c-Numerical radius isometries on matrix algebras and triangular matrix algebras

Linear Algebra and its Applications 466 (2015) 160–181

Contents lists available at ScienceDirect

Linear Algebra and its Applications www.elsevier.com/locate/laa

c-Numerical radius isometries on matrix algebras and triangular matrix algebras Kong Chan Department of Mathematics, University of Hong Kong, Hong Kong

a r t i c l e

i n f o

Article history: Received 21 December 2013 Accepted 9 October 2014 Available online xxxx Submitted by P. Semrl

a b s t r a c t Let c = (c1 , . . . , cn )t ∈ Rn and Mn be the set of n ×n complex matrices. For any A ∈ Mn , define the c-numerical range and the c-numerical radius of A by Wc (A) =

MSC: 15A04 15A60 47A12 Keywords: Isometry Generalized numerical ranges Preserver problems

 n 

ci Axi , xi  : {x1 , . . . , xn }

i=1



is an orthonormal set in Cn

and   wc (A) = max |z| : z ∈ Wc (A) , respectively. Let Tn be the set of n × n upper triangular matrices. When wc (·) is a norm on Mn , mappings T on Mn (or Tn ) satisfying   wc T (A) − T (B) = wc (A − B) for all A, B are characterized. As an intermediate step, we also characterize additive c-numerical range preservers on Mn (or Tn ). © 2014 Elsevier Inc. All rights reserved.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.laa.2014.10.012 0024-3795/© 2014 Elsevier Inc. All rights reserved.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

161

1. Introduction Let Mn be the set of n × n complex square matrices. The numerical range and the numerical radius of A ∈ Mn are defined by     W (A) = Ax, x : x ∈ Cn , x = 1 and w(A) = max |z| : z ∈ W (A) respectively. These concepts have applications in pure and applied mathematics. The readers could refer to [10,12–14] for more information. There are many generalizations of the numerical range. When c = (c1 , . . . , cn )t ∈ Cn , the c-numerical range and the c-numerical radius of A ∈ Mn are defined by Wc (A) =

 n 

 ci Axi , xi  : {x1 , . . . , xn } is an orthonormal set in Cn

i=1

and   wc (A) = max |z| : z ∈ Wc (A) respectively. All of the above are special cases of C-numerical ranges and C-numerical radii of A. For any A, C ∈ Mn , the C-numerical range and radius of A are defined by     WC (A) = tr CUAU ∗ : U ∈ Mn is unitary and   wC (A) = max |z| : z ∈ WC (A) respectively. The readers may refer to [16] for more information about this subject. It is always of interest to characterize mappings with some special properties such as leaving certain functions, subsets or relations invariant. If the mappings are assumed to be linear, the problems are often called linear preserver problems. One may see [11, 21,23] and their references for more information. There is a considerable interest about linear preservers of different generalized numerical ranges or radii. A map T is said to be c-numerical range preserving if   Wc T (A) = Wc (A) for all A in the domain. The readers may refer to [17] for a survey about this topic. There is also interest in studying preservers with milder conditions than being linear. For example, given a norm  · on a domain X, what is the form of an isometry T (with no linearity assumption but sometimes surjectivity assumption) such that T (A) −T (B) = A − B for all A, B ∈ X? Some problems of this type can be found in [1,2,5,8,9].

162

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

Let Tn be the set of n ×n upper triangular matrices. In [19], based on the results in [15], Li and Šemrl characterize numerical radius isometries on different spaces including Mn and Tn . For c ∈ Rn and wc (·) being a norm, we characterize c-numerical radius isometries in Section 3, i.e., mappings T on Mn (or Tn ) such that   wc T (A) − T (B) = wc (A − B) for all A, B ∈ Mn (or Tn ). As an intermediate step, we characterize additive c-numerical range preservers on the respective spaces in Section 2. Note that complex linear c-numerical radius isometries on Mn and Tn were characterized in [20] and [7] respectively. Thus our results can also be viewed as generalizations to these results. We introduce some notations and state the assumptions in this paper. (i) Assume c = (c1 , . . . , cn )t ∈ Rn . Without loss of generality, we also assume that c1 ≥ c2 ≥ · · · ≥ cn . (ii) The notation A¯ is used to denote the conjugation of A, i.e., the matrix obtained by taking complex conjugates of each entry in A. (iii) Hn , Dn and Un are respectively the sets of Hermitian matrices, diagonal matrices and unitary matrices in Mn . (iv) Ejk , 1 ≤ j, k ≤ n, are elements of the standard basis of Mn . (v) To avoid trivial considerations, we always assume n ≥ 2. (vi) Note that Wc (A) = WC (A) for a Hermitian C with c1 , . . . , cn as its eigenvalues. Thus we will sometimes write Wc (A) as WC (A) for C ∈ Dn with diagonal entries c1 , . . . , cn . We will also write the elements of Wc (A) as tr(CUAU ∗ ) for U ∈ Un when such expressions ease the arguments. We list some basic properties about the c-numerical range and c-numerical radius which will be used in our proofs (most of them can be found in [16] and the references therein). n (I) The function wc (·) is a norm on Mn if and only if i=1 ci = 0 and not all ci ’s are equal. (II) The set Wc (A) is convex for all A ∈ Mn . (III) Let I be the identity matrix in Mn and α, β ∈ C. We have Wc (αA + βI) = n αWc (A) + β( i=1 ci ). (IV) For any U ∈ Un , Wc (A) = Wc (At ) = Wc (UAU ∗ ) = Wc (A). (V) Suppose the ci ’s are not all equal. Then Wc (A) is a singleton if and only if A is a scalar matrix. (VI) Suppose the ci ’s are not all equal. Then Wc (A) ⊂ R if and only if A is Hermitian.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

163

(VII) Suppose A ∈ Hn has eigenvalues a1 ≥ a2 ≥ · · · ≥ an . By [20, Lemma 2.1], we have

Wc (A) =

n 

ci an−i+1 ,

n 

i=1

 ci ai .

i=1

Suppose X ∈ Mn and the eigenvalues of 12 (X + X ∗ ) are λ1 ≥ λ2 ≥ · · · ≥ λn . The n above result implies that the lines L1 = {z ∈ C : Re(z) = i=1 ci λn−i+1 } and n L2 = {z ∈ C : Re(z) = i=1 ci λi } are the left and right vertical supporting lines of Wc (X). 2. Additive c-numerical range preservers In this section, we prove two results concerning additive c-numerical range preservers. They are Theorems 2.1 and 2.2. Theorem 2.1. Case 1: Suppose wc (·) is a norm. An additive mapping T : Mn → Mn satisfies   Wc T (A) = Wc (A)

for all A ∈ Mn

if and only if there exists U ∈ Un such that T has one of the following forms: (i) T (A) = U φ(A)U ∗

for all A ∈ Mn ,

(ii) ci + cn−i+1 are equal T (A) =

or

for i = 1, . . . , n

2 (tr A)I − U φ(A)U ∗ n

and

for all A ∈ Mn

with φ taking the forms φ(X) = X or φ(X) = X t . n Case 2: Suppose i=1 ci = 0 and not all c1 , . . . , cn are equal. An additive mapping T : Mn → Mn satisfies   Wc T (A) = Wc (A)

for all A ∈ Mn

if and only if there exist U ∈ Un and a real linear functional f on Mn such that T has one of the following forms: (i) T (A) = U φ(A)U ∗ + f (A)I (ii) ci + cn−i+1 = 0

for all A ∈ Mn ,

for i = 1, . . . , n

T (A) = f (A)I − U φ(A)U

∗

and

for all A ∈ Mn

with φ taking the forms φ(X) = X or φ(X) = X t .

or

164

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

Proof. (⇐) Note that the mappings stated in the theorem are complex linear. They are c-numerical range preserving by [20, Theorems 5.1 and 5.2]. (⇒) Case 1. Since T is additive and continuous, T is real linear. In fact T is complex linear. The proof is adapted from [2, Lemma 4]. By property (VI), for any Hermitian A ∈ Mn , T (A) is Hermitian. Note that as Wc (iA) lies on the imaginary axis, Wc (−iT (iA)) ⊆ R. Hence T (iA) = iB for some Hermitian B ∈ Mn . Note that Wc ((1 + i)A) = Wc (T ((1 + i)A)) = Wc (T (A) + iB). Let C ∈ Dn with c1 , . . . , cn as diagonal entries. For any U ∈ Un , there exists a V ∈ Un such that         tr CU T (A) + iB U ∗ = tr CV (1 + i)A V ∗         tr CUT (A)U ∗ + i tr CUBU ∗ = tr CVAV ∗ + i tr CVAV ∗ . As tr(CUT (A)U ∗ ), tr(CUBU ∗ ), tr(CVAV ∗ ) are all real numbers, we can deduce that tr(CU (T (A) − B)U ∗ ) = 0 for all U ∈ Un . Thus   WC T (A) − B = 0. Note that wC (·) is a norm, so T (iA) = iB = iT (A). By [20, Theorem 5.1], T has the form (i) or (ii). Case 2. Similar techniques from the proof of [20, Theorem 5.2] can be used to deduce the proof here. Note that there is a printing mistake in (c) (ii) of the theorem mentioned, which should be T (A) = −U A+ U ∗ + f (A)I for all A ∈ Mn . 2 The additive numerical range preserving maps on Tn were characterized in [15, Theorem 11]. It is interesting to note that some of them are neither complex linear nor conjugate linear. One may wonder if such maps are also additive c-numerical range preserving maps on Tn for general c ∈ Rn . Theorem 2.2 shows that this is indeed the case. Moreover, for some particular c ∈ Rn , there are additive c-numerical range preservers which are not numerical range preserving (see Case 1 (ii) and Case 2 (ii) of Theorem 2.2). We define two maps as in [15] and employ the same notations for easy comparison and reference. Let F be the flip on Tn , i.e. F (A) = Af = EAt E, where E = E1n + E2,n−1 + · · · + En1 , or [F (A)]jk = an+1−k,n+1−j . The map G is defined by ⎛a

11

⎜ 0 G:⎜ ⎝ ... 0

a12

··· A11

⎛a a1n ⎞ 11 ⎜ 0 ⎟ ⎟ → ⎜ . ⎝ .. ⎠

a1n

··· Af11

a12 ⎞ ⎟ ⎟, ⎠

0

here A11 ∈ Tn−1 is the corresponding principal submatrix of A. Let Gn be the group of additive mappings from Tn to Tn , generated by diagonal unitary similarities (i.e., mappings A → UAU ∗ with a unitary U ∈ Dn ), the flip F and the map G, with composition as the group operation.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

165

Theorem 2.2. Case 1: Suppose wc (·) is a norm. An additive mapping T : Tn → Tn satisfies   Wc T (A) = Wc (A)

for all A ∈ Tn

if and only if (i) it belongs to Gn ;

or

(ii) ci + cn−i+1 are equal T (A) =

for i = 1, . . . , n

2 (tr A)I − η(A) n

and

for all A ∈ Tn

with η being a member of Gn . n Case 2: Suppose i=1 ci = 0 and not all c1 , . . . , cn are equal. An additive mapping T : Tn → Tn satisfies   Wc T (A) = Wc (A)

for all A ∈ Tn

if and only if there exists a real linear functional f on Tn such that for all A ∈ Tn ;

(i) T (A) = η(A) + f (A)I (ii) ci + cn−i+1 = 0

for i = 1, . . . , n

T (A) = f (A)I − η(A)

or

and

for all A ∈ Tn

with η being a member of Gn . Remark. Note that F 2 = G2 = the identity map. Thus maps generated by F and G are obtained by applying the two maps alternately. When n = 2, the image of the matrix A = by F and G has one of the following forms:  (1)

a11 0

a12 a22



 ,

(2)

a11 0

a12 a22



 a11

a12 0 a22

 ,

(3)

a22 0



∈ T2 under a mapping generated

a12 a11



 ,

or (4)

a22 0

a12 a11

 .

When n ≥ 3, the readers can check that the image of a matrix A = [aij ] ∈ Tn under a mapping generated by F and G has one of the following general forms: (1) A itself; (2) F (A);

166

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

(3) a matrix B = [bαβ ] ∈ Tn with (app , ap+1,p+1 , . . . , ann , a11 , a22 , . . . , ap−1,p−1 ) as its diagonal entries (2 ≤ p ≤ n). The off-diagonal elements are such that (for r < s)  brs =

ajk , akj ,

if brr = ajj , bss = akk and j < k, if brr = ajj , bss = akk and j > k;

(4) a matrix B = [bαβ ] ∈ Tn with (app , ap−1,p−1 , . . . , a11 , ann , an−1,n−1 , . . . , ap+1,p+1 ) as its diagonal entries. The off-diagonal elements are such that (for r < s)  brs =

akj , ajk ,

if brr = ajj , bss = akk and j > k, if brr = ajj , bss = akk and j < k.

These facts have already been observed in the proof of [15, Proposition 15]. They will be used in the proof of Theorem 2.2. The following lemma can be deduced from property (VII) in the last section. Lemma 2.3. Let A, B ∈ Mn and c ∈ Rn be nonzero. The following statements are equivalent. (i) Wc (A) = Wc (B); (ii) Wc (μA + (μA)∗ ) = Wc (μB + (μB)∗ ) for all μ ∈ C with |μ| = 1. Now we can give the proof of Theorem 2.2. Proof. The proof of Case 2 is similar to the proof of [20, Theorem 5.2]. In the following, we focus on the proof of Case 1. (⇐) By direct checking, we know that diagonal unitary similarities and the map F are additive and c-numerical range preserving. By Theorem 2.1, the map A→

2 (tr A)I − A n

is also additive and c-numerical range preserving when ci + cn−i+1 are equal for all i = 1, . . . , n. It is easy to see that the map G is additive. We only need to check that G is c-numerical range preserving. For any A = [aij ] ∈ Tn and μ ∈ C with |μ| = 1, let ⎛

2 Re(μa11 ) μa12 μa13 ⎜ μa12 2 Re(μa22 ) μa23 ⎜ ⎜ μa13 ∗ μa 2 Re(μa 23 33 ) A1 = μA + μ ¯A = ⎜ ⎜ . . . .. .. .. ⎝ μa1n μa2n μa3n

··· ··· ··· .. .

μa1n μa2n μa3n .. .

· · · 2 Re(μann )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

167

and A2 = μG(A) + μ ¯G(A)∗ ⎛ 2 Re(μa11 ) μa1n ⎜ μ ¯ a 2 Re(μa 1n nn ) ⎜ ⎜ μ μan−1,n = ⎜ ¯a1,n−1 ⎜ . .. . ⎝ . . μ ¯a12

μa1,n−1 μan−1,n 2 Re(μan−1,n−1 ) .. .

μa2n

··· ··· ··· .. .

μa12 μa2n μa2,n−1 .. .

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

· · · 2 Re(μa22 )

μa2,n−1

By Lemma 2.3, we only need to prove that Wc (A1 ) = Wc (A2 ). Let P be the n permutation matrix E11 + j=2 Ej,n+2−j (note that P = P ∗ ) and U be the diagonal matrix with 1 in the first diagonal entry and μ2 in all other diagonal entries. Then one can directly check that A2 = (UPA1 PU ∗ )t . Therefore, by property (IV), Wc (A2 ) = Wc ((UPA1 PU ∗ )t ) = Wc (A1 ). (⇒) We begin the proof of this part with the following lemma. Lemma 2.4. The mapping T is complex linear on Dn and T (D∗ ) = T (D)∗ for any diagonal matrix D in Tn . Proof of Lemma 2.4. For any D ∈ Dn , write D = A + iB with A, B ∈ Dn and both A, B being Hermitian. Since T is c-numerical range preserving, both T (A), T (B) are also Hermitian and hence T (A), T (B) ∈ Dn . Thus T (Dn ) ⊂ Dn . The proof technique of the only if part in Case 1 of Theorem 2.1 can now be applied to show that T is complex linear on Dn . Lastly, we have T (D∗ ) = T ((A + iB)∗ ) = T (A) − iT (B) = T (D)∗ . 2 For any A ∈ Tn , write B = A + A∗ . Define Φ(B) = T (A) + T (A)∗ . Using techniques similar to those in the proof of [18, Theorem 3.1] (see also [24, Theorem 2.1]), we can show that Φ : Hn → Hn is well-defined, real linear, and c-numerical range preserving for all elements in Hn . We can deduce from [20, Theorem 5.1] that Φ has one of the following forms: (i) Φ(B) = U φ(B)U ∗

for all B ∈ Hn ;

(ii) ci + cn−i+1 are equal Φ(B) =

or

for i = 1, . . . , n and

2 (tr B)I − U φ(B)U ∗ n

for all B ∈ Hn ;

with U ∈ Un and φ taking the forms φ(X) = X or φ(X) = X t .

168

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

If Φ is in the form (ii), we have   2 (tr B)I − T (A) + T (A)∗ n   ∗  2 2 (tr A)I − T (A) + (tr A)I − T (A) = n n   ∗ = T  (A) + T  (A) ,

U φ(B)U ∗ =

where T  (A) = n2 (tr A)I − T (A) for all A ∈ Tn . It is easy to see that T  is additive. Since WC (T ( n2 (tr A)I)) = WC ( n2 (tr A)I) = n2 (tr A)(tr C) for any A ∈ Tn , T ( n2 (tr A)I) = 2 2 n (tr A)I. Suppose α = ci + cn−i+1 for i = 1, . . . , n. Then WC ( n (tr A)I − T (A)) = WC (T ( n2 (tr A)I − A)) = WC ( n2 (tr A)I − A) = α tr A − WC (A) = WαI−C (A) = WC (A). Thus T  is c-numerical range preserving. It remains to prove that an additive c-numerical range preserving map T satisfying   T (A) + T (A)∗ = U φ A + A∗ U ∗

(1)

for all A ∈ Tn is generated by members of Gn . Now we look at the structure of U . Since Ejj ∈ Hn , we have T (Ejj ) ∈ Hn for j = 1, . . . , n by property (VI). By (1), T (Ejj ) = U Ejj U ∗ = uj u∗j , where uj stands for the j-th column of the unitary matrix U . Note that T (Ejj ) ∈ Dn , thus all off-diagonal entries of the matrix uj u∗j are zero. Therefore, uj has only one nonzero entry. As this is true for all j = 1, . . . , n, we conclude that U = DP for a permutation matrix P and D = diag(d1 , . . . , dn ) with |d1 | = · · · = |dn | = 1. Without loss of generality, replace T by D∗ T (·)D in (1). We have   T (A) + T (A)∗ = P φ A + A∗ P t

(2)

for some permutation matrix P ∈ Mn . From (2), for any 1 ≤ j ≤ n, there exists 1 ≤ k ≤ n such that T (Ejj ) = Ekk . By Lemma 2.4, we have T (aEjj ) = aEkk for any a ∈ C. Write A as [ajk ] ∈ Tn . The diagonal entries of T (A) is a permutation of a11 , a22 , · · · , ann from the above discussion. We now determine the entries of T (A) above the main diagonal. For r < s, assume T (Ejj ) = Err and T (Ekk ) = Ess . An analysis of (2) shows that the (r, s)-th entry of T (A) is determined as follows.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

169

Case I: When φ(A) = A, the entry is ajk if j < k or akj if j > k. Case II: When φ(A) = At , the entry is akj if j > k or ajk if j < k. If n = 2, these maps are exactly the maps generated by maps F and G as mentioned in the remark after Theorem 2.2. For the remaining part of this proof, we consider n ≥ 3. Suppose A = [aαβ ] ∈ Tn has all entries equal zero except the entries in a principal submatrix lying on the j-th, k-th and l-th rows and columns (with 1 ≤ j < k < l ≤ n). The image of A under T would be a matrix B = [bγδ ] ∈ Tn with all entries equal zero except those entries in a principal submatrix lying on the r-th, s-th and t-th rows and columns (for some 1 ≤ r < s < t ≤ n). If Case I occurs, this principal submatrix is in one of the following six forms: ⎛

ajj (A1 ) ⎝ 0 0 ⎛ ajj (B1 ) ⎝ 0 0

ajk akk 0 ajl all 0

⎞ ajl akl ⎠ , all ⎞ ajk akl ⎠ , akk

⎛

akk (A2 ) ⎝ 0 0 ⎛ akk (B2 ) ⎝ 0 0

akl all 0 ajk ajj 0

⎞ ajk ajl ⎠ , ajj ⎞ akl ajl ⎠ , all

⎛

all (A3 ) ⎝ 0 0 ⎛ all (B3 ) ⎝ 0 0

ajl ajj 0 akl akk 0

⎞ akl ajk ⎠ , akk ⎞ ajl ajk ⎠ . ajj

If Case II occurs, this principal submatrix is in one of the following six forms: ⎛

ajj ⎝ (C1 ) 0 0 ⎛ ajj (D1 ) ⎝ 0 0

ajl all 0 ajk akk 0

⎞ ajk akl ⎠ , akk ⎞ ajl akl ⎠ , all

⎛

⎞ akk ajk akl ajj ajl ⎠ , (C2 ) ⎝ 0 0 0 all ⎛ ⎞ akk akl ajk all ajl ⎠ , (D2 ) ⎝ 0 0 0 ajj

⎛

⎞ all akl ajl (C3 ) ⎝ 0 akk ajk ⎠ , 0 0 ajj ⎛ ⎞ all ajl akl (D3 ) ⎝ 0 ajj ajk ⎠ . 0 0 akk

Let Tn be a subset of Tn such that A ∈ Tn if all entries of A equal zero except the entries in a principal submatrix lying on the j-th, k-th and l-th rows and columns (for some fixed 1 ≤ j < k < l ≤ n). Define 12 maps on Tn according to the forms A1 , A2 , A3 , B1 , . . . , D3 . For example, suppose A ∈ Tn has the following nonzero principal submatrix ⎛

ajj ⎝ 0 0

ajk akk 0

⎞ ajl akl ⎠ . all

The map A3 : Tn → Tn is such that A3 (A) ∈ Tn with all entries equal zero except those entries in a principal submatrix lying on the r-th, s-th and t-th rows and columns (for

170

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

some 1 ≤ r < s < t ≤ n). This principal submatrix has the form ⎛

all ⎝ 0 0

ajl ajj 0

⎞ akl ajk ⎠ . akk

Lemma 2.5. The maps A1 , A2 , A3 , C1 , C2 and C3 are c-numerical range preserving for all elements in Tn . If T : Tn → Tn is an additive c-numerical range preserving map satisfying (2), then its restriction to Tn cannot be equal to any one of the maps B1 , B2 , B3 , D1 , D2 and D3 . Remark. The readers can compare these maps with the maps in [15, Lemma 13]. Proof of Lemma 2.5. Note that maps A1 , A2 , A3 , C1 , C2 , C3 are related to maps generated by F and G, and hence they are c-numerical range preserving according to the “if” part of this theorem. The remaining step is to show that the maps of type B and type D are not the restriction of T mentioned in the lemma. Denote by M the principal submatrix of the first 3 rows and columns. Without loss of generality, assume the domains and ranges of all 12 maps are M. Note that each of B1 , B2 , B3 , D1 , D2 and D3 is the composition of the map τ : M → Tn , defined by ⎛

a11 ⎝ 0 0

a12 a22 0

⎞ ⎛ a13 a11 a23 ⎠ → ⎝ 0 a33 0

a12 a22 0

⎞ a13 a23 ⎠ ⊕ 0, a33

following C1 , C2 , C3 , A1 , A2 and A3 respectively. Thus the proof of this lemma is finished if we can show that the restriction to M of any additive c-numerical range T satisfying (2) cannot be equal to τ . 3π Assume on the contrary the restriction map is τ . For simplicity, write μ = e 4 i and π ν = e 4 i. Case 1: n= 3.    Let Z =

01ν 00 1 00 0

. Then T (Z) =

01ν ¯ 00 1 00 0

. According to Lemma 2.3,

    ∗  Wc μZ + (μZ)∗ = Wc μT (Z) + μT (Z) . By solving det(μZ + (μZ)∗ − λI) = 0, we have λ3 − 3λ = 0 √ √ λ = 3, 0, − 3. Hence the eigenvalues of μZ + (μZ)∗ are x1 =

√ √ 3, x2 = 0, and x3 = − 3.

(3)

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

171

Similarly, by solving det(μT (Z) + (μT (Z))∗ − λI) = 0, we have λ3 − 3λ + 2 = 0 λ = 1, 1, −2. Hence the eigenvalues of μT (Z) + (μT (Z))∗ are y1 = 1, y2 = 1, and y3 = −2. By (3) and property (VII), x1 c1 + x2 c2 + x3 c3 = y1 c1 + y2 c2 + y3 c3

and

x3 c1 + x2 c2 + x1 c3 = y3 c1 + y2 c2 + y1 c3 . These 2 equations imply c1 = c2 = c3 . This is contradictory to the fact that wc (·) is a norm. Case 2: n ≥ 4. The arguments are similar to Case 1. However, we let Z ∈ Tn be ⎛

⎞ 0 1 ν ⎝ 0 0 1 ⎠ ⊕ K ⊕ 0, 0 0 0 where K ∈ R and 0 ∈ Tn−4 . Since T satisfies (2), the number K will be mapped to a diagonal entry outside M. Note that (3) is still valid for this Z. By solving det(μZ + (μZ)∗ − λI) = 0, we have   λn−4 (λ − K) λ3 − 3λ = 0. √ √ Hence the eigenvalues of μZ + (μZ)∗ are x1 = 3, x2 = 0, x3 = − 3, K, and n − 4 zeroes. Similarly, by solving det(μT (Z) + (μT (Z))∗ − λI) = 0, we have   λn−4 (λ − K) λ3 − 3λ + 2 = 0. Hence the eigenvalues of μT (Z) + (μT (Z))∗ are y1 = 1, y2 = 1, y3 = −2, K, and n − 4 zeroes. Assume K = K1 < −2. By (3) and property (VII), x1 c1 + x2 c2 + x3 cn−1 + K1 cn = y1 c1 + y2 c2 + y3 cn−1 + K1 cn

and

K1 c1 + x3 c2 + x2 cn−1 + x1 cn = K1 c1 + y3 c2 + y2 cn−1 + y1 cn . √ Assume K = K2 and 0 > K2 > − 3. By (3) and property (VII) again,

172

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

x1 c1 + x2 c2 + K2 cn−1 + x3 cn = y1 c1 + y2 c2 + K2 cn−1 + y3 cn

and

x3 c1 + K2 c2 + x2 cn−1 + x1 cn = y3 c1 + K2 c2 + y2 cn−1 + y1 cn . The above 4 equations imply c1 = c2 = cn−1 = cn . This is contradictory to the fact that wc (·) is a norm. 2 We can further determine the form of T using information from the above lemma. Note that for any A = [ajk ] ∈ Tn , the diagonal entries of T (A) is a permutation of a11 , a22 , · · · , ann . Suppose Case I occurs, i.e., φ is the identity map. We look at the relative position of a11 and ann in the diagonal entries of T (A). We will say that an entry in the upper left corner is in front of an entry in the lower right corner. If a11 is in front of ann in the diagonal entries of T (A), because A1 is a c-numerical range preserving map while B1 and B2 cannot occur, all ajj (j = 2, 3, . . . , n − 1) lie in between a11 and ann . Similar considerations about the maps A1 and B1 show that all all ’s are arranged in ascending order. Thus T is the identity map. If ann is in front of a11 in the diagonal entries of T (A), then there is no other element lying between ann and a11 by the consideration of B3 . Since A2 and A3 are c-numerical range preserving, there is some p (1 ≤ p ≤ n) such that app , ap+1,p+1 , . . . , ann all lie before a11 while a11 , a22 , . . . , ap−1,p−1 all lie after ann . By the consideration of B1 and B2 , the diagonal entries have to be arranged as app , ap+1,p+1 , . . . , ann , a11 , a22 , . . . , ap−1,p−1 . This informs us that T is in the form (3) mentioned in the remark after Theorem 2.2. Thus T is generated by maps F and G. Suppose Case II occurs, i.e., φ is the transpose map. Similar arguments as above inform us that T is either in the forms (2) or (4) mentioned in the remark after Theorem 2.2. Hence T is also generated by maps F and G in this case. 2 3. c-Numerical radius isometries The main result in this section is the following theorem. Theorem 3.1. Suppose wc (·) is a norm on Mn and K = Mn or Tn . A mapping T : K → K satisfies   wc T (A) − T (B) = wc (A − B)

for all A, B ∈ K

if and only if there exist a fixed R ∈ Tn and λ ∈ C with |λ| = 1 such that λψ ◦ T + R ¯ is c-numerical range preserving on K, with ψ taking the forms ψ(X) = X or ψ(X) = X.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

173

n Let c0 = i=1 ci . We assume c0 > 0 in this section, otherwise we consider −c instead of c. We state two lemmas before we give the proof of the above theorem. The following lemma is analogous to characterizations of I and iI by their numerical radii [19, Lemma 1]. Lemma 3.2. Let K = Mn or Tn . Suppose X, Y ∈ K are such that wc (xX + yY ) ≤ c0 whenever x, y ∈ R satisfy x2 + y 2 ≤ 1. Then the following two conditions are equivalent. (a) There exists a complex unit λ such that (X, Y ) = λ(I, ±iI). (b) For any A ∈ K, there are x0 , y0 ∈ R with x20 + y02 = 1 and wc (x0 X + y0 Y + A) = c0 + wc (A). Proof. (a) ⇒ (b). Note that for any x, y ∈ R with x2 + y 2 ≤ 1, A ∈ K and U ∈ Un , | tr(CU (xX + yY + A)U ∗ )| ≤ | tr(CU (xX + yY )U ∗ )| + | tr(CUAU ∗ )| ≤ c0 + wc (A). Note that there exists V ∈ Un such that tr(CVAV ∗ ) = (a + ib)wc (A) for some a, b ∈ R with a2 +b2 = 1. Suppose Y = iλI. Let x0 +iy0 = λ(a +ib). Then it is straightforward to show that x20 + y02 = 1 and | tr(CV (x0 X + y0 Y + A)V ∗ )| = c0 + wc (A). If Y = −iλI, then we consider x0 −iy0 = λ(a +ib). We can also show that x20 +y02 = 1 and wc (x0 X +y0 Y +A) = c0 + wc (A). (b) ⇒ (a). We first prove the case for K = Mn . Claim 1. For any fixed V ∈ Un , | tr(CV ∗ XV )| = | tr(CV ∗ YV )| = c0 . Moreover, tr(CV ∗ YV ) = i tr(CV ∗ XV ) or tr(CV ∗ YV ) = −i tr(CV ∗ XV ). Proof of Claim 1. For any fixed V ∈ Un , define Vθ = eiθ VCV ∗ (∈ K) for any θ ∈ R. Then, by assumption, there exist xθ , yθ ∈ R with x2θ + yθ2 = 1 such that wc (xθ X + yθ Y + Vθ ) = c0 + wc (Vθ ). Note that          tr CU (xθ X + yθ Y + Vθ )U ∗  ≤ tr CU (xθ X + yθ Y )U ∗  + tr CUV θ U ∗  ≤ c0 + wc (Vθ ) for any U ∈ Un . Hence there exist Uθ ∈ Un and γ ∈ C with |γ| = 1 such that   tr CU θ (xθ X + yθ Y )Uθ∗ = γc0

(4)

  tr CU θ Vθ Uθ∗ = γwc (Vθ )   tr CU θ VCV ∗ Uθ∗ = e−iθ γwc (C).

(5)

and

Note that tr(AB) = A, B ∗  is the usual inner product on Mn . Hence, by (5) and the Cauchy–Schwarz inequality,

174

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

   1 1 wc (C) =  C, Uθ VCV ∗ Uθ∗  ≤ C, C 2 · Uθ VCV ∗ Uθ∗ , Uθ VCV ∗ Uθ∗ 2   = tr C 2 ≤ wc (C). Hence C and Uθ VCV ∗ Uθ∗ are linearly dependent. Thus there exists μ ∈ C such that C = μUθ VCV ∗ Uθ∗ . By taking trace on both sides of this equation, we see that μ = 1. Hence Uθ∗ CU θ = VCV ∗ .

(6)

From (5) and (6), we can deduce that γ = eiθ . Let tr(CV ∗ XV ) = x1 + ix2 c0

and

tr(CV ∗ YV ) = y1 + iy2 , c0

(7)

where x1 , x2 , y1 , y2 ∈ R. By (4),   tr CU θ (xθ X + yθ Y )Uθ∗ = eiθ c0   tr CV ∗ (xθ X + yθ Y )V = eiθ c0 (by (6))     ∗ ∗ −1 iθ xθ c−1 0 tr CV XV + yθ c0 tr CV YV = e      x1 y1 xθ cos θ = . x2 y2 yθ sin θ Note that for any (cos θ, sin θ)t ∈ R2 , we can find (xθ , yθ )t ∈ R2 with x2θ + yθ2 = 1 x y  satisfying the above equation. The mapping Λ induced by the matrix x12 y12 is surjective and hence bijective, it maps the unit sphere of R2 onto itself. Hence Λ is an isometry of R2 with the form     cos ρ − sin ρ cos ρ sin ρ Λ= or Λ = (8) sin ρ cos ρ sin ρ − cos ρ ∗

for some ρ ∈ R. Hence | tr(CVc0 XV ) | = |x1 + ix2 | = | cos ρ + i sin ρ| = 1. Similarly, tr(CV ∗ YV ) = − sin ρ c0 −i tr(CVc0 XV ) . Thus Claim 1

|tr(CV ∗ YV )| = c0 . Moreover, by (7) and (8), we have either ∗

i cos ρ = i tr(CVc0 XV ) or proved. 2

∗

tr(CV YV ) c0

= sin ρ − i cos ρ =

∗

+ is

Since V is an arbitrary unitary matrix, all elements in Wc (X) have the constant modulus c0 . As Wc (X) is a convex set in C, X is a singleton set. Therefore, X = λI for some complex unit λ by property (V). Hence Y = ±iλI. Now we prove the case for K = Tn . Note that for general V ∈ Un , eiθ VCV ∗ may not be in Tn . But if V is a permutation matrix, then VCV ∗ ∈ Tn . Thus the arguments in Claim 1 can be used to prove the following facts.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

175

Claim 2. For any fixed permutation P ∈ Mn , (i) | tr(CP t XP)| = | tr(CP t YP)| = c0 . (ii) tr(CP t YP) = i tr(CP t XP) or tr(CP t YP) = −i tr(CP t XP). Our next task is to show that X and Y are diagonal matrices. We adapt some ideas in [18, Lemma 4.2]. For any matrix Z ∈ Mn , let Z[p, q] ∈ M2 be the principal submatrix of Z lying in the p-th and q-th rows and columns. Suppose xkl = 0 for some k < l. We choose a permutation matrix P so that X  = t P XP = [xij ] differs from X by interchanging the first row and column with those of the k-th, and also interchanging the l-th row and column with those of the n-th. Thus we n have x11 = xkk , xnn = xll , x1n = xkl , xn1 = 0. Note that | j=1 cj xjj | = c0 by Claim 2. Let c = (c1 , cn ). For any Z in M2 , it is not hard to see that Wc (Z) = (c1 − cn )W (Z) + cn tr(Z).

(9)

We can deduce from [16, (1.6)] that Wc (X  [1, n]) is an elliptical disk with c1 x11 +cn xnn    as one of the foci. Thus there exists a unitary U ∈ M2 such that tr c01 c0n U X  [1, n]U ∗ is a complex number c1 x11 +cn xnn +z (around the focus c1 x11 +cn xnn ) with the property that     n n           cj xjj + z  >  cj xjj .      j=1

j=1

Let V ∈ Mn be obtained from the identity matrix I by replacing I[1, n] with U . Then   n   n              c0 ≥ wc (X) = wc X ≥ tr CVX  V ∗  =  cj xjj + z  >  cj xjj  = c0 ,     j=1

j=1

which is a contradiction. Therefore, X is a diagonal matrix. Similar arguments show that Y is also a diagonal matrix.   The next task is to show that x11 = xnn and y11 = ynn . Let B = 00 20 ∈ M2 . Consider the matrix A ∈ Mn with all entries equal 0 except a1n = 2. We have Wc (I + A) = c0 + Wc (A)    = c0 + conv ∪ Wc1 (B) + Wc2 (0) : ct1 , ct2 = ct P,  P a permutation matrix (by [16, (4.2)])   (by (9)) = c0 + conv ∪ (cj − ck )W (B) : cj , ck are entries of c = c0 + (c1 − cn )W (B),

176

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

where the last equality follows from the fact that W (B) is a circular disk with center 0 and radius 1 (see, e.g., Example 1 in [10, Chapter 1]). Thus Wc (I + A) is a circular disk with center c0 and radius c1 − cn . By the above information, for any θ ∈ R, eiθ (I + A) attains its c-numerical radius at some V ∈ Un (depending on θ) if and only if eiθ (c0 + c1 − cn ) = tr(CV (eiθ (I + A))V ∗ ). Let V = [vpq ] ∈ Un and vk (k = 1, 2, . . . , n) be the column vectors of Vθ , i.e. vk = (v1k , v2k , . . . , vnk )t . Then   c0 + c1 − cn = tr CV (I + A)V ∗ = c0 + 2  c1 − cn = 2 Re

n 

cj vj1 vjn

j=1 n 

 cj vj1 vjn .

j=1

Let P = {j ∈ {1, 2, . . . , n} : Re(vj1 vjn ) > 0} and N = {j ∈ {1, 2, . . . , n} : Re(vj1 vjn ) < 0}. Then 

c1 − cn = 2

cj Re(vj1 vjn ) +

j∈P



 cj Re(vj1 vjn )

j∈N

    Re(vj1 vjn ) + cn Re(vj1 vjn ) ≤ 2 c1 j∈P

= 2(c1 − cn )

j∈N



Re(vj1 vjn ) + cn

j∈P

= 2(c1 − cn )





Re(vj1 vjn ) +

j∈P



 Re(vj1 vjn )

j∈N

Re(vj1 vjn ) + cn Rev1 , vn 

j∈P

= 2(c1 − cn )



Re(vj1 vjn ).

