Linear maps preserving the closure of numerical

Linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest

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Jianlian Cui LMAM School of Mathematical Sciences, Peking University, Beijing 100871, China [email protected] Jinchuan Hou Department of Mathematics, Shanxi Teachers University, Linfen, 041004, China Abstract: Let N be a maximal atomic nest on Hilbert space H and AlgN denote the associated nest algebra. We prove that a weakly continuous and surjective linear map Φ : AlgN →AlgN preserves the closure of numerical range if and only if there exists a unitary operator U ∈ B(H) such that Φ(T ) = U T U ∗ for every T ∈AlgN or Φ(T ) = U T tr U ∗ for every T ∈AlgN , where T tr denotes the transpose of T relative to an arbitrary but fixed base of H. As applications, we get the characterizations of the numerical range or numerical radius preservers on AlgN . The surjective linear maps on the diagonal algebras of atomic nest algebras preserving the closure of numerical range or preserving the numerical range (radius) are also characterized. Key Words: Linear preservers, maximal atomic nest, nest algebras, numerical range (radius) 2000 MRC: 47A12, 47B49. 1. Introduction and Preliminaries Let H be a Hilbert space over complex field C with inner products denoted by h·, ·i and B(H) denote the von Neumann algebra of all bounded linear operators on H. Let F(H) denote the ideal of all finite rank linear operators in B(H). For every A in B(H), the numerical range and the numerical radius of A are defined, respectively, by W (A) = {hAx, xi | x ∈ H and kxk = 1}, w(A) = sup{|λ| | λ ∈ W (A)}. 1

This work is supported by NNSFC (10071046) and PNSFS (981009). 1

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It is well known that w(·) is a norm on B(H) and that this norm is equivalent to the usual operator norm (see [7]). These concepts and their generalizations have been studied extensively because of their connections and applications to many different areas (see, e.g. Chapter 1 of [8]). So it is not surprised that there has been considerable interest in characterizing those linear maps on operator algebras which preserve the numerical range or numerical radius. For instance, one may see [1-2, 6, 9-12]. It is shown in [11] and [6] (resp., in [1]) that every surjective linear map preserving numerical range (resp., numerical radius) from B(H) onto itself is either a ∗-automorphism or a ∗-anti-automorphism (resp., either a ∗-automorphism or a ∗-anti-automorphism multiplied by a scalar of modulus one). It is interesting to ask how to characterize numerical range preservers and numerical radius preservers on nest algebras. Up to now, there has been only one paper [10] discussing this kind of linear preserver problems on nest algebras and only for a special case on finite dimensional spaces, i.e., for n × n upper triangular complex matrix algebras Tn . In [10], ˇ by induction on n, Li, Semrl and Soares proved that for every positive integer n and every linear map Φ : Tn → Tn , Φ preserves numerical range if and only if either there exists a unitary diagonal matrix U such that Φ(T ) = U T U ∗ for every T ∈ Tn or there exists an anti-diagonal unitary matrix U such that Φ(T ) = U T tr U ∗ for every T ∈ Tn , where the anti-diagonal matrix means it has nonzero entries only in positions (i, j) with i + j = n + 1 for 1 ≤ i, j ≤ n. If Φ has one of the above forms, we call Φ is of standard form. It is also showed in [10] that if Φ preserves numerical radius, then Φ is a multiple of standard form by a scalar with modulus one. Some characterizations of such linear maps on diagonal matrix algebras were obtained in [10], too. In this paper, our purpose is to extend the results in [10] to infinite dimensional case, i.e., atomic nest algebra case. Note that, in general, the numerical range of an operator acting on infinite dimensional space is not closed. We, in Section 2, consider more generally the linear maps preserving the closure of numerical range on atomic nest algebras. Since the induction method does not work here, we blend some ideas from [11] and [10] together and develop a new approach, which is fit for both cases of infinite dimension and finite dimension. Firstly, we employ an idea from [11] to give a characterization of multiples of rank one projections in nest algebras, and then, we show that every linear surjective map preserving the closure of numerical range on an atomic nest algebra must preserve the disjoint projections of rank one in both directions (see Lemma 2.2). By adopting some proof techniques used in [10], we prove that such linear maps on the nest algebras with maximal atomic nest preserve nilpotents of rank one; furthermore we get an order isomorphism or an anti-order isomorphism of the nest, and this allows us to prove that such a linear map is of standard form by constructing a desired unitary operator (see

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Theorem 2.5). In Section 3, applying the results in Section 2, we obtain a characterization of the linear maps preserving numerical range or numerical radius on the nest algebras with maximal atomic nest (see Theorem 3.3); we also extend the results concerning diagonal matrix algebras to those on the diagonal algebras of atomic nest algebras (see Theorem 2.4 and 3.4). Recall that a nest on H is a chain N of closed (under norm topology) subspaces of H containing {0} and H, which is closed under the formation of arbitrary closed linear W V span (denoted by ) and intersection (denoted by ). AlgN denotes the associated nest algebra, which is the set of all operators T in B(H) such that T N ⊆ N for every element T N ∈ N . We denote AlgF N =AlgN F(H), the set of all finite rank operators in AlgN . Notation “⊂” denotes proper contained relation between subsets. The self-adjoint operator algebra DN =AlgN ∩ (AlgN )∗ is called the diagonal algebra of AlgN . For N ∈ N , define W V N− = {M ∈ N | M ⊂ N } and N+ = {M ∈ N | N ⊂ M }. We define 0− = 0 and H+ = H. As usual, N ⊥ is the orthogonal complement of N . If N ª N− = N ∩ (N− )⊥ 6= 0, we say N ª N− is an atom of N . A nest N on H is said to be atomic if H is spanned by its atoms, and to be maximal if all its atoms are one dimensional. Let N is a closed linear subspace in H, PN will denote the (orthogonal) projection onto N along N ⊥ . Set x, f ∈ H, the rank-1 operator defined by y 7−→ hy, f ix will be denoted by x ⊗ f . Now we recall some basic results on numerical range and numerical radius that are useful in our study. One may see [7] and Chapter 1 of [8] for more information. Proposition 1.1. [7] Let A ∈ B(H). (1) W (A) = W (Atr ), where Atr stands for the transpose of A relative to an arbitrary but fixed orthonormal base of H. (2) W (A) = W (U AU ∗ ) for any unitary U. (3) W (λA) = λW (A) for any λ ∈ C. (4) W (λI + A) = λ + W (A) for any λ ∈ C. Proposition 1.2. [7] The numerical range à of A ∈!B(H) is always convex. In particλ1 b ular, if A ∈ M2 (C) is unitarily similar to , then W (A) is an elliptical disk 0 λ2 with λ1 and λ2 as foci, and length of minor axis equal to |b| , where M2 (C) denotes the 2 × 2 complex matrix algebra. Proposition 1.3. [7] Let A ∈ B(H). Then W (A) = {λ} if and only if A = λI. Proposition 1.4. [7, 8] Suppose that N ⊂ H be a closed subspace and A ∈ B(H). Then W (PN A|N ) ⊆ W (A) and w(PN A|N ) ≤ w(A).

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Proposition 1.5. [8, §1.2.9] Suppose that A ∈ Mn (C) such that A + A∗ have λn and λ1 as the largest and smallest eigenvalues, respectively. Then [λ1 , λn ] = {z + z | z ∈ W (A)}. Proposition 1.6. [7] If A ∈ B(H) is unitarily similar to A1 ⊕ A2 , then W (A) = conv{W (A1 ) ∪ W (A2 )}. 2. Main results In this section, we discuss the linear maps on nest algebras or on the diagonal algebras of nest algebras which preserve the closure of numerical ranges. In particular, we give a characterization of linear surjective maps preserving the closure of the numerical range on the diagonal algebras of atomic nest algebras or on the nest algebras with maximal atomic nest. We start with several lemmas. For the proof of the first lemma, we borrow the idea from [11]. Lemma 2.1. Let N be an atomic nest on H and A ∈AlgN be positive. Then it is a scalar multiple of a projection of rank one if and only if the following is true: for any positive operators B and C ∈AlgN such that A = B + C, both B and C must be scalar multiples of A. Proof. First of all, we prove “only if” part. Let A ∈AlgN be a scalar multiple of a rank-1 projection. Then there exists an element N ∈ N and a unit vector x ∈ N ªN− such that A = rx ⊗ x for some non-negative real number r. It is clear that we may assume r > 0. If A = B + C for some positive operators B, C ∈AlgN , then, for any y ∈ H orthogonal to the vector x, we have 0 = hAy, yi = hBy, yi + hCy, yi, which forces hBy, yi = hCy, yi = 0, and consequently, B = u ⊗ x and C = v ⊗ x for some u, v ∈ H. Now the positivity of B and C yields that u = sx and v = tx for some non-negative real numbers s and t, and hence, both B and C are scalar multiples of A. We prove the “if” part by inducing a contradiction from assuming that A is not a scalar multiple of a rank-1 projection. We consider it in two cases. Case 1. A is a (strictly) positive multiple of a projection P of rank greater than one (the possibility P = I is included). Then we can write P = Q + R, where Q and R are nonzero projections. Put A = rP = rQ + rR. By the assumption, rQ and rR are scalar multiples of A, say rQ = αA and rR = βA. It leads to R = 0 or Q = 0, a contradiction. Case 2. A is not a scalar multiple of any projection. Then A has at least two strictly positive points r and s in its spectrum. Assume 0 < r < s. Let P be the spectral projection

