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Maps preserving the local spectrum of skew-product of operators
Linear Algebra and its Applications 485 (2015) 58–71
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Maps preserving the local spectrum of skew-product of operators Z. Abdelali ∗ , A. Achchi, R. Marzouki University Mohammed V, Faculty of Sciences, Department of Mathematics, Rabat, Morocco
a r t i c l e
i n f o
Article history: Received 18 May 2015 Accepted 17 July 2015 Available online xxxx Submitted by P. Semrl MSC: primary 47B49 secondary 47A10, 47A11 Keywords: Nonlinear preservers Local spectrum
a b s t r a c t Let H and K be infinite dimensional complex Hilbert spaces and let B (H ) be the algebra of all bounded linear operators on H . Let σT (h) denote the local spectrum of an operator T ∈ B (H ) at any vector h ∈ H , and fix two nonzero vectors h0 ∈ H and k0 ∈ K . We show that if a map ϕ : B (H ) → B (K ) has a range containing all operators of rank at most two and satisfies σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 ) for all T, S ∈ B (H ), then there exist two unitary operators U and V in B (H , K ) such that U h0 = αk0 for some nonzero α ∈ C and ϕ(T ) = U T V ∗ for all T ∈ B (H ). We also described maps ϕ : B (H ) → B (K ) satisfying σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) for all T, S ∈ B (H ), and with the range containing all operators of rank at most four. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (Z. Abdelali), [email protected] (A. Achchi), [email protected] (R. Marzouki). http://dx.doi.org/10.1016/j.laa.2015.07.019 0024-3795/© 2015 Elsevier Inc. All rights reserved.
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1. Introduction Several results on linear or additive preserver problems have been extended to the setting of nonlinear preservers, and, in many cases, their extensions demonstrated to be nontrivial. In particular, the problem of characterizing maps preserving certain functions, subsets, relations and properties of product of matrices or operators has attracted the attention of several authors; see for instance [3,11,14,15,20,17,21] and the references therein. Motivated by problems concerning local automorphisms, L. Molnár characterized, in [17], maps preserving different spectra of operator or matrix products, and his results have been extended by several authors. In [15], J. Hou and Q.H. Di described, in particular, maps preserving the numerical range of products of Hilbert space operators. In [20], maps preserving the nilpotency of operator products are characterized. T. Miura and D. Honma gave in [16] a generalization of peripherally-multiplicative surjections between standard operator algebras. In [2], G. An and J. Hou described multiplicative maps ϕ on B (H ), the algebra of all bounded linear operators on a complex Hilbert space H , that satisfy T ∗ S = 0 if and only if ϕ(T )∗ ϕ(S) = 0 for all S, T ∈ B (H ). While in [22], W. Zhang and J. Hou characterized maps preserving peripheral spectrum of Jordan semi-triple product T S ∗ T of operators. The problem of characterizing linear or additive maps on B (X), the algebra of all bounded linear operators on a complex Banach space X, preserving local spectra was initiated by A. Bourhim and T.J. Ransford in [9], and continued by several authors; see for instance [8,10–13] and the references therein. In [10], J. Bračič and V. Müller characterized surjective linear and continuous mappings on B (X) preserving the local spectrum and local spectral radius at a fixed nonzero vector x0 of X, and thus extending the main results of [8] to infinite-dimensional Banach spaces. While in [13], C. Costara showed, in particular, that every surjective linear map on B (X) preserving the local spectral radius at a fixed nonzero vector of X is automatically continuous. Furthermore, in [12], C. Costara described surjective linear maps on B (X) which preserve operators of local spectral radius zero at points of X. His result has been extended in [4] to the nonlinear setting where A. Bourhim and J. Mashreghi gave a complete description of surjective (not necessarily linear) maps on B (X) preserving the difference of operators with local spectral radius zero at points of X. In [5], A. Bourhim and J. Mashreghi showed that surjective map ϕ on B (X) preserving the local spectrum of product of operators at a fixed nonzero vector x0 ∈ X if and only if ϕ is a positive or negative multiple automorphism and x0 is an eigenvector of the intertwining operator. In [6], they completely describe the form of surjective maps on B (X) preserving the local spectrum of triple product of operators at a nonzero fixed vector of X. To have a good chronological handling of the preserving problems, see the excellent survey by A. Bourhim and J. Mashreghi [7]. Let H and K be two infinite-dimensional complex Hilbert spaces. Let B (H ) be the algebra of all bounded linear operators on H . In this paper, we follow the same path of studies by considering general local spectra preservers, and characterize maps
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ϕ : B (H ) → B (K ) that preserves the local spectrum at fixed nonzero vector of the skew-double product T S ∗ , and under the mild assumption on the ranges of the maps ϕ contain operators with rank at most two. In particular, we will show that such a map ϕ is a bijective isometry of the form ϕ(T ) = U T V ∗ (T ∈ B (H )) with U h0 = αk0 for some nonzero scalar α. We also obtain a similar result by characterizing maps ϕ preserving the local spectrum at fixed nonzero vector of the skew-triple product T S ∗ T . Our proofs are influenced by ideas from [5,6] but they use new ingredients that will be established in Section 4 and Section 5. 2. Main results Throughout this paper, H and K are two infinite-dimensional complex Hilbert spaces. As usual B (H , K ) denotes the space of all bounded linear operators from H into K . When H = K we simply write B (H ) instead of B (H , H ). The inner product of H or K will be denoted by , if there is no confusion. Let F (H ) denote the ideal of all finite rank operators on H . For a positive integer n, let Fn (H ) be the set of all operators of B (H ) of rank at most n. For an operator T ∈ B (H , K ), let T ∗ denote as usual its adjoint (T h, k = h, T ∗ k, (h, k) ∈ H × K ). The local resolvent set, ρT (x), of an operator T ∈ B (H ) at a point x ∈ H is the union of all open subsets U of the complex plane C for which there is an analytic function φ : U → H such that (T − λ)φ(λ) = x (λ ∈ U ). Clearly ρT (x) contains the resolvent set ρ(T ) of T , but this containment could be proper. The local spectrum of T at x is defined by σT (x) := C\ρT (x), and thus it is a closed subset (possibly empty) of σ(T ), the spectrum of T . In fact, for every operator T ∈ F (H ), σT (x) = ∅ for all nonzero vectors x in H . The books by P. Aiena [1] and by K.B. Laursen, M.M. Neumann [19] provide an excellent exposition as well as a rich bibliography of the local spectral theory. The following theorems are the main results of this paper. Their proofs will be presented in Section 4 and Section 5. Theorem 2.1. Let h0 ∈ H and k0 ∈ K be two fixed nonzero vectors and ϕ : B (H ) → B (K ) a map such that its range contains F2 (K ). Then ϕ satisfies σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 )
(T, S ∈ B (H )),
(2.1)
if and only if there exist two unitary operators U and V in B (H , K ) and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T V ∗ for all T ∈ B (H ).
(2.2)
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The following result is a variant of the above one. Theorem 2.2. Let h0 ∈ B (H ) and k0 ∈ B (K ) be two fixed nonzero vectors ϕ : B (H ) → B (K ) a map such that its range contains F4 (K ). Then ϕ satisfies σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) (T, S ∈ B (H )),
(2.3)
if and only if there exist a unitary operator U : H → K and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T U ∗
(2.4)
for all T ∈ B (H ). Remark 2.3. (1) If H and K are isomorphic, then they are isometrically isomorphic. Using U ϕU ∗ instead of ϕ for an appropriate unitary operator U : K → H , the statements of Theorems 2.1 and 2.2 can be reduced to the case when H = K and h0 = k0 . Although, the fact that “H and K are isomorphic” is one of the conclusions of the main results. (2) We point out that the description of maps ψ from B (H ) into B (K ) satisfying F2 (K ) ⊆ ψ(B (H )) and σT ∗ S (h0 ) = σψ(T )∗ ψ(S) (k0 ) (resp. F4 (K ) ⊆ ψ(B (H )) and σT ∗ ST ∗ (h0 ) = σψ(T )∗ ψ(S)ψ(T )∗ (k0 )), for all T, S ∈ B (H ), can be obtained by applying Theorem 2.1 (resp. Theorem 2.2) to ϕ given by ϕ(T ) = ψ(T ∗ )∗ , T ∈ B (H ). 3. Preliminaries In this section, we fix some notions and exhibit some tools on the local spectral theory and some essential results needed for the proof of our main results. The first lemma summarizes some basic properties of the local spectrum that will be often used in the sequel. Lemma 3.1. For an invertible operator A ∈ B (H ), a vector x ∈ H and a nonzero scalar α ∈ C, the following statements hold. (1) σT (αx) = σT (x) and σαT (x) = ασT (x) for all T ∈ B (H ). (2) σAT A−1 (Ax) = σT (x) for all T ∈ B (H ). For a nonzero h0 ∈ H and an operator T ∈ B (H ), we use a useful notation defined by A. Bourhim and J. Mashreghi in [5,6] by
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σT∗ (h0 )
:=
{0}
if σT (h0 ) = {0}
σT (h0 ) \ {0}
if σT (h0 ) = {0}.
(3.1)
Therein, they also gave some essential lemmas and theorems which are useful tools to establish our main results. For our purpose, we state these results only in the case of Hilbert spaces. Lemma 3.2. (See [5,6].) Let h0 be a nonzero vector in H . For any vectors x, y and z in H , the following statements hold. (1) ∗ σx⊗z (h0 )
=
{0}
if h0 , z = 0
{x, z}
if h0 , z = 0.
∗ ∗ ∗ (2) σ(x+y)⊗z (h0 ) = σx⊗z (h0 ) + σy⊗z (h0 ). (3) For all rank one operators R and all T, S ∈ B (H ), we have ∗ ∗ ∗ • σ(T +S)R (h0 ) = σT R (h0 ) + σSR (h0 ), and ∗ ∗ ∗ • σR(T +S)R (h0 ) = σRT R (h0 ) + σRSR (h0 ).
Next lemma provides some local spectral identity principles. Lemma 3.3. (See [5, Theorem 3.2] and [6, Theorem 3.1].) For a nonzero vector h0 in H and two operators A and B in B (H ), the following statements are equivalent. (1) (2) (3) (4) (5) (6) (7)
A = B. σAT (h0 ) = σBT (h0 ) for all T ∈ B (H ). σAT (h0 ) = σBT (h0 ) for all T ∈ F1 (H ). ∗ ∗ σAT (h0 ) = σBT (h0 ) for all T ∈ F1 (H ). σT AT (h0 ) = σT BT (h0 ) for all T ∈ B (H ). σT AT (h0 ) = σT BT (h0 ) for all T ∈ F1 (H ). σT∗ AT (h0 ) = σT∗ BT (h0 ) for all T ∈ F1 (H ).
Finally, we close this section with the following lemma that gives a local spectral characterization of rank one operators of B (H ). Lemma 3.4. (See [5, Theorem 4.1] and [6, Theorem 3.2].) Let R be a nonzero bounded linear operator on H , and h0 a nonzero vector in H . Then the following statements are equivalent. (1) R has rank one. ∗ (2) σRT (h0 ) contains at most one element for all T ∈ B (H ). ∗ (3) σRT (h0 ) contains at most one element for all T ∈ F2 (H ).
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(4) σT∗ RT (h0 ) is a singleton for all operators T ∈ B (H ). (5) σT∗ RT (h0 ) contains at most one element for all T ∈ F4 (H ). Notice that the equivalence with the fifth statement is clear by the proof of [6, Theorem 3.2]. 4. Proof of Theorem 2.1 For the proof of Theorem 2.1, we need the following essential lemmas. Lemma 4.1. Let h0 ∈ H and k0 ∈ K be two nonzero fixed vectors and A and B be two bijective linear operators from H into K , and ϕ : F1 (H ) → F1 (K ) the map defined by ϕ(x ⊗ y) := Ax ⊗ By for all x, y ∈ H . If σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 )
(4.1)
for all rank one operators T and S in B (H ), then A and B are bounded unitary operators multiplied by positive scalars λ and μ such that λμ = 1. Moreover, Ah0 = αk0 for some nonzero scalar α ∈ C. Proof. Let x, y, u and v be four vectors in H , and let us first show that v, yx, u = Bv, ByAx, Au.
(4.2)
Note that (4.1) applied to x ⊗ y and u ⊗ v entails that v, yσx⊗u (h0 ) = σv,y(x⊗u) (h0 ) = σBv,By(Ax⊗Au) (k0 ) = Bv, ByσAx⊗Au (k0 ),
(4.3)
and let us show that h0 , u = 0 ⇐⇒ k0 , Au = 0.
(4.4)
Indeed, if h0 , u = 0 and k0 , Au = 0, then (4.3), applied when x = y = v = h0 , implies that ∗ ∗ (h0 ) = σBh (k0 ) = {0}. h0 2 h0 , u = σh 0 ,h0 (h0 ⊗u) 0 ,Bh0 (Ah0 ⊗Au)
This arises a contradiction and shows that if k0 , Au = 0 then h0 , u = 0. Conversely, if h0 , u = 0 and k0 , Au = 0, apply (4.3) when x = A−1 k0 and y = v = B −1 k0 so that ∗ ∗ {0} = σB k0 2 k0 , Au . −1 k ,B −1 k (A−1 k ⊗u) (h0 ) = σk ,k (k ⊗Au) (k0 ) = 0 0 0 0 0 0
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This contradiction shows that if h0 , u = 0 then k0 , Au = 0. Therefore, (4.4) is established. By (4.4) and the fact that ∗ σx⊗u (h0 )
=
{0}
if h0 , u = 0
{x, u} if h0 , u = 0,
(4.5)
we see that (4.2) holds provided that h0 , u = 0. Now, if h0 , u = 0, note that h0 , u + h0 = h0 2 = 0 and apply (4.2) twice when u is replaced by h0 and by h0 + u. We get v, yx, h0 = Bv, ByAx, Ah0 , and v, yx, h0 + u = Bv, ByAx, A(h0 + u). By expanding the second identity and using the first one, we deduce that (4.2) holds in this case too. 1 Now, let v0 be a fixed unit vector in H and let λ = Bv . By (4.2), we have 0 Ax 2 = Ax, Ax = λ2 x, x = λ2 x 2 for all x ∈ H . Hence, A/λ is an isometry and thus it is a unitary operator in B (H , K ) since it is supposed to be bijective. By the same way, fix a unit vector u0 in H and take 1 μ = Au , and note that B/μ is a unitary operator in B (H , K ). Finally, by (4.2), we 0 see that λμ = 1. To finish the proof of this lemma, let us show that h0 and A−1 k0 are linearly dependent. If not, there exist a vector u ∈ H such that h0 , u = 1 and A−1 k0 , u = 0, and by Lemma 3.2 ∗ ∗ {1} = σh∗0 ⊗u (h0 ) = σ(h (h0 ) = σ(h ∗ (h0 ) 0 ⊗u)(h0 ⊗u) 0 ⊗u)(u⊗h0 ) ∗ ∗ = σϕ(h ∗ (k0 ) = σ(Ah ⊗Bu)(Au⊗Bh )∗ (k0 ) 0 ⊗u)ϕ(u⊗h0 ) 0 0 ∗ ∗ (k0 ) = σBh (k0 ) = σ(Ah 0 ⊗Bu)(Bh0 ⊗Au) 0 ,Bu(Ah0 ⊗Au) ∗ ∗ −1 k0 ) = σBh ∗ (k0 ) = σλ2 Bh ,Bu(h ⊗u) (A 0 ,BuA(h0 ⊗u)A 0 0
= {0}. This contradiction shows that h0 and A−1 k0 are linearly dependent; as desired. 2 Lemma 4.2. Let h0 ∈ H and k0 ∈ K be two nonzero fixed vectors, and let C and D be two bijective linear operators from H to K , and ϕ : F1 (H ) → F1 (K ) the map defined by ϕ(x ⊗ y) = Cy ⊗ Dx for all x, y ∈ H . Then there are rank one operators T, S ∈ F1 (H ) such that σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 ).
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Proof. Assume by the way of contradiction that σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 ) for all rank one operators T, S ∈ F1 (H ). Choose a nonzero vector y1 ∈ K such that k0 , y1 = 0. Since the vectors x = C −1 y1 and h0 are nonzero, there exists a nonzero y ∈ H such that h0 , y = 0 and x, y = 1. By Lemma 3.2, and since x ⊗ y is idempotent, we have ∗ ∗ {1} = σx⊗y (h0 ) = σ(x⊗y)(x⊗y) (h0 ) ∗ ∗ = σ(x⊗y)(y⊗x) ∗ (h0 ) = σ(Cy⊗Dx)(Cx⊗Dy)∗ (k0 ) ∗ (k0 ) = σ(Cy⊗Dx)(Dy⊗Cx) ∗ (k0 ) = σDy,Dx(Cy⊗Cx) ∗ (k0 ) = {0}. = σDy,Dx(Cy⊗y 1)
This yields to a contradiction and the lemma is therefore proved. 2 Now, we are in a position to prove Theorem 2.1. Proof of Theorem 2.1. Assume that there exist two unitary operators U and V in B (H , K ) and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T V ∗ for all T ∈ B (H ). We have σϕ(T )ϕ(S)∗ (k0 ) = σU T V ∗ V S ∗ U ∗ (k0 ) = σU T S ∗ U ∗ (k0 ) = σT S ∗ (h0 ) for all T, S ∈ B (H ), and (2.1) is established. Conversely, assume that ϕ satisfies (2.1) and let us show that ϕ takes the desired form. The proof breaks down into three steps. Step 1. ϕ is a one to one map preserving rank one operators in both directions. We first show ϕ is a one to one map and ϕ(0) = 0. Take two operators A, B ∈ B (H ) such that ϕ(A) = ϕ(B), and note that σAS ∗ (h0 ) = σϕ(A)ϕ(S)∗ (k0 ) = σϕ(B)ϕ(S)∗ (k0 ) = σBS ∗ (h0 ) for all S ∈ B (H ). Lemma 3.3 tells us that A = B, and thus ϕ is a one to one. Second, we show that ϕ(0) = 0. To do that, just observe that σϕ(0)ϕ(S)∗ (k0 ) = σ0×S ∗ (h0 ) = {0} = σ0×ϕ(S)∗ (k0 ) for all S ∈ B (H ). As the range of ϕ contains all rank one operators, Lemma 3.3 entails that ϕ(0) = 0. Third, we show that ϕ preserves rank one operators in both direction. Let R be a ∗ rank one operator, and note that ϕ(R) = 0 and that σRT ∗ (h0 ) has at most one element ∗ for all T ∈ B (H ), and so is σϕ(R)ϕ(T )∗ (k0 ). As the range of ϕ contains F2 (K ), we see
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∗ that σϕ(R)S (k0 ) has at most one element for all operators S ∈ F2 (K ). By Lemma 3.4, we see that ϕ(R) is rank one. Conversely, assume that ϕ(R) is rank one for some operator R ∈ B (H ), and note that ∗ R = 0 and that σϕ(R)ϕ(S) ∗ (k0 ) has at most one element for all S ∈ B (H ). Therefore, ∗ σRS ∗ (h0 ) has at most one element for all S ∈ B (H ). Again Lemma 3.4 tells us that R is rank one.
Step 2. ϕ is linear. First we show that ϕ is additive. Let R be a rank one operator, and note that, by the previous step, ϕ(R) is a rank one operator too. Let T and S two operators in B (H ), and note that, by applying the statement (c) of Lemma 3.2, we have ∗ ∗ σϕ(T +S)ϕ(R)∗ (k0 ) = σ(T +S)R∗ (h0 ) ∗ = σT∗ R∗ (h0 ) + σSR ∗ (h0 ) ∗ ∗ = σϕ(T )ϕ(R)∗ (k0 ) + σϕ(S)ϕ(R)∗ (k0 ) ∗ = σ(ϕ(T )+ϕ(S))ϕ(R)∗ (k0 )
for all rank one operators R ∈ B (H ). By Lemma 3.3, we conclude that ϕ(T + S) = ϕ(T ) + ϕ(S) for all T, S ∈ B (H ), and ϕ is additive; as desired. Now, let us show that ϕ is homogeneous. Indeed, take a scalar λ ∈ C and an operator T ∈ B (H ), and note that σϕ(λT )ϕ(R)∗ (k0 ) = σ(λT )R∗ (h0 ) = λσT R∗ (h0 ) = λσϕ(T )ϕ(R)∗ (k0 ) = σλϕ(T )ϕ(R)∗ (k0 ) for all rank one operators R ∈ B (H ). By Lemma 3.3, we see that ϕ(λT ) = λϕ(T ) for all T ∈ B (H ) and λ ∈ C. Hence, ϕ is linear. Step 3. ϕ takes the desired form “(2.2)”. By the previous steps, ϕ : F (H ) → F (K ) is a bijective linear map preserving rank one operators in both directions. By [18], either there are two bijective linear operators A, B : H → K such that
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ϕ(x ⊗ y) = Ax ⊗ By,
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(4.6)
or there are two bijective linear operators C, D : H → K such that ϕ(x ⊗ y) = Cy ⊗ Dx.
(4.7)
By Lemma 4.2, ϕ takes only the form (4.6) when it is restricted on F (H ). While Lemma 4.1 tells us that we may and shall assume that A = U and B = V for some unitary operators U and V on B (H , K ), and that Ah0 = U h0 = αk0 for some nonzero scalar α ∈ C. So, for every rank one operator R ∈ B (H ) and every operator T ∈ B (H ), we have σϕ(T )ϕ(R)∗ (k0 ) = σT R∗ (h0 ) 1 = σT R∗ ( h0 ) α = σT R∗ (U ∗ k0 ) = σU T R∗ U ∗ (k0 ) = σU T V ∗ V R∗ U ∗ (k0 ) = σU T V ∗ ϕ(R)∗ (k0 ). Since ϕ bijectively maps F1 (H ) to F1 (K ), Lemma 3.3 shows that ϕ(T ) = U T V ∗ for all T ∈ B (H ), and the proof is therefore complete. 2 5. Proof of Theorem 2.2 Just as for the proof of Theorem 2.1, we need some auxiliary lemmas to prove Theorem 2.2. These lemmas are variant of Lemma 4.1 and Lemma 4.2, and their proofs are somehow similar to the ones of Lemma 4.1 and Lemma 4.2. Lemma 5.1. Let h0 ∈ H and k0 ∈ K be two nonzero vectors of H and A and B be two bijective linear operators from H into K , and ϕ : F1 (H ) → F1 (K ) defined by ϕ(x ⊗ y) := Ax ⊗ By for all x, y ∈ H . If ϕ satisfies σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 )
(5.1)
for all rank one operators T, S ∈ F1 (H ), then there is a unitary operator U ∈ B (H , K ) and a positive scalar μ such that A = μU and B = μ1 U . Moreover, U h0 = αk0 for some nonzero scalar α ∈ C.
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Proof. From (5.1), we get that ∗ ∗ x, uv, yσ(x⊗y) (h0 ) = Ax, AuBv, Byσ(Ax⊗By) (k0 )
(5.2)
for all x, y, u, v ∈ H . As in the proof of Lemma 4.1, one shows that h0 , y = 0 ⇐⇒ k0 , By = 0. This together with (5.2) and Lemma 3.2-(1), we infer that if h0 , y = 0, then x, uv, yx, y = Ax, AuBv, ByAx, By
(5.3)
for all x, y, u, v ∈ H . Now, if h0 , y = 0, (5.3) is true when replacing y by y + λh0 for all nonzero scalar λ, we get an equality between two polynomial functions of λ. Thus, (5.3) is true without any restriction on x, y, u, v ∈ H. Next, let us show that the mappings A and B are continuous. Fix a nonzero vector 2 x,y x ∈ H , and set v = y := B −1 Ax and δx := y By4 . From (5.3), we get that Ax, Au = δx x, u for all u ∈ H . This obviously shows that u → Ax, Au is continuous, and thus, since x is an arbitrary vector in H and A is bijective, the closed graph theorem implies that A itself is continuous. Similarly, we can show that B is continuous, and we therefore omit the details here. Now, let us show that there is a unitary operator U ∈ B (H , K ) and a positive scalar μ such that A = μU and B = μ1 U . To do this, we rewrite (5.3) as follows x, uv, yx, y = A∗ Ax, uB ∗ Bv, yB ∗ Ax, y
(5.4)
for all x, y, u, v ∈ H . We show at first that A∗ Ax and x are linearly dependent for every x ∈ H . Indeed, assume by the way of contradiction that there exists a nonzero vector x1 ∈ H such that A∗ Ax1 and x1 are linearly independent, and let y1 be a nonzero vector in H such that x1 , y1 = 1 and A∗ Ax1 , y1 = 0. From (5.4), we get that 0 = y1 2 = A∗ Ax1 , y1 By1 2 Ax1 , By1 = 0. This contradiction shows that A∗ A = αIH for some positive scalar α. By a similar way, we show that B ∗ B = βIH and B ∗ A = γIH for some positive scalars β and γ. Moreover, observe that (5.4) implies obviously that αβγ = 1. Moreover, such a scalar γ must be 1. Indeed, since A and B are inversibles, we have AB ∗ = γIK , AA∗ = αIK and BB ∗ = βIK and thus αβIH = A∗ AB ∗ B = γ 2 IH .
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This shows that αβ = γ 2 , and γ 3 = αβγ = 1. Therefore γ = 1 and B ∗ A = IH . √ Now, take U = μ1 A where μ = α and note that U is a unitary operator and U = μB; as desired. Finally, let us show that U h0 and k0 are linearly dependent. Assume by way of contradiction that they are linearly independent, and let u be a vector in H such that h0 , u = 1 and U −1 k0 , u = 0. We have, by Lemma 3.2 and since h0 ⊗ u is idempotent ∗ {1} = σh∗0 ⊗u (h0 ) = σ(h (h0 ) 0 ⊗u)(h0 ⊗u)(h0 ⊗u) ∗ (h0 ) = σ(h ∗ 0 ⊗u)(u⊗h0 ) (h0 ⊗u) ∗ (k0 ) = σϕ(h ∗ 0 ⊗u)ϕ(u⊗h0 ) ϕ(h0 ⊗u) ∗ = σ(U h0 ⊗U u)(U u⊗U h0 )∗ (U h0 ⊗U u) (k0 ) ∗ = σU (h0 ⊗u)(h0 ⊗u)(h0 ⊗u)U ∗ (k0 ) ∗ = σ(h (U −1 k0 ) = {0}. 0 ⊗u)
This arises to a contradiction, and shows that U h0 and k0 are linearly dependent. The lemma is therefore proved. 2 Lemma 5.2. Let h0 ∈ H and k0 ∈ K are two nonzero fixed vectors, and let C and D be two bijective linear operators from H into K , and ϕ : F1 (H ) → F1 (K ) defined by ϕ(x ⊗ y) = Cy ⊗ Dx for all x, y ∈ B (H ). If then there are rank one operators T, S ∈ F1 (H ) such that σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ). Proof. Assume by the way of contradiction that σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) for all rank one operators T, S ∈ F1 (H ) and choose a nonzero vector y1 ∈ K such that k0 , y1 = 0 and x = D−1 y1 . Since x and h0 are nonzero vectors, there exists y ∈ H such that h0 , y = 0 and x, y = 1. We therefore have ∗ ∗ {1} = σx⊗y (h0 ) = σ(x⊗y)(x⊗y)(x⊗y) (h0 ) ∗ ∗ = σ(x⊗y)(y⊗x) ∗ (x⊗y) (h0 ) = σ(Cy⊗Dx)(Cx⊗Dy)∗ (Cy⊗Dx) (k0 ) ∗ (k0 ) = σ(Cy⊗Dx)(Dy⊗Cx)(Cy⊗Dx) ∗ (k0 ) = σCy,CxDy,Dx(Cy⊗Dx) ∗ = σCy,CxDy,Dx(Cy⊗y (k0 ) = {0}. 1)
This leads to a contradiction and the lemma is therefore proved. 2 Let us now establish the proof of Theorem 2.2.
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Proof of Theorem 2.2. Assume that there exists a unitary operator U in B (H , K ) and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T U ∗ for all T ∈ B (H ). We have σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) = σU T U ∗ U S ∗ U ∗ U T U ∗ (k0 ) = σU T S ∗ T U ∗ (k0 ) = σT S ∗ T (h0 ) for all T, S ∈ B (H ), and (2.3) is established. Conversely, assume that ϕ satisfies (2.3) and let us show that ϕ takes the desired form. The proof breaks down into three steps. Following the same lines as in the proof of Theorem 2.1, we can shows that ϕ is a one-to-one linear map preserving rank one operators in both directions as its range contains F4 (K ). Thus, by [18] and Lemma 5.2, ϕ takes the form (4.6) when it is restricted on F (H ). While Lemma 5.1 tells us that there are a unitary operator U of B (H , K ) and a nonzero scalar α ∈ C such that U h0 = αk0 and ϕ(T ) = U T U ∗ for all T ∈ F (H ). It follows that for every rank one operator R ∈ B (H ) and every operator T ∈ B (H ), we have ∗ ∗ σϕ(R)U T ∗ U ∗ ϕ(R) (k0 ) = σU RU ∗ U T ∗ U ∗ U RU ∗ (k0 ) ∗ = σU RT ∗ RU ∗ (δk0 ) ∗ = σRT ∗ R (h0 ) ∗ = σϕ(R)ϕ(T )∗ ϕ(R) (k0 ).
Since this equality holds for any rank one operator R ∈ B (H ), by Lemma 3.3 we get that ϕ(T )∗ = U T ∗ U ∗ for all T ∈ B (H ), and ϕ has the desired form (2.4). The proof is then complete. 2 Acknowledgement Thanks are due to the referee for his/her careful reading of the manuscript, and some helpful comments. References [1] P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, 2004. [2] G. An, J. Hou, Rank-preserving multiplicative maps on B (X), Linear Algebra Appl. 342 (2002) 59–78. [3] R. Bhatia, P. Šemrl, A. Sourour, Maps on matrices that preserve the spectral radius distance, Studia Math. 134 (2) (1999) 99–110. [4] A. Bourhim, J. Mashreghi, Local spectral radius preservers, Integral Equations Operator Theory 76 (1) (2013) 95–104. [5] A. Bourhim, J. Mashreghi, Maps preserving the local spectrum of product of operators, Glasg. Math. J. (2014).
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[6] A. Bourhim, J. Mashreghi, Maps preserving the local spectrum of triple product of operators, Linear Multilinear Algebra 63 (4) (2015) 765–773. [7] A. Bourhim, J. Mashreghi, A survey on preservers of spectra and local spectra, Contemp. Math. 638 (2013). [8] A. Bourhim, V.G. Miller, Linear maps on Mn (C) preserving the local spectral radius, Studia Math. 188 (1) (2008) 67–75. [9] A. Bourhim, T. Ransford, Additive maps preserving local spectrum, Integral Equations Operator Theory 55 (2006) 377–385. [10] J. Bračič, V. Müller, Local spectrum and local spectral radius of an operator at a fixed vector, Studia Math. 194 (2) (2009) 155–162. [11] J.T. Chan, C.K. Li, N.S. Sze, Mappings preserving spectra of products of matrices, Proc. Amer. Math. Soc. 135 (2007) 977–986. [12] C. Costara, Linear maps preserving operators of local spectral radius zero, Integral Equations Operator Theory 73 (1) (2012) 7–16. [13] C. Costara, Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector, Arch. Math. 95 (6) (2010) 567–573. [14] J.L. Cui, J.C. Hou, Maps leaving functional values of operator products invariant, Linear Algebra Appl. 428 (2008) 1649–1663. [15] J.C. Hou, Q.H. Di, Maps preserving numerical range of operator products, Proc. Amer. Math. Soc. 134 (2006) 1435–1446. [16] T. Miura, D. Honma, A generalization of peripherally-multiplicative surjections between standard operator algebras, Cent. Eur. J. Math. 7 (3) (2009) 479–486. [17] L. Molnár, Some characterizations of the automorphisms of B(H) and C(X), Proc. Amer. Math. Soc. 130 (1) (2002) 111–120. [18] M. Olmadič, P. Šemrl, Additive mappings preserving operators of rank one, Linear Algebra Appl. 182 (1993) 239–256. [19] K.B. Laursen, M.M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monograph, New Series, vol. 20, 2000. [20] C.K. Li, P. Šemrl, N.S. Sze, Maps preserving the nilpotency of products of operators, Linear Algebra Appl. 424 (2007) 222–239. [21] M. Wang, L. Fang, G. Ji, Linear maps preserving idempotency of products or triple Jordan products of operators, Linear Algebra Appl. 429 (2008) 181–189. [22] W. Zhang, J. Hou, Maps preserving peripheral spectrum of Jordan semi-triple products of operators, Linear Algebra Appl. 435 (2011) 1326–1335.
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Linear Algebra and its Applications www.elsevier.com/locate/laa
Maps preserving the local spectrum of skew-product of operators Z. Abdelali ∗ , A. Achchi, R. Marzouki University Mohammed V, Faculty of Sciences, Department of Mathematics, Rabat, Morocco
a r t i c l e
i n f o
Article history: Received 18 May 2015 Accepted 17 July 2015 Available online xxxx Submitted by P. Semrl MSC: primary 47B49 secondary 47A10, 47A11 Keywords: Nonlinear preservers Local spectrum
a b s t r a c t Let H and K be infinite dimensional complex Hilbert spaces and let B (H ) be the algebra of all bounded linear operators on H . Let σT (h) denote the local spectrum of an operator T ∈ B (H ) at any vector h ∈ H , and fix two nonzero vectors h0 ∈ H and k0 ∈ K . We show that if a map ϕ : B (H ) → B (K ) has a range containing all operators of rank at most two and satisfies σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 ) for all T, S ∈ B (H ), then there exist two unitary operators U and V in B (H , K ) such that U h0 = αk0 for some nonzero α ∈ C and ϕ(T ) = U T V ∗ for all T ∈ B (H ). We also described maps ϕ : B (H ) → B (K ) satisfying σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) for all T, S ∈ B (H ), and with the range containing all operators of rank at most four. © 2015 Elsevier Inc. All rights reserved.
* Corresponding author. E-mail addresses: [email protected] (Z. Abdelali), [email protected] (A. Achchi), [email protected] (R. Marzouki). http://dx.doi.org/10.1016/j.laa.2015.07.019 0024-3795/© 2015 Elsevier Inc. All rights reserved.
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1. Introduction Several results on linear or additive preserver problems have been extended to the setting of nonlinear preservers, and, in many cases, their extensions demonstrated to be nontrivial. In particular, the problem of characterizing maps preserving certain functions, subsets, relations and properties of product of matrices or operators has attracted the attention of several authors; see for instance [3,11,14,15,20,17,21] and the references therein. Motivated by problems concerning local automorphisms, L. Molnár characterized, in [17], maps preserving different spectra of operator or matrix products, and his results have been extended by several authors. In [15], J. Hou and Q.H. Di described, in particular, maps preserving the numerical range of products of Hilbert space operators. In [20], maps preserving the nilpotency of operator products are characterized. T. Miura and D. Honma gave in [16] a generalization of peripherally-multiplicative surjections between standard operator algebras. In [2], G. An and J. Hou described multiplicative maps ϕ on B (H ), the algebra of all bounded linear operators on a complex Hilbert space H , that satisfy T ∗ S = 0 if and only if ϕ(T )∗ ϕ(S) = 0 for all S, T ∈ B (H ). While in [22], W. Zhang and J. Hou characterized maps preserving peripheral spectrum of Jordan semi-triple product T S ∗ T of operators. The problem of characterizing linear or additive maps on B (X), the algebra of all bounded linear operators on a complex Banach space X, preserving local spectra was initiated by A. Bourhim and T.J. Ransford in [9], and continued by several authors; see for instance [8,10–13] and the references therein. In [10], J. Bračič and V. Müller characterized surjective linear and continuous mappings on B (X) preserving the local spectrum and local spectral radius at a fixed nonzero vector x0 of X, and thus extending the main results of [8] to infinite-dimensional Banach spaces. While in [13], C. Costara showed, in particular, that every surjective linear map on B (X) preserving the local spectral radius at a fixed nonzero vector of X is automatically continuous. Furthermore, in [12], C. Costara described surjective linear maps on B (X) which preserve operators of local spectral radius zero at points of X. His result has been extended in [4] to the nonlinear setting where A. Bourhim and J. Mashreghi gave a complete description of surjective (not necessarily linear) maps on B (X) preserving the difference of operators with local spectral radius zero at points of X. In [5], A. Bourhim and J. Mashreghi showed that surjective map ϕ on B (X) preserving the local spectrum of product of operators at a fixed nonzero vector x0 ∈ X if and only if ϕ is a positive or negative multiple automorphism and x0 is an eigenvector of the intertwining operator. In [6], they completely describe the form of surjective maps on B (X) preserving the local spectrum of triple product of operators at a nonzero fixed vector of X. To have a good chronological handling of the preserving problems, see the excellent survey by A. Bourhim and J. Mashreghi [7]. Let H and K be two infinite-dimensional complex Hilbert spaces. Let B (H ) be the algebra of all bounded linear operators on H . In this paper, we follow the same path of studies by considering general local spectra preservers, and characterize maps
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ϕ : B (H ) → B (K ) that preserves the local spectrum at fixed nonzero vector of the skew-double product T S ∗ , and under the mild assumption on the ranges of the maps ϕ contain operators with rank at most two. In particular, we will show that such a map ϕ is a bijective isometry of the form ϕ(T ) = U T V ∗ (T ∈ B (H )) with U h0 = αk0 for some nonzero scalar α. We also obtain a similar result by characterizing maps ϕ preserving the local spectrum at fixed nonzero vector of the skew-triple product T S ∗ T . Our proofs are influenced by ideas from [5,6] but they use new ingredients that will be established in Section 4 and Section 5. 2. Main results Throughout this paper, H and K are two infinite-dimensional complex Hilbert spaces. As usual B (H , K ) denotes the space of all bounded linear operators from H into K . When H = K we simply write B (H ) instead of B (H , H ). The inner product of H or K will be denoted by , if there is no confusion. Let F (H ) denote the ideal of all finite rank operators on H . For a positive integer n, let Fn (H ) be the set of all operators of B (H ) of rank at most n. For an operator T ∈ B (H , K ), let T ∗ denote as usual its adjoint (T h, k = h, T ∗ k, (h, k) ∈ H × K ). The local resolvent set, ρT (x), of an operator T ∈ B (H ) at a point x ∈ H is the union of all open subsets U of the complex plane C for which there is an analytic function φ : U → H such that (T − λ)φ(λ) = x (λ ∈ U ). Clearly ρT (x) contains the resolvent set ρ(T ) of T , but this containment could be proper. The local spectrum of T at x is defined by σT (x) := C\ρT (x), and thus it is a closed subset (possibly empty) of σ(T ), the spectrum of T . In fact, for every operator T ∈ F (H ), σT (x) = ∅ for all nonzero vectors x in H . The books by P. Aiena [1] and by K.B. Laursen, M.M. Neumann [19] provide an excellent exposition as well as a rich bibliography of the local spectral theory. The following theorems are the main results of this paper. Their proofs will be presented in Section 4 and Section 5. Theorem 2.1. Let h0 ∈ H and k0 ∈ K be two fixed nonzero vectors and ϕ : B (H ) → B (K ) a map such that its range contains F2 (K ). Then ϕ satisfies σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 )
(T, S ∈ B (H )),
(2.1)
if and only if there exist two unitary operators U and V in B (H , K ) and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T V ∗ for all T ∈ B (H ).
(2.2)
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The following result is a variant of the above one. Theorem 2.2. Let h0 ∈ B (H ) and k0 ∈ B (K ) be two fixed nonzero vectors ϕ : B (H ) → B (K ) a map such that its range contains F4 (K ). Then ϕ satisfies σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) (T, S ∈ B (H )),
(2.3)
if and only if there exist a unitary operator U : H → K and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T U ∗
(2.4)
for all T ∈ B (H ). Remark 2.3. (1) If H and K are isomorphic, then they are isometrically isomorphic. Using U ϕU ∗ instead of ϕ for an appropriate unitary operator U : K → H , the statements of Theorems 2.1 and 2.2 can be reduced to the case when H = K and h0 = k0 . Although, the fact that “H and K are isomorphic” is one of the conclusions of the main results. (2) We point out that the description of maps ψ from B (H ) into B (K ) satisfying F2 (K ) ⊆ ψ(B (H )) and σT ∗ S (h0 ) = σψ(T )∗ ψ(S) (k0 ) (resp. F4 (K ) ⊆ ψ(B (H )) and σT ∗ ST ∗ (h0 ) = σψ(T )∗ ψ(S)ψ(T )∗ (k0 )), for all T, S ∈ B (H ), can be obtained by applying Theorem 2.1 (resp. Theorem 2.2) to ϕ given by ϕ(T ) = ψ(T ∗ )∗ , T ∈ B (H ). 3. Preliminaries In this section, we fix some notions and exhibit some tools on the local spectral theory and some essential results needed for the proof of our main results. The first lemma summarizes some basic properties of the local spectrum that will be often used in the sequel. Lemma 3.1. For an invertible operator A ∈ B (H ), a vector x ∈ H and a nonzero scalar α ∈ C, the following statements hold. (1) σT (αx) = σT (x) and σαT (x) = ασT (x) for all T ∈ B (H ). (2) σAT A−1 (Ax) = σT (x) for all T ∈ B (H ). For a nonzero h0 ∈ H and an operator T ∈ B (H ), we use a useful notation defined by A. Bourhim and J. Mashreghi in [5,6] by
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σT∗ (h0 )
:=
{0}
if σT (h0 ) = {0}
σT (h0 ) \ {0}
if σT (h0 ) = {0}.
(3.1)
Therein, they also gave some essential lemmas and theorems which are useful tools to establish our main results. For our purpose, we state these results only in the case of Hilbert spaces. Lemma 3.2. (See [5,6].) Let h0 be a nonzero vector in H . For any vectors x, y and z in H , the following statements hold. (1) ∗ σx⊗z (h0 )
=
{0}
if h0 , z = 0
{x, z}
if h0 , z = 0.
∗ ∗ ∗ (2) σ(x+y)⊗z (h0 ) = σx⊗z (h0 ) + σy⊗z (h0 ). (3) For all rank one operators R and all T, S ∈ B (H ), we have ∗ ∗ ∗ • σ(T +S)R (h0 ) = σT R (h0 ) + σSR (h0 ), and ∗ ∗ ∗ • σR(T +S)R (h0 ) = σRT R (h0 ) + σRSR (h0 ).
Next lemma provides some local spectral identity principles. Lemma 3.3. (See [5, Theorem 3.2] and [6, Theorem 3.1].) For a nonzero vector h0 in H and two operators A and B in B (H ), the following statements are equivalent. (1) (2) (3) (4) (5) (6) (7)
A = B. σAT (h0 ) = σBT (h0 ) for all T ∈ B (H ). σAT (h0 ) = σBT (h0 ) for all T ∈ F1 (H ). ∗ ∗ σAT (h0 ) = σBT (h0 ) for all T ∈ F1 (H ). σT AT (h0 ) = σT BT (h0 ) for all T ∈ B (H ). σT AT (h0 ) = σT BT (h0 ) for all T ∈ F1 (H ). σT∗ AT (h0 ) = σT∗ BT (h0 ) for all T ∈ F1 (H ).
Finally, we close this section with the following lemma that gives a local spectral characterization of rank one operators of B (H ). Lemma 3.4. (See [5, Theorem 4.1] and [6, Theorem 3.2].) Let R be a nonzero bounded linear operator on H , and h0 a nonzero vector in H . Then the following statements are equivalent. (1) R has rank one. ∗ (2) σRT (h0 ) contains at most one element for all T ∈ B (H ). ∗ (3) σRT (h0 ) contains at most one element for all T ∈ F2 (H ).
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(4) σT∗ RT (h0 ) is a singleton for all operators T ∈ B (H ). (5) σT∗ RT (h0 ) contains at most one element for all T ∈ F4 (H ). Notice that the equivalence with the fifth statement is clear by the proof of [6, Theorem 3.2]. 4. Proof of Theorem 2.1 For the proof of Theorem 2.1, we need the following essential lemmas. Lemma 4.1. Let h0 ∈ H and k0 ∈ K be two nonzero fixed vectors and A and B be two bijective linear operators from H into K , and ϕ : F1 (H ) → F1 (K ) the map defined by ϕ(x ⊗ y) := Ax ⊗ By for all x, y ∈ H . If σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 )
(4.1)
for all rank one operators T and S in B (H ), then A and B are bounded unitary operators multiplied by positive scalars λ and μ such that λμ = 1. Moreover, Ah0 = αk0 for some nonzero scalar α ∈ C. Proof. Let x, y, u and v be four vectors in H , and let us first show that v, yx, u = Bv, ByAx, Au.
(4.2)
Note that (4.1) applied to x ⊗ y and u ⊗ v entails that v, yσx⊗u (h0 ) = σv,y(x⊗u) (h0 ) = σBv,By(Ax⊗Au) (k0 ) = Bv, ByσAx⊗Au (k0 ),
(4.3)
and let us show that h0 , u = 0 ⇐⇒ k0 , Au = 0.
(4.4)
Indeed, if h0 , u = 0 and k0 , Au = 0, then (4.3), applied when x = y = v = h0 , implies that ∗ ∗ (h0 ) = σBh (k0 ) = {0}. h0 2 h0 , u = σh 0 ,h0 (h0 ⊗u) 0 ,Bh0 (Ah0 ⊗Au)
This arises a contradiction and shows that if k0 , Au = 0 then h0 , u = 0. Conversely, if h0 , u = 0 and k0 , Au = 0, apply (4.3) when x = A−1 k0 and y = v = B −1 k0 so that ∗ ∗ {0} = σB k0 2 k0 , Au . −1 k ,B −1 k (A−1 k ⊗u) (h0 ) = σk ,k (k ⊗Au) (k0 ) = 0 0 0 0 0 0
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This contradiction shows that if h0 , u = 0 then k0 , Au = 0. Therefore, (4.4) is established. By (4.4) and the fact that ∗ σx⊗u (h0 )
=
{0}
if h0 , u = 0
{x, u} if h0 , u = 0,
(4.5)
we see that (4.2) holds provided that h0 , u = 0. Now, if h0 , u = 0, note that h0 , u + h0 = h0 2 = 0 and apply (4.2) twice when u is replaced by h0 and by h0 + u. We get v, yx, h0 = Bv, ByAx, Ah0 , and v, yx, h0 + u = Bv, ByAx, A(h0 + u). By expanding the second identity and using the first one, we deduce that (4.2) holds in this case too. 1 Now, let v0 be a fixed unit vector in H and let λ = Bv . By (4.2), we have 0 Ax 2 = Ax, Ax = λ2 x, x = λ2 x 2 for all x ∈ H . Hence, A/λ is an isometry and thus it is a unitary operator in B (H , K ) since it is supposed to be bijective. By the same way, fix a unit vector u0 in H and take 1 μ = Au , and note that B/μ is a unitary operator in B (H , K ). Finally, by (4.2), we 0 see that λμ = 1. To finish the proof of this lemma, let us show that h0 and A−1 k0 are linearly dependent. If not, there exist a vector u ∈ H such that h0 , u = 1 and A−1 k0 , u = 0, and by Lemma 3.2 ∗ ∗ {1} = σh∗0 ⊗u (h0 ) = σ(h (h0 ) = σ(h ∗ (h0 ) 0 ⊗u)(h0 ⊗u) 0 ⊗u)(u⊗h0 ) ∗ ∗ = σϕ(h ∗ (k0 ) = σ(Ah ⊗Bu)(Au⊗Bh )∗ (k0 ) 0 ⊗u)ϕ(u⊗h0 ) 0 0 ∗ ∗ (k0 ) = σBh (k0 ) = σ(Ah 0 ⊗Bu)(Bh0 ⊗Au) 0 ,Bu(Ah0 ⊗Au) ∗ ∗ −1 k0 ) = σBh ∗ (k0 ) = σλ2 Bh ,Bu(h ⊗u) (A 0 ,BuA(h0 ⊗u)A 0 0
= {0}. This contradiction shows that h0 and A−1 k0 are linearly dependent; as desired. 2 Lemma 4.2. Let h0 ∈ H and k0 ∈ K be two nonzero fixed vectors, and let C and D be two bijective linear operators from H to K , and ϕ : F1 (H ) → F1 (K ) the map defined by ϕ(x ⊗ y) = Cy ⊗ Dx for all x, y ∈ H . Then there are rank one operators T, S ∈ F1 (H ) such that σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 ).
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Proof. Assume by the way of contradiction that σT S ∗ (h0 ) = σϕ(T )ϕ(S)∗ (k0 ) for all rank one operators T, S ∈ F1 (H ). Choose a nonzero vector y1 ∈ K such that k0 , y1 = 0. Since the vectors x = C −1 y1 and h0 are nonzero, there exists a nonzero y ∈ H such that h0 , y = 0 and x, y = 1. By Lemma 3.2, and since x ⊗ y is idempotent, we have ∗ ∗ {1} = σx⊗y (h0 ) = σ(x⊗y)(x⊗y) (h0 ) ∗ ∗ = σ(x⊗y)(y⊗x) ∗ (h0 ) = σ(Cy⊗Dx)(Cx⊗Dy)∗ (k0 ) ∗ (k0 ) = σ(Cy⊗Dx)(Dy⊗Cx) ∗ (k0 ) = σDy,Dx(Cy⊗Cx) ∗ (k0 ) = {0}. = σDy,Dx(Cy⊗y 1)
This yields to a contradiction and the lemma is therefore proved. 2 Now, we are in a position to prove Theorem 2.1. Proof of Theorem 2.1. Assume that there exist two unitary operators U and V in B (H , K ) and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T V ∗ for all T ∈ B (H ). We have σϕ(T )ϕ(S)∗ (k0 ) = σU T V ∗ V S ∗ U ∗ (k0 ) = σU T S ∗ U ∗ (k0 ) = σT S ∗ (h0 ) for all T, S ∈ B (H ), and (2.1) is established. Conversely, assume that ϕ satisfies (2.1) and let us show that ϕ takes the desired form. The proof breaks down into three steps. Step 1. ϕ is a one to one map preserving rank one operators in both directions. We first show ϕ is a one to one map and ϕ(0) = 0. Take two operators A, B ∈ B (H ) such that ϕ(A) = ϕ(B), and note that σAS ∗ (h0 ) = σϕ(A)ϕ(S)∗ (k0 ) = σϕ(B)ϕ(S)∗ (k0 ) = σBS ∗ (h0 ) for all S ∈ B (H ). Lemma 3.3 tells us that A = B, and thus ϕ is a one to one. Second, we show that ϕ(0) = 0. To do that, just observe that σϕ(0)ϕ(S)∗ (k0 ) = σ0×S ∗ (h0 ) = {0} = σ0×ϕ(S)∗ (k0 ) for all S ∈ B (H ). As the range of ϕ contains all rank one operators, Lemma 3.3 entails that ϕ(0) = 0. Third, we show that ϕ preserves rank one operators in both direction. Let R be a ∗ rank one operator, and note that ϕ(R) = 0 and that σRT ∗ (h0 ) has at most one element ∗ for all T ∈ B (H ), and so is σϕ(R)ϕ(T )∗ (k0 ). As the range of ϕ contains F2 (K ), we see
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∗ that σϕ(R)S (k0 ) has at most one element for all operators S ∈ F2 (K ). By Lemma 3.4, we see that ϕ(R) is rank one. Conversely, assume that ϕ(R) is rank one for some operator R ∈ B (H ), and note that ∗ R = 0 and that σϕ(R)ϕ(S) ∗ (k0 ) has at most one element for all S ∈ B (H ). Therefore, ∗ σRS ∗ (h0 ) has at most one element for all S ∈ B (H ). Again Lemma 3.4 tells us that R is rank one.
Step 2. ϕ is linear. First we show that ϕ is additive. Let R be a rank one operator, and note that, by the previous step, ϕ(R) is a rank one operator too. Let T and S two operators in B (H ), and note that, by applying the statement (c) of Lemma 3.2, we have ∗ ∗ σϕ(T +S)ϕ(R)∗ (k0 ) = σ(T +S)R∗ (h0 ) ∗ = σT∗ R∗ (h0 ) + σSR ∗ (h0 ) ∗ ∗ = σϕ(T )ϕ(R)∗ (k0 ) + σϕ(S)ϕ(R)∗ (k0 ) ