Ax = (A), so all the inequalities ae in fact equalities. Fo the second inequality to be equality implies that v = x has stictly positive enties. Fo the fist inequality to be equality implies that thee exists µ C with µ = 1 such that D ÃD = µa. If D = diag (w), then à = A (µww ) as desied. Lemma 2.3. Let w, z be complex numbes of modulus one. Then () () = w = z o w = z. 0 w 0 z Poof. Let v = [ x f(θ) = max { () ([ ]) ] t e iθ/2 1 y and f(θ) = 0 e iθ = 0 e iθ/2. Then v [ ] } e iθ/2 1 0 e iθ/2 v : v = 1 = max{ cos(θ/2) + xy + i( y 2 x 2 ) sin(θ/2) : x 2 + y 2 = 1} = max{ cos(θ/2) + i(y 2 x 2 ) sin(θ/2) + xy : x 2 + y 2 = 1; x, y 0}. Let y = cos(α/2), x = sin(α/2), 0 α π. We get Let t = sin α. We get f(θ) = max{ cos(θ/2) + i(cos α) sin(θ/2) + (1/2) sin α : α [0, π]}. { } f(θ) = max 1 sin 2 (θ/2)t 2 + t/2 : 0 t 1. Since 0 e iθ is not nomal, its numeical ange contains the eigenvalue 1 in its inteio, so f(θ) > 1. Thus the maximum is attained at some t > 0, and hence f is stictly inceasing on [ π, 0]. Since f( θ) = f(θ), the esult follows. 4

5 Lemma 2.4. Let w, z be complex numbes of modulus one. Then () () = w = z o w = z. w 0 z 0 () () 1 e iθ/2 Poof. Let f(θ) = e iθ = 0 e iθ/2 and v = 0 f(θ) = max{ v 1 e iθ/2 e iθ/2 v : v = 1} 0 = max{ x 2 + 2Re(xy)e iθ/2 : v = 1} = max{ x 2 + 2xye iθ/2 : x 2 + y 2 = 1; x, y 0}. [ x y] t. Then fo θ ( π, 0), Since e iθ has unimodula deteminant, one eigenvalue has modulus at least one. 0 As this matix is not nomal (except when e iθ is eal, which we ve excluded), this eigenvalue lies in the inteio of the numeical ange, so f(θ) > 1. Thus the maximum does not occu when x = 0 o y = 0, and hence f is stictly inceasing on [ π, 0]. Since f( θ) = f(θ), the esult follows. Lemma 2.5. Let w, z be complex numbes of modulus one. Then w = 0 0 z w = z o w = z. 1 Poof. Let A ψ = 0 0 e iψ, f(ψ) = (A ψ ), and Then S = {v = [ e iθ c] t : a, b 0, θ [ π, π], a 2 + b 2 + c 2 = 1}. f(ψ) = max v S v A ψ v = max v S a2 + abe iθ + c(a + be i(ψ θ) ) = max v S a2 + abe iθ + c a + be i(ψ θ). Fo ψ ( π, 0), the maximum is attained at some θ 0 [ π, 0] and fo some a 0, b 0, c 0 0. (If b 0 = 0 o c 0 = 0, then f(ψ) = max{a 2 + ab : a 2 + b 2 = 1} = (1 + 2)/2. But λ max ((A ψ + A ψ)/2) is the lagest oot of p(λ) = λ 3 λ 2 3λ/4 + (1 cos ψ)/4; since p((1 + 2)/2) < 0 and p(2) > 0, f(ψ) λ max > (1 + 2)/2, giving a contadiction.) Thus fo ɛ sufficiently small, f(ψ) = a a 0 b 0 e iθ 0 + c 0 a 0 + b 0 e i(ψ θ 0) < a a 0 b 0 e i(θ 0+ɛ) + c 0 a 0 + b 0 e i((ψ+ɛ) (θ 0+ɛ)) f(ψ + ɛ) 5

6 if θ 0 < 0 and f(ψ) = a a 0 b 0 + c 0 a 0 + b 0 e iψ < a a 0 b 0 + c 0 a 0 + b 0 e i(ψ+ɛ) f(ψ + ɛ) if θ 0 = 0. Thus, f is stictly inceasing on [ π, 0]. Since f( ψ) = f(ψ), the esult follows. 2.2 Poof of Theoem 1.1 Assume that (A B) = (φ(a) φ(b)) fo all A, B M n. By Lemma 2.1, thee ae nonzeo µ ij C such that {φ(e ij ) : 1 i, j n} = {µ ij E ij : 1 i, j n}. Step 1. Thee exists a pemutation matix P such that the mapping X P t φ(x)p will map E jj to µ jj E jj with µ jj = 1 fo all j = 1,..., n. Suppose, by way of contadiction, φ(e 11 ) = µ 1 E s fo some s. Then, since 1 = (E 11 E 11 ) = (µ 1 E s µ 1 E s ), µ 1 = 2. Similaly, if φ(e 12 ) = µ 2 E pq then µ 2 = 1 if p q and µ 2 = 1/ 2 if p = q. Let X = E 11 + E 12. Since (X E ij ) = (φ(x) φ(e ij )) fo all i, j, we see that φ(x) = ξ 1 E s + ξ 2 E pq, whee ξ 1 = 2 and But then we obtain the contadiction ξ 2 = { 1/ 2 if p = q, 1 if p q = (X X) = (ξ 2 1E s + ξ 2 2E pq ) = { o 1 if p = q 1, 2, o if p q. 5 2 So, φ(e 11 ) = µ jj E jj fo some j. Since 1 = (E 11 E 11 ) = (µ 2 jj E jj), we see that µ jj = 1. A simila conclusion holds fo φ(e kk ) fo all k, and ou assetion follows. Step 2. Without loss of geneality, eplace φ by the mapping X P t φ(x)p, so φ(e jj ) = µ jj E jj fo all j. Moeove, fo p q, φ(e pq ) = µ pq E s fo some s and, since 1 = (E pq E pq ) = (µ 2 pqe s ), we have µ pq = 1. We show that E s = E pq o E qp. Let X = E pp +E pq. Since (X E ij ) = (φ(x) φ(e ij ) fo all i, j, we have φ(x) = ξ 1 E pp +ξ 2 E s whee ξ 1 = ξ 2 = 1. Since = (X X) = (ξ 2 1E pp + ξ 2 2E s ), E s must lie in the pth ow o pth column. Simila consideation of Y = E qq + E pq implies E s lies in the qth ow o qth column, so φ(e pq ) = µ pq E pq o µ pq E qp with µ pq = 1 as desied. Step 3. We show that φ(j) = J R whee R R = µ( x i x j ) with µ = x 1 = = x n = 1. Since {φ(e ij ) : 1 i, j n} = {µ ij E ij : 1 i, j n} fo some complex units µ ij, and (E ij ) = (J E ij ) = (φ(j) φ(e ij )), we see that φ(j) = J R fo some unimodula R. Now, 6

7 n = (J J) = (φ(j) φ(j)) = (J R R). Applying Lemma 2.2 with A = J and U = R R, we get the desied conclusion. Step 4. Intelude of fou items. Replace φ by the mapping X R ( 1) φ(x), whee R ( 1) is the Schu invese of R having the (i, j)th enty equal to ij 1. We may then assume that φ(j) = J, and (φ(a)) = (φ(a) J) = (A J) = (A) fo all A M n. By using a ij = (A E ij ) = (φ(a) φ(e ij )), we have φ(a) = A σ U fo some unimodula U. Hee A σ is the matix obtained by pefoming some (pehaps none) local tanspositions (swapping the a pq and a qp enties fo some p q). Define an equivalence elation A B if B = e iθ D AD fo some diagonal unitay D and eal numbe θ. Note that A B if and only if B = A (e iθ ww ) fo some eal θ and unimodula vecto w. Some simple popeties of this elation ae: 1. If A B, then A B and W (A) = W (B). 2. If A 1 B 1 and A 2 B 2, then A 1 A 2 B 1 B 2. Suppose B has positive enties in a pincipal 2 2 submatix, and zeo enties eveywhee else. Then (B t ) = (B) = (φ(b)) = (B U) o (B t U) fo some unimodula U. By Lemma 2.2, φ(b) B o B t, depending on what φ does to the off-diagonal element. We shall epeatedly use this fact. Step 5. We chaacteize the action of φ on all A suppoted on a paticula 2 2 pincipal submatix. To simplify notation, we let n = 2. If φ(e 12 ) = µe 21, eplace φ with X φ(x) t fo this step. If A has two o moe zeo enties, then φ(a) A A, so suppose A has at most one zeo enty. We conside thee cases. a b Case i: A = fo some a, b, d 0. We can wite A 0 d 0 d e iδ and φ(a) [ ] a b 1/ a 1/ b 0 d e iθ whee δ, θ [ π, π]. Let B =, so φ(b) B. Then / d () 0 e iδ = (A B) = (φ(a) φ(b)) = () 0 e iθ. By Lemma 2.3, θ = ±δ, whence φ(a) A o A. To show that eithe φ(a) A fo all A = o φ(a) A fo all A =, let 0 d 0 d C =. By eplacing φ with φ if necessay, we may assume φ(c) C. Suppose B = 0 i c d whee a, b, c, d > 0. Eithe φ(b C) B C o B C. In the latte case, () = (C (C B)) = (φ(c) φ(c B)) 0 d () = (C C B) =, 0 d 7

8 contadicting Lemma 2.2. Hence φ(b C) B C fo any positive matix B. Define A, B as at the outset of this Case (i), and suppose φ(a) A. Then () 0 ie iδ = (A C B) = (φ(a) φ(c B)) () = (A C B) = 0 ie iδ, whence, by Lemma 2.3, δ = 0. Thus A A, so in fact φ(a) A fo all A of the fom, as 0 d desied. a b Case ii: A = fo some a, b, c 0. We can wite A c 0 c e iγ and φ(a) 0 a b 1/ a 1/ b c e iθ whee γ, θ [ π, π]. Let B =, so φ(b) B. Then 0 1/ c 1 () () e iγ = (A B) = (φ(a) φ(b)) = 0 e iθ. 0 By Lemma 2.4, θ = ±γ, whence φ(a) A o A. [ Using ] an agument simila to that in Case i, we may conclude that eithe φ(a) A fo all A =, o φ(a) A fo all such A. c 0 We now show that, assuming φ(a) A fo all A in Case i, φ(a) A[ fo all ] A in [ Case ] ii. By way of contadiction, suppose φ(a) A fo all A in Case ii. Let X =, Y =, i i 0 i Z =. We wite φ(x) i 0 ie iγ ie iδ fo some γ, δ ( π, π]. Then () () = (X Y ) = (φ(x) φ(y )) = (φ(x) Y ) = e iδ whence δ = 0 by Lemma 2.3. Since φ(z) Z, () () = (X Z) = (φ(x) φ(z)) = (φ(x) Z) = 1 0 e iγ 0 whence γ = π by Lemma 2.4. But then ]) () ([ 2 < = (X) = (φ(x)) = = 2 i i i i gives the desied contadiction. (Note 1 + i is an eigenvalue of X and X is not nomal, so 1 + i lies in the inteio of W (X) and (X) > 2. Meanwhile φ(x) is equivalent to 2 times a unitay matix.) It follows that φ(a) A fo all A =, as desied. c 0 8

9 Case iii: We still suppose φ(a) A fo all A in Case i. If A has a zeo in the fist ow, we may use aguments simila to those[ in the ] fist two [ cases to ] conclude [ that φ(a) ] A. Now 1/a 1/b 1/a 1/b suppose A has no zeo enties. Let A =, Y =, and Z =. We wite c d 0 1/d 1/c 0 φ(a) whee γ, δ [ π, π]. Then, since φ(y ) Y, ce iγ de iδ () = (A Y ) = (φ(a) φ(y )) = 0 1 () 0 e iδ whence δ = 0. Simila consideation of (A Z) eveals γ = 0, and thus φ(a) A. Step 6. We have shown that fo all A suppoted on a given 2 2 pincipal submatix, we have φ(a) A τ whee A τ is one of: A, A, A t, o A. We shall show that φ has the same type of behavio on all 2 2 pincipal submatices. Case a: Conjugation. Suppose now that φ(a) A fo A suppoted on the (p, q)-submatix, and φ(a) A fo A suppoted on the (, s)-submatix with = p o q. We show that this gives a contadiction. Without loss of geneality, we take p = 1, q = = 2, s = 3, and wite all matices as 3 3. w 1 0 w 1 0 Let w = exp(iπ/4), A = 1 w 1 and wite φ(a) a wb 1 whee a, b, c, d ae 0 1 w 0 c wd complex numbes of modulus one. Using Lemmas 2.3, 2.4, and (A B) = (φ(a) φ(b)) = (φ(a) B) w 1 0 w 1 0 fo B = 0 w 0 o 1 0 0, we have a = b = 1. Using Lemmas 2.3, 2.4, and (A B) = (φ(a) φ(b)) = (φ(a) B) fo B = 0 w 1 o 0 w 1, we have d = 1 and c = w 4 = w Since A is a nomal matix with eigenvalues w and w ± 2, we have (A) = w + 2 = 5. On the othe hand, φ(a) wi = UNU whee 1/ 2 0 1/ U = / 2 0 1/ and N = Since W (N) is the unit disk, (φ(a)) = 2 (A), giving the desied contadiction. The same agument shows that we cannot have φ(a) A t fo A suppoted on the (p, q)- submatix, and φ(a) A fo A suppoted on the (, s)-submatix with = p o q. 9

10 Case b: Tansposition. Let p < q <. We show that eithe φ(e ij ) = E ij fo all i < j in {p, q, }, o φ(e ij ) = E ji fo all i < j in {p, q, }. Suppose, by way of contadiction, this is not tue. Without loss of geneality, we take p = 1, q = 2, = 3; wite all matices as 3 3; and assume that φ(e 12 ) = E 12, φ(e 13 ) = E 13, and φ(e 23 ) = E 32. By Case a, and by eplacing φ with φ if needed, we have φ(a) A fo all A suppoted on the (1, 2)- o (1, 3)-submatix. Meanwhile φ(a) A t o A fo all A suppoted on the (2, 3)-submatix. 1 1 Let w = e 2πi/3, A = w w 1 and wite φ(a) aw bw cw fo some unit scalas 1 w w d e fw a, b, c, d, e, f. By using Lemmas 2.3, 2.4, and (A B) = (φ(a) φ(b)) = (φ(a) B) fo B = 0 w 0, w 0 0,,, o 0, we have a = b = d = f = 0 0 w , and c = e oew. 1 Let A ψ = 0 0 e iψ 1 and B ψ =. A diect computation shows that (A ψ ) = 0 e iψ 0 (B ψ ). If we wite φ(a 0 ) B θ, Lemma 2.5 and (φ(a 0 )) = (A 0 ) imply φ(a 0 ) B 0. Then Lemma 2.5 and (A A 0 ) = (φ(a) B 0 ) imply e = 1, so c = 1 o w. Hence φ(a) w w w o w w 1. w w But in the fist case, (φ(a)) 1.65 < 3 and in the second, (φ(a)) 1.87 > 3. Since (A) = 3, we have the desied contadiction. Case c: Combining pevious cases. By eplacing φ with φ t if necessay, we may assume φ(e 12 ) = E 12. Applying Case b with p = 1 and q = 2, we have φ(e 1 ) = E 1 fo all. Applying Case b with p = 1, we have φ(e q ) = E q fo all q <. It follows that given any 2 2 pincipal submatix, eithe φ(a) A o φ(a) A fo all A suppoted on said submatix. By eplacing φ with φ if necessay, we may assume φ(a) A fo A suppoted on the (1, 2)- submatix. By Case a, it follows that φ(a) A fo A suppoted on the (1, p)-submatix fo any p, and hence on the (p, q)-submatix fo any q as well. We conclude that φ(a) A fo any A suppoted on any 2 2 pincipal submatix. (Moe geneally, φ(a) = A U fo some unimodula U.) Step 7. We show φ(a) A fo a special class of matices (see Lemma 2.5) suppoted on a 3 3 pincipal submatix. 10

11 1 We simplify notation by taking n = 3. Conside A ψ = 0 0 e iψ, and wite φ(a ψ ) e iθ whee ψ, θ [ π, π]. By Lemma 2.5, θ = ±ψ, so φ(a ψ ) A ψ o A ψ. To ule out φ(a ψ ) A ψ, we suppose, by way of contadiction, that φ(a π/4 ) A π/4. Let w w A = 1 w 1 and wite φ(a) w c. Hee w = exp(iπ/4) and a, b, c, d, e, f ae unit w d e fw scalas. By using Lemmas 2.3, 2.4, and (A B) = (φ(a) φ(b)) = (φ(a) B) w 1 0 w 1 0 w 0 1 w 0 1 fo B = 0 w 0, 1 0 0,,, o 0 w 1, we have a = b = d = 0 0 w f = 1 and e = c. By using Lemma 2.5, (A A π/4 ) = (φ(a) φ(a π/4 )) = (φ(a) A π/4 ), 1 and noting φ(a) c/w, we have c = w 2. Since A is a nomal matix with eigenvalues w + 2, w 1, we have (A) = 2 + exp(iπ/4). On the othe hand, φ(a) is equivalent to a nomal matix with eigenvalues w, w ± 3, so (φ(a)) = 3 + exp(iπ/4) < (A), giving a contadiction. Thus φ(a π/4 ) A π/4. If φ(a ψ ) A ψ, then, using the notation in the poof of Lemma 2.5, f(π/4 + ψ) = (A π/4 A ψ ) = (φ(a π/4 ) φ(a ψ )) = f(π/4 ψ). By Lemma 2.5, ψ = ψ mod 2π, whence A ψ A ψ. Thus φ(a ψ ) A ψ fo all ψ, as desied. Step 8. We show that φ(a) A fo any A whose suppot is a 3 3 pincipal submatix. c c Fist let B = d have positive enties. Wite φ(a ψ B) 0 0 de iψ e iβ. Then c 0 0 d = (A ψ A ψ B) c = (φ(a ψ ) φ(a ψ B)) = 0 0 de iβ so β = 0 by Lemma 2.2, and φ(a ψ B) A ψ B. 11

12 and Now suppose A = [a ij ] has no zeo enties. We wite a 11 a 12 a 13 A a 21 e iα 21 a 22 e iα 22 a 23 e iα 23 a 31 e iα 31 a 32 e iα 32 a 33 e iα 33 a 11 a 12 a 13 φ(a) a 21 e iα 21 e iθ 21 a 22 e iα 22 e iθ 22 a 23 e iα 23 e iθ 23 a 31 e iα 31 e iθ 31 a 32 e iα 32 e iθ 32 a 33 e iα 33 e iθ 33 whee α ij, θ ij [ π, π]. By using Lemmas 2.3, 2.4, and (A B) = (φ(a) B) whee B is a matix of the fom in case (i) o (ii) of Step 5, and whose nonzeo enties ae ecipocals of those of A, we have θ ij = 0 fo all (i, j) (2, 3) o (3, 2), and θ 23 = θ 32. Now let B = A ( 1) (absolute value and invese opeations ae enty-wise). Using the notation in the poof of Lemma 2.5, so θ 23 = 0. Ou assetion follows. Step 9. We conside n n matices. f(0) = (A B A α23 ) = (φ(a) φ(b A α23 )) = (φ(a) B A α23 ) = f(θ 23 ), Fist conside an n n matix A such that A ij 0 i, j I = {i 1,..., i k } whee 1 i 1 < < i k n and k 3. Let A = [a ij ] and wite φ(a) [a ij e iθ ij ] whee θ ij [ π, π] fo all i, j and θ i1 j = 0 fo all j I. Let i 1 < p < q with p, q I. Let B be suppoted on the 3 3 pincipal submatix on (i 1, p, q) with nonzeo enties 1/a ij. Since (A B) = (φ(a) B), Lemma 2.2 implies θ pq = θ qp = θ pp = θ qq = θ pi1 = θ qi1 = 0. Thus φ(a) A if A s suppot is a pincipal submatix. In paticula, φ(a) A if A has no zeo enties. Let A be an n n matix such that A + A t is ieducible. Wite φ(a) = A R whee R is a unimodula matix. Define B ij = A ij /A ij if A ij 0 and B ij = 1 othewise. Then ( A ) = (A B) = (φ(a) φ(b)) = (φ(a) B) = ( A R) so it follows by Lemma 2.2 that φ(a) A. Finally, in the most geneal case, let Q be a pemutation such that Q t AQ = A 1 A k, whee each A j is indecomposable and so A j + A j t is ieducible. Without loss of geneality, we take Q = I to simplify notation. Wite φ(a) = A R whee R is a unimodula matix, and let R j be the submatix of R coesponding to A j. Define B j to be the n n matix whose suppot is the pincipal submatix undelying A j, and whose nonzeo enties ae eithe (A j ) pq /(A j ) pq, if (A j ) pq 0, o 1. Then fo each j ( A j ) = (A B j ) = (φ(a) φ(b j )) = (A B j R) = ( A j R j ), 12

13 so, by Lemma 2.2, we may (by edefining those enties of R j coesponding to zeo enties fo A j if needed) assume R j = λ j w j w j fo some unit scala λ j and unimodula vecto w j. Let D j = diag (w j ). Then φ(a) = A R = k j=1a j R j = k j=1λ j D j A j D j = DAE whee D = k i=1 D j and E = k i=1 λ jd j. 2.3 Poof of Theoem 1.2 Assume W (A B) = W (φ(a) φ(b)) fo all A, B M n. Thus (A B) = (φ(a) φ(b)) and so φ has one of the foms in Theoem 1.1. By eplacing φ with X R ( 1) φ(p XP t ) o X R ( 1) φ(p X t P t ), we may assume that φ(x) = D X XE X o D X XE X whee D X, E X ae diagonal unitaies such that D X E X commutes with X. Note that if X is indecomposable, then D X E X = λi and so φ(x) = λd X XDX o φ(x) = λd XXDX fo some unit scala λ. Step 1. Fixing J. We have φ(j) = λd J JDJ. Replacing φ with X D J φ(x)d J, we may assume φ(j) = λj. Since [0, n] = W (J J) = W (φ(j) φ(j)) = λ 2 W (J), we have λ = ±1. Replacing φ with X λj φ(x), we may assume φ(j) = J, and W (φ(a)) = W (φ(a) J) = W (A J) = W (A) fo all A. Note that φ still has one of the foms φ(x) = D X XE X o D X XE X fo all X. Step 2. Conjugation. ( ) 1 1 Let X = i n 2, so X is an indecomposable nomal matix such that W (X) is eithe the line segment joining i and 2 (if n = 2) o the tiangle with vetices at 0, i, 2. Since W (λd X XDX ) = λw (X) W (X) fo any complex unit λ, we must have φ(x) = D XXE X fo all X. Fo the next 3 steps, we assume A is an indecomposable matix, so φ(a) = λd A AD A and W (A) = W (φ(a)) = λw (A) fo some complex unit λ. We shall show that we can take λ = 1 in each case (i.e., φ(a) A, whee we define an equivalence elation A B if A = DBD fo some diagonal unitay D.) Step 3. Nonnegative indecomposable matices. If A is a nonnegative indecomposable matix, then H = (A + A t )/2 is ieducible and has a unique positive unit eigenvecto x such that x t Ax = (A). Since W (A) = λw (A), thee is a unit vecto v such that v Av = λx t Ax. Let D be a diagonal unitay such that D v = v. Following the poof of Lemma 2.2 (set à = A), we have v = x and D AD = µa fo some complex unit µ. Then λx t Ax = v Av = x t D ADx = µx t Ax, so µ = λ. Thus φ(a) λa A as desied. Step 4. Full matices. Suppose all of A s enties ae nonzeo. Define a positive matix B by B 11 = n A 11 1 and B ij = ((n 2 1) A ij ) 1 fo all othe (i, j)-enties. We have W (A B) = W (φ(a) φ(b)) = W (λd A AD A D B BD B) = λw ((D A D B )(A B)(D A D B ) ) = λw (A B). (2.1) 13

14 Let C = A B, and wite C ij = C ij e iθ ij. Let x be a unit vecto. Then x Cx e t 1Ce 1 ne iθ 11 x 1 2 ne iθ 11 + (i,j) (1,1) 1 n 2 1 x ix j e iθ ij n + 1. Thus W (C) lies inside a cicle of adius n + 1 about C 11. But if λ 1, then W (C) = λw (C) implies that λ k C 11 W (C) fo all k. Choose k such that 2π/3 ag λ k 4π/3. Then contadicts n 2, so λ = 1. Step 5. Abitay indecomposable matices. n + 1 λ k C 11 C 11 e 2πi/3 C 11 C 11 = 3n Let A be an abitay indecomposable matix. Define a full matix B by B ij = A ij /A ij if A ij 0, and B ij = 1 othewise. Using (2.1) we have W ( A ) = λw ( A ). By step 3, A = λd A D fo some diagonal unitay D. Wite A = A R fo some unimodula R. Then A R = λd(a R)D = λ(dad ) R, so A = λdad. Then φ(a) λa A as desied. Step 6. Geneal matices. Let Q be a pemutation such that Q t AQ = A 1 A k whee each A j is indecomposable. Without loss of geneality, we take Q = I to simplify notation. We have φ(a) = λ 1 D 1 A 1 D 1 λ k D k A k D k fo some complex units λ j and diagonal unitaies D j. The aguments in Lemma 2.2 and the peceding thee steps eadily apply to matices of the fom A = A 1 0 with A 1 indecomposable, in which case it follows φ(a 1 0) = D 1 A 1 D 1 0 fo some diagonal unitay D 1. Let B = B 1 0 0, whee B 1 is a matix of the same size as A 1 such that A 1 B 1 = A 1. Since φ(b) = DBD fo some diagonal unitay, we have W ( A 1 0) = W (A B) = W (φ(a) φ(b)) = λ 1 W ( A 1 0). The aguments in steps 3 and 5 imply A 1 0 λ 1 ( A 1 0), so A 1 λ 1 A 1 and thus A 1 λ 1 A 1. Similaly A j λ j A j fo all j, and so φ(a) A as desied. 3 Poofs fo Hemitian matices 3.1 Auxiliay esults Lemma 3.1. Suppose S H n has n(n + 1)/2 nonzeo elements such that X Y = 0 fo any X Y S. Then thee ae nonzeo scalas µ ij C such that S = {µ ij E ij + µ ij E ji : 1 i j n}. Lemma 3.2. Let f(t) = (A t ) whee 2 A t = 1 0 e it. 1 e it 0 Then f(s) = f(t) fo s, t [ π, π] if and only if s = ±t. 14

15 Poof. Since det(a t ) = 2 cos t 2 < 0 fo t (0, π], we see that A t has eigenvalues λ 1 (t) λ 2 (t) > 0 > λ 3 (t). Since det(a t zi) = z 3 + 2z 2 + 3z + 2 cos t 2, λ 1 (t) (espectively, λ 3 (t) ) clealy deceases (espectively, inceases) as t inceases fom 0 to π. Since λ 1 (π) = (1 + 17)/2 > (1 17)/2 = λ 3 (π), it follows that f(t) = λ 1 (t) and hence stictly deceases on [0, π]. Since f is even, the esult follows. 3.2 Poof of Theoem 1.1 Assume that (φ(a) φ(b)) = (A B) fo all A, B H n. Define A B if A = ±D AD fo some diagonal unitay D. Step 1. Thee is a pemutation P and complex units µ ij with µ 11,..., µ nn {1, 1} such that φ(e ij + E ji ) = P t (µ ij E ij + µ ij E ji )P fo all 1 i j n. Conside S = {E 11,..., E nn } {E ij + E ji : 1 i < j n}. Since 0 = (X Y ) = (φ(x) φ(y )) fo all X Y S and 1 = (X X) = (φ(x) φ(x)) fo all X S, Lemma 3.1 implies that the image of S unde φ is {µ 11 E 1,..., µ nn E nn } {µ ij E ij + µ ij E ji : 1 i < j n} whee µ ij = 1 and µ ii = ±1. Suppose, by way of contadiction, that φ(e 12 + E 21 ) = ±E ii fo some i. If n = 2, then (X I) = (φ(x) φ(i)) fo all X S shows that φ(i) has unit enties except fo one zeo diagonal enty. Then 1 = (I I) = (φ(i) φ(i)) = (1 + 5)/2, a contadiction. If n > 2, let Y = E 12 + E 21 + E 23 + E 32. Then, no matte what φ(e 23 + E 32 ) is, 2 (φ(y ) φ(y )) = (Y Y ) = 2, a contadiction. Thus, afte applying a pemutation similaity, we may assume that φ(e jj ) = ±E jj. Now, let Y = E ii +E jj +E ij +E ji. Since 2 = (Y Y ) = (φ(y ) φ(y )), we must have φ(e ij + E ji ) = µ ij E ij + µ ij E ji fo some unit µ ij, as desied. Step 2. The conclusion of the theoem holds fo ieducible nonnegative matices and matices with nonzeo suppot on a 2 2 pincipal submatix. By Lemma 2.2 and (J) = (J J) = (φ(j) φ(j)), we have φ(j) = J R whee R is a unimodula hemitian and R R = ( x i x j ) with x 1 = = x n = 1. By eplacing φ with A φ(j) ( 1) φ(a), we may assume φ(j) = J and consequently, (A) = (φ(a)) fo all A. By Lemma 2.2, φ(a) A fo all ieducible nonnegative A. Suppose A has nonzeo suppot on a 2 2 pincipal submatix. We can wite A b d whee a, b > 0, and φ(a) whee µ = ±1. Since (φ(a)) = (A), we must have µ = 1 b µd and φ(a) A (apply Lemma 2.2). Step 3. The conclusion of the theoem holds fo matices with nonzeo suppot in a 3 3 pincipal submatix. 15

16 Suppose A has nonzeo suppot on a given 3 3 pincipal submatix. We can wite c A b d e it c and φ(a) b αd µ c e it f c µ βf whee a, b, c, > 0, t R, α, β {1, 1}, and µ = 1. Setting X = 0,, o /a 1/b 1/c 1/b 0 1/, and using (X A) = (φ(x) φ(a)) = (X φ(a)), Lemma 2.2, and Lemma 1/c 1/ gives α = β = 1 and µ = e it o e it. Hence φ(a) A o Ā. 1 Let C = 1 0 i. By eplacing φ with φ if needed, we may assume φ(c) C. Let B be 1 i 0 any matix with positive enties. If φ(b C) B C, then (C B C) = (φ(c) φ(b C)) = (C B C) = ( C B) contadicts Lemma 2.2. Thus φ(b C) B C fo any positive matix B. Now if φ(a) Ā then, witing B = A ( 1) and using the notation in Lemma 3.2, f(t + π/2) = (A B C) = (φ(a) φ(b C)) = (Ā B C) = f( t + π/2), whence e it R and φ(a) A Ā. Step 4. We have shown that fo all A suppoted on a given 3 3 pincipal submatix, eithe φ(a) A o φ(a) Ā. Suppose, by way of contadiction, that φ(a) A fo A suppoted on one 3 3 submatix and φ(a) Ā fo A suppoted on a diffeent 3 3 submatix. Without loss of geneality, we may suppose φ(a) A fo all A suppoted on the (1, 2, 3)- o (1, 3, 4)-submatix and φ(a) Ā fo all A suppoted on the (2, 3, 4)-submatix. We wite i 1 A = 1 i, φ(a) 1 α µ 1 1 µ β ν ν γ whee α, β, γ {±1} and µ = ν = 1. Using (A X) = (φ(a) φ(x)) = (φ(a) X) and Lemma 2.2 when X is a (0, 1)-matix with nonzeo suppot on a 2 2 pincipal submatix implies α = β = γ = 1. When X is the Schu invese of the leading 3 3 pincipal submatix of A, we get µ = i. When X is the Schu invese of the (1, 3, 4)-submatix of A, we get ν = 1. When X is the Schu invese of the (2, 3, 4)-submatix of A, we have (A X) = (φ(a) φ(x)) = (φ(a) X), contadicting Lemma 2.2. Thus, by eplacing φ with φ if needed, we may assume that φ(a) A fo all A with suppot in the leading 3 3 pincipal submatix, and hence φ(a) A fo all A suppoted on any 3 3 pincipal submatix. 16

17 The est of the poof is exactly the same as step 9 fo complex matices, and we conclude φ(a) A fo all A H n. The poof of Theoem 1.2 follows the analogous poof in the complex case. Refeences [1] Z. Bai and J. Hou, Numeical adius distance peseving maps on B(H), Poc. Ame. Math. Soc. 132 (2004), [2] J.T.Chan, Numeical adius peseving opeatos on B(H), Poc. Ame. Math. Soc. 123 (1995), [3] J.T. Chan, C.K. Li and N.S. Sze, Mappings on matices: Invaiance of functional values of matix poducts, J. Austalian Math. Soc., to appea. [4] W.S. Cheung, S. Fallat, and C.K. Li, Multiplicative Peseves on Semigoups of Matices, Linea Algeba Appl. 355 (2002), [5] J. Cui and J. Hou, Non-linea numeical adius isometies on atomic nest algebas and diagonal algebas, J. Func. Anal. 206 (2004), [6] G. Dolina and P. Šeml, Deteminant peseving maps on matix algebas, Linea Algeba Appl. 348 (2002), [7] J. Hou and Q. Di, Maps peseving numeical ange of opeato poducts, Poc. Ame. Math. Soc. to appea. [8] G. Lešnjak, Additive peseves of numeical ange, Linea Algeba Appl. 345 (2002), [9] C.K. Li, Linea opeatos peseving the numeical adius of matices, Poc. Ame. math. Soc. 99 (1987), [10] C.K. Li and S. Piece, Linea peseve poblems, Ame. Math. Monthly 108 (2001), [11] C.K. Li, L. Rodman and P. Šeml, Linea maps on selfadjoint opeatos peseving invetibility, positive definiteness, numeical ange, Canad. Math. Bull., 46 (2003), [12] C.K. Li, P. Šeml and G. Soaes, Linea opeatos peseving the numeical ange (adius) on tiangula matices, Linea and Multilinea Algeba 48 (2001), [13] L. Molná, Some chaacteizations of the automophisms of B(H) and C(X), Poc. Ame. Math. Soc. 130 (2002), [14] V. Pellegini, Numeical ange peseving opeatos on matix algebas, Studia Math. 54 (1975),