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of A relative to the interval [0, r]. Then B = AP and C = A(I − P ) are nonzero positive operators with sum A. So, by the assumption, C = tA for some scalar t. Thus we have tC = tA(I − P ) = C(I − P ) = C, which forces t = 1. Hence C = A and consequently, B = 0, contradicting to the fact that B is nonzero. We say that projections P and Q are disjoint if P Q = QP = 0. Lemma 2.2. Let N be an atomic nest on H and Φ : AlgN →AlgN (or Φ : DN → DN ) be a surjective linear map. If Φ preserves the closure of numerical range, then Φ preserves the disjoint projections of rank one in both directions. Proof. It is easy to see that Φ is injective since Φ preserves the closure of numerical range. Note that, for any finite rank operator F , we have W (F ) = W (F ). Let P ∈AlgN be any rank-1 projection. Then [0, 1] = W (P ) = W (P ) = W (Φ(P )), and hence, Φ(P ) ≥ 0. Set Φ(P ) = B + C, where B and C are positive operators in AlgN , then P = Φ−1 (B) + Φ−1 (C) and Φ−1 (B) and Φ−1 (C) are positive. By Lemma 2.1, there is a real number r ∈ [0, 1] such that Φ−1 (B) = rP and Φ−1 (C) = (1 − r)P. So B = rΦ(P ) and C = (1 − r)Φ(P ) by the linearity of Φ. Applying Lemma 2.1 again, we see that Φ(P ) is a positive multiple of a projection of rank one. So W (Φ(P )) = W (Φ(P )) = [0, 1], and hence, Φ(P ) is a rank-1 projection. Now suppose that rank-1 projections P , Q ∈AlgN are disjoint. Let Φ(P ) = x ⊗ x and Φ(Q) = y ⊗ y. If x and y belong to the different atoms of N , then it is clear that Φ(P ) and Φ(Q) are disjoint. Otherwise, there exists N ∈ N with dim(N ª N− ) ≥ 2 such that x, y ∈ N ª N− . Since {|hz, xi|2 − |hz, yi|2 | |z| = 1} = W (Φ(P − Q)) = W (P − Q) = [−1, 1], and since |hz, xi|2 − |hz, yi|2 takes maximal value 1 only when z and x are linear dependent and z⊥y, we must have hx, yi = 0. So Φ(P ) and Φ(Q) are disjoint. Similarly, since Φ−1 preserves numerical range, Φ−1 also preserves the disjoint projections of rank one. Hence Φ preserves the disjoint projections of rank one in both directions. The proof the case Φ : DN → DN is just the same. Recall that N is atomic means that the Hilbert space H is spanned by its atoms, i.e., we can write H = ⊕i∈J Hi , where {Hi | i ∈ J} is the set of all atoms of N . It is clear in this case we have DN = ⊕i∈J B(Hi ). To characterize the linear maps preserving the closure of numerical range on AlgN or DN , the following lemma is needed.

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Lemma 2.3. Suppose that N is an atomic nest on H with {Hi | i ∈ J} the set of all its atoms. Let Φ : AlgN →AlgN (or Φ : DN → DN ) be a surjective linear map. If Φ preserves the closure of numerical range, then there exists a 1-1 and onto map θ : J → J such that for every i ∈ J, Φ(B(Hi )) = B(Hθ(i) ). Proof. Let i ∈ J, by Lemma 2.2, we need only to consider the case that the dimension of Hi is greater than one. Take any orthogonal unit vectors u, v ∈ Hi and λ ∈ C, by Lemma 2.2, there exist orthogonal unit vectors x and y such that Φ(u ⊗ u) = x ⊗ x and Φ(v ⊗ v) = y ⊗ y. Put Φ((u + λv) ⊗ (u + λv)) = yλ ⊗ yλ , where yλ ∈ Hj for some j ∈ J and kyλ k2 = |λ|2 + 1. If dim H = 2, then yλ is a linear combination of the vectors x and y, and hence, both x and y are in Hj . Otherwise, there is a nonzero vector z lying in some atom of N orthogonal to both x and y. Since Φ is surjective and Φ preserves the disjoint projections of rank one, there is a nonzero vector w belonging to some atom of N orthogonal to both u and v such that Φ(w ⊗ w) = z ⊗ z. We get from the orthogonality of w and u + λv that z⊥yλ . We conclude that again yλ is a linear combination of x and y. Hence x and y must belong to the same atom Hj . That is, Φ maps disjoint projections of rank one in B(Hi ) into those of B(Hj ). Let θ(i) = j. Because Φ−1 also preserves the closure of numerical range, θ must be one to one from J onto itself. We claim that Φ(B(Hi )) ⊆ B(Hθ(i) ). Let A be any positive operator in B(Hi ). If Φ(A) ∈ / B(Hθ(i) ), then there is i1 6= i such that Pθ(i1 ) Φ(A) Pθ(i1 ) 6= 0, where Pi denotes the projection onto Hi . Thus, there exists a unit vector x ∈ Hθ(i1 ) and a positive number r such that Pθ(i1 ) Φ(A) Pθ(i1 ) ≥ rx ⊗ x. By Lemma 2.2, we can pick u ∈ Hi1 so that Φ(u ⊗ u) = x ⊗ x. Then W (A − ru ⊗ u) ∩ (−∞, 0) 6= ∅ since A ∈ B(Hi ) and u ∈ Hi1 . However, W (Φ(A − ru ⊗ u)) ⊂ [0, +∞) since Φ(A) − rx ⊗ x ≥ 0, which contradicts to the assumption that Φ preserves the closure of numerical range. So Φ(A) ∈ B(Hθ(i) ) and therefore, for every operator A in B(Hi ), we have Φ(A) ∈ B(Hθ(i) ). That is, Φ(B(Hi )) ⊆ B(Hθ(i) ). Similarly, we have Φ−1 (B(Hθ(i) )) ⊆ B(Hi ). Hence for every i ∈ J, Φ(B(Hi )) = B(Hθ(i) ). Note that, if K is a closed subspace of H spanned by some atoms of a nest N , then for every T ∈ DN , we have PK T = T PK , i.e., K is a reductive subspace of T . Theorem 2.4. Let N be an atomic nest on H and Φ : DN → DN be a surjective linear map. Then Φ preserves the closure of numerical range if and only if there exists a space decomposition H = K1 ⊕ K2 with Kl spanned by some atoms of N , and there exist unitary operators Ul ∈ B(Kl ) such that Φ(T ) = U1 T1 U1∗ ⊕ U2 T2tr U2∗ for every T ∈ DN , where Tl = PKl T ( l = 1, 2) and T2tr denotes the transpose of T2 relative to an arbitrary but fixed base of K2 .

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Proof. The sufficiency is clear. Now we prove the necessity. Firstly note that Φ : DN → DN is a bijective linear map since Φ preserves the closure of numerical range. By Lemma 2.3, there exists a 1-1 and onto map θ : J → J such that for every i ∈ J, the restriction Φi of Φ to B(Hi ) is a bijective linear map preserving the numerical radius from B(Hi ) onto B(Hθ(i) ). By [1], for every i ∈ J, there exists a scalar ci of modulus one and a unitary operator Ui : Hi → Hθ(i) such that either Φi (Ti ) = ci Ui Ti Ui∗ for every Ti ∈ B(Hi ) or Φi (Ti ) = ci Ui Titr Ui∗ for every Ti ∈ B(Hi ), where Titr is the transpose of Ti relative to an arbitrary but fixed orthonormal base of Hi . Note that Φi is positive, so we must have ci = 1. Let J1 = {i ∈ J | Φi (Ti ) = Ui Ti Ui∗ for every Ti ∈ B(Hi )} and J2 = {i ∈ J | Φi (Ti ) = Ui Titr Ui∗ for every Ti ∈ B(Hi )}, then J = J1 ∪ J2 . For every T ∈ DN , we P P⊕ P⊕ P⊕ have T = T1 ⊕ T2 , where T1 = ⊕ s∈J1 Ts ∈ s∈J1 B(Hs ) and T2 = t∈J2 Tt ∈ t∈J2 P⊕ P B(Ht ). Put U1 = ⊕ U and U = U , then U = U ⊕ U ∈ B(H) is unitary and 1 2 2 s∈J1 s t∈J2 t P⊕ P⊕ ∗ tr ∗ ∗ tr ∗ Φ(T ) = ( s∈J1 Us Ts Us )⊕( t∈J2 Ut Tt Ut ) = U1 T1 U1 ⊕U2 T2 U2 . The proof is completed. Next we give our main result which characterizes the linear maps preserving the closure of numerical range on nest algebras with maximal atomic nest. This kind of nests contains many cases that one may meet. For example, let H be a separable infinite dimensional W Hilbert space with orthonormal bases {ei | i ∈ } and {uk | k ∈ }. Let Ni = {ej , j ≤ i} W and Mk = {uj , j ≤ k}. Then, N = {0, H, Ni , i ∈ }, M = {0, H, Mk , k ∈ } and L = {0, H, Mk⊥ , k ∈ } are the simplest ones of the maximal atomic nests for infinite dimensional case.

Theorem 2.5. Let N be a maximal atomic nest on H and Φ :AlgN →AlgN be a weakly continuous and surjective linear map. Then Φ preserves the closure of numerical range if and only if one of the following holds true. (1) There exists a unitary operator U ∈ B(H) satisfying U (N ) = N such that Φ(T ) = U T U ∗ for every operator T ∈AlgN . (2) There exists a unitary operator U ∈ B(H) satisfying U (N ⊥ ) = N such that Φ(T ) = U T tr U ∗ for every operator T ∈AlgN , where T tr denote the transpose of T relative to an arbitrary but fixed orthonormal base of H. Moreover, in this case N ⊥ and N are unitarily similar. Proof. It is clear that we need only prove the necessity. Let {Hi | i ∈ I} be the set of all atoms of N . For each i ∈ I, pick out a unit vector ei ∈ Hi , then {ei | i ∈ I} is an orthonormal base of H since N is maximal atomic. For i, j ∈ I, we say i < j if for any N ∈ N , ej ∈ N will imply that ei ∈ N. We denote Hi,j the two dimensional subspace spanned by ei and ej , and Pi,j the projection onto Hi,j . If T ∈ B(H), we say that hT ej , ei i is the entry of T at position (i, j). By Lemma 2.2, Φ preserves the disjoint projections of

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rank one. So, for any p ∈ I, we have Φ(ep ⊗ ep ) = eθ(p) ⊗ eθ(p) , where θ : I → I is the 1-1 and onto map as in Lemma 2.3. For any i ∈ I, let Vi = {T ∈AlgN | hT ei , eα i = hT eα , ei i = 0 for any α ∈ I}, then Vi are the subspaces of AlgN . Claim 1. For any i, k ∈ I and i < k, there exists some αik ∈ C with |αik | = 1 such that either Φ(ei ⊗ ek ) = αik eθ(i) ⊗ eθ(k) or Φ(ei ⊗ ek ) = αik eθ(k) ⊗ eθ(i) . Firstly, we prove that Φ(es ⊗ et ) ∈ Vθ(p) holds for any p ∈ I and every es ⊗ et ∈ Vp . Let A(µ) = ep ⊗ ep + µes ⊗ et with |µ| = 1, then 1 = w(A(µ)) = w(Φ(A(µ))). If h[Φ(es ⊗ et )]eθ(p) , eθ(p) i 6= 0, then we can take µ ∈ C with |µ| = 1 such that h[Φ(ep ⊗ ep + µes ⊗ et )]eθ(p) , eθ(p) i > 1. Consequently, w(Φ(A(µ))) = 1 < w(Φ(A(µ))), a contradiction. Without loss of generality, we may assume that θ(p) < θ(k). So, Ã ! 1 b B = PHθ(p),θ(k) Φ(ep ⊗ ep + µes ⊗ et )|Hθ(p),θ(k) = 0 a with a and b the entries of Φ(ep ⊗ ep + es ⊗ et ) at positions (θ(k), θ(k)) and (θ(p), θ(k)), respectively. Now, if Φ(es ⊗ et ) ∈ / Vθ(p) , then b 6= 0 for some k 6= p. By Proposition 1.2, W (B) is a nondegenerate elliptical disk with 1 as a focus. Thus we have w(Φ(ep ⊗ ep + es ⊗ et )) = 1 < w(B) ≤ w(Φ(ep ⊗ ep + es ⊗ et )), which is a contradiction. Hence Φ(es ⊗ et ) ∈ Vθ(p) for every rank-1 element es ⊗ et ∈ Vp . Now it is clear that Φ(es ⊗et ) has nonzero entries only in positions (θ(s), θ(s)), (θ(t), θ(t)) and (θ(s), θ(t)) or (θ(t), θ(s)). Without loss of generality, we may assume θ(s) < θ(t). Hence, for some α, β and δ ∈ C, Φ(es ⊗ et ) = αeθ(s) ⊗ eθ(s) + βeθ(t) ⊗ eθ(t) + δeθ(s) ⊗ eθ(t) . By Proposition 1.2, W (es ⊗ et ) is a circular disk centered at 0 with radius 12 and W (Φ(es ⊗ et )) is the convex hull of 0 and the elliptical disk with α and β as foci, and length of minor axis equal to |δ|. It follows from W (es ⊗et ) = W (es ⊗ et ) = W (Φ(es ⊗ et )) = W (Φ(es ⊗et )) that α = β = 0 and |δ| = 1. Claim 2. θ : J → J is either an order isomorphism or an anti-order isomorphism. Otherwise, there must be elements l, j, k ∈ J such that l < j, l < k while θ(k) < θ(l) < θ(j). Firstly assume that l < k < j. Let S = ek ⊗ ek + el ⊗ ek + el ⊗ ej + ek ⊗ ej , then by Claim 1 and Lemma 2.2, we have Φ(S) = eθ(k) ⊗ eθ(k) + αeθ(k) ⊗ eθ(l) + βeθ(l) ⊗ eθ(j) + γeθ(k) ⊗ eθ(j) with |α| = |β| = |γ| = 1. It is easily checked that S +S ∗ has eigenvalues −1, 0 and 3. Since W (S) = W (Φ(S)), by proposition 1.5, Φ(S) + Φ(S)∗ has −1 and 3 as the smallest and the

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largest eigenvalues, respectively. Let M be the subspace spanned by {eθ(k) , eθ(l) , eθ(j) }, then det((Φ(S) + Φ(S)∗ + I)|M ) = 0 and |α| = |β| = |γ| = 1 implies that αβγ = 1. Next, let T = (1 + i)el ⊗ ek + iel ⊗ ej + (1 − i)ek ⊗ ej , where i is the imaginary unit, then Φ(T ) = (1 + i)αeθ(k) ⊗ eθ(l) + iβeθ(l) ⊗ eθ(j) + γ(1 − i)eθ(k) ⊗ eθ(j) . Since T + T ∗ has √ √ eigenvalues − 5, 0 and 5 and W (T ) = W (Φ(T )), by Proposition 1.5 again, Φ(T )+Φ(T )∗ √ √ has − 5 and + 5 as the smallest and the largest eigenvalues, respectively. However, by √ using the fact αβγ = 1, one sees that det((Φ(T ) + Φ(T )∗ − 5I)|M ) = −4 6= 0, which is a contradiction. Now assume that l < j < k. Let S = el ⊗ ej + el ⊗ ek + ej ⊗ ek + ej ⊗ ej , then Φ(S) = eθ(j) ⊗ eθ(j) + αeθ(k) ⊗ eθ(l) + βeθ(l) ⊗ eθ(j) + γeθ(k) ⊗ eθ(j) for some α, β, γ ∈ C with |α| = |β| = |γ| = 1. By a similar argument as above, we have αβγ = 1. Furthermore, let √ √ T = (1 + i)el ⊗ ej + iel ⊗ ek + (1 − i)ej ⊗ ek , then T + T ∗ has eigenvalues − 5, 0 and 5 and Φ(T ) = iαeθ(k) ⊗ eθ(l) + (1 + i)βeθ(l) ⊗ eθ(j) + γ(1 − i)eθ(k) ⊗ eθ(j) . Again, one can get √ a contradiction by checking det((Φ(T ) + Φ(T )∗ − 5I)|M ) = −4 6= 0. Claim 3. Let αij be in the Claim 1. For any i, k, j ∈ J with i < k < j, we have αij = αik αkj . Assume that θ : J → J is an order isomorphism. Let A = ei ⊗ ek + ek ⊗ ej + ei ⊗ ej , then, by Claim 1, Φ(A) = αik eθ(i) ⊗ eθ(k) + αkj eθ(k) ⊗ eθ(j) + αij eθ(i) ⊗ eθ(j) with |αik | = |αkj | = |αij | = 1. One can check that A + A∗ has the smallest eigenvalue −1 and the largest eigenvalue 2. Since W (A) = W (Φ(A)), we have Φ(A) + Φ(A)∗ has −1 and 2 as the smallest and the largest eigenvalues, respectively. Let L be the subspace spanned by {eθ(k) , eθ(i) , eθ(j) }, then it follows from det((Φ(A) + Φ(A)∗ + I)|L ) = 0 and |αik αkj αij | = 1 that αik αkj αij = 1. Hence αij = αik αkj . Assume that θ : J → J is an anti-order isomorphism, then, by a similar argument, one can prove that αij = αik αkj holds true, too. Now we are ready to show that one of the statements (1) and (2) in the theorem is true. Fix some i0 ∈ J. Let U ∈ B(H) be an operator defined by ( U ei =

αii0 eθ(i) if i ≤ i0 , αi0 i eθ(i) if i > i0 .

It is clear that U is unitary. If θ : J → J is an order isomorphism, then U (N ) = N . For every ei ⊗ ej ∈AlgN , since

Φ(ei ⊗ ej ) = αij eθ(i) ⊗ eθ(j)

   αii0 αji0 eθ(i) ⊗ eθ(j) if i ≤ j ≤ i0 , = αii0 αi0 j eθ(i) ⊗ eθ(j) if i ≤ i0 ≤ j,   αi0 i αi0 j eθ(i) ⊗ eθ(j) if i0 ≤ i ≤ j,

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we have Φ(ei ⊗ ej ) = U ei ⊗ U ej . For any rank-1 operator x ⊗ y ∈AlgF N , there exists i0 P P⊕ such that x = ⊕ i≤i0 ξi ei and y = j≥i0 ηj ej , so x⊗y =(

⊕ X i≤i0

ξi ei ) ⊗ (

⊕ X

η j ej ) =

j≥i0

⊕ X X

ξi ηj ei ⊗ ej .

i≤i0 j≥i0

It is well known that w(·) is a norm on B(H) and that this norm is equivalent to the usual operator norm (see [7]), hence Φ is bounded. This, together with Claim 1 and Lemma 2.2, implies that X X Φ(x ⊗ y) = ξi ηj αij eθ(i) ⊗ eθ(j) = ξi ηj U ei ⊗ U ej = U x ⊗ U y. i≤i0 ≤j

i≤i0 ≤j

Thus, Φ(F ) = U F U ∗ holds for every F ∈AlgF N . Up to now, we have not used the weak continuity of Φ. Because, by assumption, Φ is weakly continuous and the set AlgF N is weakly dense in AlgN , Φ(T ) = U T U ∗ holds for every T ∈AlgN . If θ : J → J is an anti-order isomorphism, then it is clear that U (N ⊥ ) = N . Hence N and N ⊥ are similar. By Claim 3, we have Φ(ei ⊗ ej ) = αij eθ(j) ⊗ eθ(i) = U ej ⊗ U ei . It follows that Φ(x ⊗ y) =

X

ξi ηj αij eθ(j) ⊗ eθ(i) =

i≤i0 ≤j

X

ξi ηj U ej ⊗ U ei = U Jy ⊗ U Jx,

i≤i0 ≤j

P P where J is a conjugate linear operator defined by J( i∈J ξi ei ) = i∈J ξi ei . It is clear that J 2 = I and for all x, y ∈ H, hJx, yi = hJy, xi. So Φ(x ⊗ y) = U J(y ⊗ x)JU ∗ , hence for all F ∈AlgF N , Φ(F ) = U JF ∗ JU ∗ . Let T tr denote the transpose of T relative to the base {ei }i∈J . Then we have X X hT tr x, yi = ξi ηj hT tr ei , ej i = ξi ηj hT ej , ei i = hT Jy, xi = hJT ∗ Jx, yi, i,j∈∆

i,j∈∆

that is, JT ∗ J = T tr . Since Φ and the transpose operation are weakly continuous, for all T ∈AlgN , we get Φ(T ) = U T tr U ∗ . Remark 2.6. If N and N ⊥ are not unitarily similar, in particular, if they are not order isomorphic to each other, then the case (2) will not occur. 3. Applications: numerical range (radius) preservers In this section, we apply the results in Section 2 to the discussion of numerical range (or radius) preservers. It is clear that Theorem 2.4 and Theorem 2.5 are true for numerical range preservers. For instance, the following result is an immediate consequence of Theorem 2.5.

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Theorem 3.1. Let Φ : AlgN →AlgN be a weakly continuous and surjective linear map. Then Φ preserves the numerical range if and only if one of the following holds true. (1) There exists a unitary operator U ∈ B(H) satisfying U (N ) = N such that Φ(T ) = U T U ∗ for every operator T ∈AlgN . (2) There exists a unitary operator U ∈ B(H) satisfying U (N ⊥ ) = N such that Φ(T ) = U T tr U ∗ for every operator T ∈AlgN , where T tr denote the transpose of T relative to an arbitrary but fixed base of H. Moreover, in this case N ⊥ and N are unitarily similar. The remain of this section is devoted to characterize the linear maps preserving numerical radius. We begin with a lemma which describes scalar multiple of identity operator I in terms of numerical radius. Let Λ = {λ ∈ C | |λ| = 1}. Lemma 3.2. Let N be an atomic nest on a Hilbert space H. Then, an operator A ∈AlgN is a scalar multiple of I if and only if for every B ∈AlgN , there is a λ ∈ Λ such that w(A + λB) = w(A) + w(B). Proof. It is clear that if A is a scalar multiple of I, then A satisfies the condition. Now assume that for every B ∈AlgN , there is a λ ∈ Λ such that w(A + λB) = w(A) + w(B). Let {Hi | i ∈ J} be the set of all atoms of N . We will show that A is a scalar multiple of I by three steps. Step 1. For every Hi and x ∈ Hi with kxk = 1, we have |hAx, xi| = w(A). Assume, on the contrary, that there exist Hi and x ∈ Hi with kxk = 1 such that |hAx, xi| < w(A). Let B = x ⊗ x, then it is clear that w(B) = 1. Fix any r such that |hAx, xi| < r < w(A), then we can find an ε > 0 such that |hAy, yi| < r whenever ky − xk < ε. So if there exists α ∈ Λ such that ky − αxk < ε, then we must have |hAy, yi| < r and hence |h(A + λB)y, yi| < |hAy, yi| + |hBy, yi| < r + |hx, yi|2 ≤ r + 1 whenever kyk = 1. Now, suppose that y ∈ H satisfies kyk = 1 and for every α ∈ Λ, ky − αxk ≥ ε. Then ε2 ≤ hy − αx, y − αxi = 2 − 2Rehy, αxi

for every α ∈ Λ.

It follows that |hy, xi| ≤ 1 − 12 ε2 . Let k = min{r + 1, w(A) + 1 − 12 ε2 }, then for every λ ∈ Λ and y ∈ H with kyk = 1, we have |h(A + λB)y, yi| ≤ |hAy, yi| + |hy, xi| ≤ k. Hence w(A + λB) < w(A) + w(B), a contradiction. So, for every i ∈ J, we have W (Pi A|Hi ) ⊂ {λ | |λ| = w(A)}. It follows from the convexity of the numerical ranges (Proposition 1.2) that W (Pi A|Hi ) is a singleton, so, by Proposition 1.3, Pi A|Hi = αi Ii for some αi ∈ C with |αi | = w(A).

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P Step 2. We prove that A = ⊕ i∈J αi Ii . Assume that there exists some i, j ∈ J with i < j such that B0 = Pi A|Hj 6= 0. Then there exist unit vectors e ∈ Hi and u ∈ Hj so that b = hB0 u, ei = 6 0. Let à ! αi Ii B0 B= . 0 αj Ij ! ! à αi b αi b ) is an elliptical disk with αi and αj as ) ⊂ W (B) and W ( Since W ( 0 αj 0 αj foci, and the length of minor axis equal to |b|, it follows that w(A) = |αi | < w(B) ≤ w(A), a contradiction. Step 3. There is a α ∈ C such that for every i ∈ J, αi = α. For T ∈AlgN , put Ã

ST = {x ∈ H | |hT x, xi| = w(T ) and kxk = 1}. For any i, j ∈ J with i < j, let B = x ⊗ y, where x ∈ Hi and y ∈ Hj with |x| = kyk = 1, we have √ 2 SB = {µx + δy | |µ| = |δ| = } 6= ∅. 2 Then the condition w(A + λB) = w(A) + w(B) for some λ ∈ Λ implies that SA ∩ SB 6= ∅. |α +α | Hence we have i 2 j = w(A) = |αi | = |αj | . It follows that αi = αj . Therefore A = αI for some α ∈ C with |α| = w(A). Now we are in a position to prove the main theorems in this section. Theorem 3.3. Let N be a maximal atomic nest on a Hilbert space H and let Φ : AlgN →AlgN be a weakly continuous and surjective linear map. Then Φ preserves numerical radius if and only if one of the following holds true. (1) There exists a unitary operator U ∈ B(H) satisfying U (N ) = N and a complex number ξ with |ξ| = 1 such that Φ(T ) = ξU T U ∗ for every operator T ∈AlgN . (2) There exists a unitary operator U ∈ B(H) satisfying U (N ⊥ ) = N and a complex number ξ with |ξ| = 1 such that Φ(T ) = ξU T tr U ∗ for every operator T ∈AlgN , where T tr denote the transpose of T relative to an arbitrary but fixed orthonormal base of H. Moreover, in this case, N and N ⊥ are unitarily similar. Proof. Assume that Φ preserves numerical radius, it is clear that Φ is injective. Furthermore, Φ−1 also preserves numerical radius. Let A = Φ(I). Then for any B ∈AlgN , there exists λ ∈ Λ such that w(A + λB) = w(I + λΦ−1 (B)) = w(I) + w(Φ−1 (B)) = w(A) + w(B).

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Thus, by Lemma 3.2, for some ξ ∈ C, Φ(I) = ξI. It follows from w(I) = w(ξI) that |ξ| = 1 since Φ preserves numerical radius. Let Ψ(·) = ξ −1 Φ(·), then Ψ(I) = I and Ψ preserves numerical radius, too. We claim that Ψ preserves the closure of the numerical range, that is, for all T ∈AlgN , W (Ψ(T )) = W (T ). For any T ∈AlgN , W (T ) is bounded and convex, so W (T ) is a compact convex subset in C. Assume that there exists a µ ∈ C such that µ ∈ W (Ψ(T ))\W (T ), then, there exists a circle with sufficiently large radius centered at a certain λ ∈ C such that W (T ) lies inside the circle, but µ lies outside the circle. Hence, for any z ∈ W (T ), we have |z − λ| < |µ − λ| . Consequently, w(T − λI) < |µ − λ| ≤ w(Ψ(T ) − λI) = w(Ψ(T − λI)) = w(T − λI), which is a contradiction. So W (Ψ(T )) ⊆ W (T ) and hence W (Ψ(T )) ⊆ W (T ). Using the same argument to Ψ−1 , we see that W (T ) ⊆ W (Ψ(T )). So for all T ∈AlgN , W (Ψ(T )) = W (T ). Now the results follow from Theorem 2.5. For diagonal algebra case we have Theorem 3.4. Let N be an atomic nest on H and Φ : DN → DN be a surjective linear map. Then Φ preserves the numerical radius if and only if there exists a space decomposition H = K1 ⊕ K2 with Kl spanned by some atoms of N , a unitary operator A ∈ DN and there exist unitary operators Ul ∈ B(Kl ) such that Φ(T ) = A1 U1 T1 U1∗ ⊕ A2 U2 T2tr U2∗ for every T ∈ DN , where Tl = PKl T ( l = 1, 2) and T2tr denotes the transpose of T2 relative to an arbitrary but fixed base of K2 . Proof. Let A = Φ(I), then w(A) = 1 and for any B ∈ DN there exists λ ∈ Λ such that w(A+λB) = w(A)+w(B). The Step 1 and Step 2 in the proof of Lemma 3.2 are still valid P if we replace AlgN by DN . It follows that A = ⊕ i∈J αi Ii with |αi | = w(A) = 1 for each P −1 i ∈ J. Let Ψ(T ) = A Φ(T ) for every T ∈ DN . Since Φ(T ) has the form Φ(T ) = ⊕ i∈J Si , P⊕ we have Ψ(T ) = i∈J αi Si and w(Ψ(T )) = supi∈J {w(αi Si )} = w(Φ(T )) = w(T ). Hence Ψ : DN → DN is unital and numerical radius preserving. A similar argument as that in the proof of Theorem 3.3 shows that Ψ preserves the closure of numerical range. Now, the proof is completed by applying Theorem 2.4. Remark 3.5. We wish to thank Professor Peter Semrl for showing his paper [10] to us.

References [1] Chan, J.T., Numerical radius preserving operators on B(H), Proc. Amer. Math. Soc. 123 (1995), 1437-1439.

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[2] Chan, J.T., Numerical radius preserving operators on C ∗ -algebras, Arch. Math. (Basel) 70 (1998), 486-488. [3] Cui, J., Hou, J., Linear preservers on upper triangular operator matrix algebras, preprint. [4] Cui, J., Hou, J., Linear preservers on nest algebras, preprint. [5] Davidson, K. R., Nest Algebra, Ritman Research Notes in Mathematics, Vol. 191, Longman, London/New York, 1988. [6] Gao, M., Numerical range preserving linear maps and spectrum preserving elementary operators on B(H), Chinese Ann. Math., 14A (3) (1993), 295-301. [7] Halmos, P.R., A Hilbert space problem book, 2nd ed., Springer-Verlag, New York, 1982. [8] Horn, R.A. and Johnson, C.R., Topics in matrix Analysis, Cambridge University Press, New York, 1991. [9] Li, C.K., Linear operators preserving the numerical radius of matrices, Proc. Amer. Math. Soc., 99 (1987), 601-608. [10] Li., C.K., Semrl, P. and Soares, G., Linear operators preserving the numerical range (radius) on triangular matrices, Linear and Multilinear Algebra 48 (2001), 281-292. [11] Omladic, M. On operators preserving the numerical range, Lin. Alg. Appl. 134 (1990), 31-51. [12] Pellegrini, V., Numerical range preserving operators on a Banach algebras, Studia Math. 54 (1975), 143-147.