(10)

j∈P

Thus  1 ≤ Re(vj1 vjn ). 2

(11)

j∈P

Since Rev1 , vn  = 0, we also have  j∈N

1 Re(vj1 vjn ) ≤ − . 2

(12)

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

177

Let wn = (w1n , . . . , wnn )t ∈ Cn be such that  wjn =

vjn , −vjn ,

if j ∈ P, if j ∈ / P,

for j = 1, 2, . . . , n. Note that wn is of norm one. By (11) and (12), 

1≤

Re(vj1 vjn ) −

j∈P



Re(vj1 vjn ) = Rev1 , wn 

j∈N

  ≤ v1 , wn  ≤ v1 wn  = 1. Thus all the above inequalities become equalities. Therefore, wn = v1 , and j∈N vj1 vjn = − 12 . As (10) becomes an equality, we also have j∈P

⇒

cj = c1

and j ∈ N

⇒

j∈P

vj1 vjn =

1 2

cj = cn .

In summary, eiθ (I + A) attains its c-numerical radius at V = [vpq ] ∈ Un if and only if V satisfies the following conditions: (i) vj1 = 0 only if cj = c1 or cj = cn ; 1 1 2 2 (ii) and j∈{1,2,...,n} |vj1 | = j∈{1,2,...,n} |vj1 | = 2 2; cj =c1 cj =cn  vj1 , if cj =c1 , (iii) vjn = −v j1 , if cj =c1 . By condition (b) of this lemma, there exist cos φ and sin φ (both depending on θ) such that the c-numerical radius of the matrix cos φX + sin φY + eiθ (I + A) is c0 + wc (I + A). Suppose this matrix attains its c-numerical radius at some unitary V depending on θ and φ. Then V has the three properties mentioned above and   tr CV (cos φX + sin φY )V ∗ = eiθ c0 .

(13)

Let vm ∈ Cn (m = 1, 2, . . . , n) be the column vectors of V = [vpq ]. Let zm ∈ Cn (m = 1, 2, . . . , n) be such that

zm

⎧ 1 ⎪ ⎨ √2 (v1 + vn ), if m = 1, if m = 1, n, = vm , ⎪ ⎩ √1 (v1 − vn ), if m = n. 2

Let ! (1) " Z1 = zij = [z1 , z2 , · · · , zn−1 , zn ] and ! (2) " Z2 = zij = [zn , z2 , z3 , · · · , zn−2 , zn−1 , z1 ].

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

178

It is easy to check that Z1 and Z2 are unitary matrices. Note that the j-th entry of z1 is non-zero only if cj = c1 . The j-th entry of zn is non-zero only if cj = cn . Let the (i, i)-th entry of the diagonal matrices X, Y and cos φX + sin φY be xi , yi and di respectively. By (13),    c0 = tr CV (cos φX + sin φY )V ∗    n  n     2  = cj dk |vjk |    j=1 k=1    n n      = dk cj |vjk |2    j=1 k=1     n  n−1  n  n        2 2 2  = d1 cj |vj1 | + dn cj |vjn | + dk cj |vjk |    j=1 j=1 j=1 k=2    n−1 n d c   d1 cn dn c1 dn cn   1 1  = + + + + dk cj |vjk |2   2  2 2 2 j=1 k=2   n   n−1   1  2 =  d1 c1 + dk cj |vjk | + dn cn 2 j=1 k=2   n   n−1     2 + d1 cn + dk cj |vjk | + dn c1   j=1 k=2   n   n  n n   (1) 2    (2) 2  1  c0 =  dk cj zjk  + dk cj zjk    2 j=1

k=1

k=1

j=1

   1  = tr CZ1 (cos φX + sin φY )Z1∗ + tr CZ2 (cos φX + sin φY )Z2∗  2  1    1  ≤ tr CZ1 (cos φX + sin φY )Z1∗  + tr CZ2 (cos φX + sin φY )Z2∗  2 2 ≤ c0 . By (13) and the above calculation, we have     eiθ c0 = tr CZ1 (cos φX + sin φY )Z1∗ = tr CZ2 (cos φX + sin φY )Z2∗ . From the second part of (14), d1 c1 +

n−1  k=2

 dk

n 

 cj |vjk |

2

+ dn cn = d1 cn +

j=1

n−1 

 dk

k=2

d1 = dn cos φ(x1 − xn ) = sin φ(yn − y1 ).

n  j=1

 cj |vjk |

2

+ dn c1

(14)

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

179

We prove by contradiction that x1 = xn and y1 = yn . Suppose this is not the case, then each of cos φ and sin φ can have at most 2 distinct values. (1) (1) Since Z1 = [zpq ] is unitary, [|zij |2 ] is a doubly stochastic matrix. By the celebrated Birkhoff’s theorem ([4], [25, Theorem 4.21]), there exist permutation matrices Pk , k = m (1) 1, . . . , m for some positive integer m, λk > 0, and k=1 λk = 1 such that [|zij |2 ] = m k k=1 λk Pk . Let Pk = [pij ] for k = 1, . . . , m. The (i, i)-th entry of the matrix Z1 (cos φX + ∗ sin φY )Z1 is equal to n 

n  (1) 2    dj zij = dj

j=1

j=1



m 

 λk pkij

=

k=1

m  k=1

 λk

n 

 dj pkij .

j=1

m

Hence the matrices k=1 λk Pk (cos φX + sin φY )Pkt and Z1 (cos φX + sin φY )Z1∗ have the same main diagonal. Therefore, 

 

 ∗

tr CZ1 (cos φX + sin φY )Z1 = tr C

m 

 λk Pk (cos φX + sin φY )Pkt

k=1

=

m 

     λk cos φ tr CP k XP tk + sin φ tr CP k YP tk .

k=1

By the above equation and the first part of (14),   eiθ c0 = tr CZ1 (cos φX + sin φY )Z1∗ =

m 

     λk cos φ tr CP k XP tk + sin φ tr CP k YP tk

k=1

=

m 

   λk (cos φ + iδk sin φ) tr CP k XP tk ,

k=1

where δk = 1 or −1. The last equality follows from Claim 2 (ii). By Claim 2 (i), all numbers (cos φ + iδk sin φ) tr(CP k XP tk ) (k = 1, . . . , m) have moduli c0 . Since the modulus of their convex combination is also c0 , so all these numbers are the same. Hence   eiθ c0 = (cos φ + iδ1 sin φ) tr CP 1 XP t1 . As θ can be any number between 0 and 2π, the left hand side of the above equation can assume infinitely many values. However, the right hand side can only assume a finite number of values because there are a finite number of choices for each of cos φ, sin φ, δ1 and P1 . We reach a contradiction. Hence x1 = xn and y1 = yn . For a fixed integer j with 2 ≤ j ≤ n − 1, let Pj be the permutation matrix such that pkk = 1 for k = j, n, pjn = 1 and pnj = 1. For any θ ∈ R, there exist cos φ and sin φ (both depending on θ) such that

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

180

    wc cos φX + sin φY + Pj eiθ (I + A)Pj = c0 + wc Pj eiθ (I + A)Pj   wc cos φX  + sin φY  + eiθ (I + A) = c0 + wc (I + A), where X  = Pj XPj and Y  = Pj Y Pj . We can use arguments similar to those above to show that x1 = xj and y1 = yj . Hence X and Y are scalar matrices. By Claim 2, we have (X, Y ) = λ(I, ±iI). 2 The following lemma is an adaptation of [22, Lemma 2] to the case that T is real linear (see also [19, Lemma 4]). We refer the readers to the mentioned papers for its proof. Lemma 3.3. Let K = Mn or Tn . If T : K → K is a real linear map satisfying (T (I), T (iI)) = (I, iI) and wc (T (A)) = wc (A) for all A ∈ K, then Wc (T (A)) = Wc (A) for all A ∈ K. Now we are ready to give the proof of Theorem 3.1. Proof. Assume K = Mn . The proof for K = Tn is similar. (⇐) Direct computation. (⇒) Let η(A) = T (A) − T (0). Then wc (η(A)) = wc (A) for every A ∈ Mn and η(0) = 0. It follows from a theorem of Charzyński ([6], [3, p. 500]) that η is real linear. Without loss of generality, we replace η by T . Note that (I, iI) satisfies Lemma 3.2 (b). By the assumption on T , the pair of matrices T (I), T (iI) also satisfies the condition in Lemma 3.2 (b), hence T (I) = λI and T (iI) is equal to either λiI or −λiI, where λ is a complex unit. Without loss of generality, we replace T by λT . Suppose T (iI) = iI. By Lemma 3.3 and Theorem 2.1, T has one of the forms stated in this theorem with ψ being the identity function. Suppose T (iI) = −iI. Then the real linear operator L defined as ¯ will satisfy L(I) = I, L(iI) = iI and wc (L(A)) = wc (A) for all A ∈ Mn . L(A) = T (A) By Lemma 3.3 and Theorem 2.1 again, we can deduce the form of L. Therefore T has ¯ 2 one of the forms stated in this theorem with ψ(X) = X. Acknowledgements The author is grateful to Dr. Jor-Ting Chan for his constant guidance and encouragement, and to the referee for numerous valuable comments leading to the substantial improvement of the manuscript. References [1] Z.F. Bai, J.C. Hou, Numerical radius distance-preserving maps on B(H), Proc. Amer. Math. Soc. 132 (2004) 1453–1461. [2] Z.F. Bai, J.C. Hou, Z.B. Xu, Maps preserving numerical radius distance on C ∗ -algebras, Studia Math. 162 (2004) 97–104.

K. Chan / Linear Algebra and its Applications 466 (2015) 160–181

181

[3] R. Bhatia, P. Šemrl, Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997) 497–504. [4] G. Birkhoff, Tres observaciones sobre el algebra lineal, Univ. Nac. Tucuman Rev. Ser. A 5 (1946) 147–150. [5] J.T. Chan, C.K. Li, N.S. Sze, Isometries for unitarily invariant norms, Linear Algebra Appl. 399 (2005) 53–70. [6] Z. Charzyński, Sur les transformations isométriques des espaces du type (F), Studia Math. 13 (1953) 94–121. [7] W.S. Cheung, C.K. Li, Linear operators preserving generalized numerical ranges and radii on certain triangular algebras of matrices, Canad. Math. Bull. 44 (2001) 270–281. [8] J.L. Cui, J.C. Hou, Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras, J. Funct. Anal. 206 (2004) 414–448. [9] M. Gonçalves, A. Sourour, Isometries of a generalized numerical radius, Linear Algebra Appl. 429 (2008) 1478–1488. [10] K.E. Gustafson, D.K.M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Springer-Verlag, New York, 1997. [11] A. Guterman, C.K. Li, P. Šemrl, Some general techniques on linear preserver problems, Linear Algebra Appl. 315 (2000) 61–81. [12] P.R. Halmos, A Hilbert Space Problem Book, 2nd ed., Springer-Verlag, New York, 1982. [13] L. Hogben (Ed.), Handbook of Linear Algebra, Chapman & Hall/CRC, Boca Raton, Florida, 2007. [14] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991. [15] G. Lešnjak, Additive preservers of numerical range, Linear Algebra Appl. 345 (2002) 235–253. [16] C.K. Li, C-numerical ranges and C-numerical radii, Linear Multilinear Algebra 37 (1994) 51–82. [17] C.K. Li, A survey on linear preservers of numerical ranges and radii, Taiwanese J. Math. 5 (2001) 477–496. [18] C.K. Li, Y.T. Poon, N.S. Sze, Linear maps transforming the higher numerical ranges, Linear Algebra Appl. 400 (2005) 291–311. [19] C.K. Li, P. Šemrl, Numerical radius isometries, Linear Multilinear Algebra 50 (2002) 307–314. [20] C.K. Li, N.K. Tsing, Linear operators that preserve the c-numerical range or radius of matrices, Linear Multilinear Algebra 23 (1988) 27–46. [21] C.K. Li, N.K. Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162–164 (1992) 217–235. [22] W.Y. Man, The invariance of C-numerical range, C-numerical radius and their dual problems, Linear Multilinear Algebra 30 (1991) 117–128. [23] S. Pierce, et al., A survey of linear preserver problems, Linear Multilinear Algebra 33 (1992) 1–129. [24] N.S. Sze, Linear operators preserving numerical ranges and radii on certain triangular matrices, Master’s thesis, The University of Hong Kong, 2002. [25] F. Zhang, Matrix Theory: Basic Results and Techniques, Springer-Verlag, New York, 1999.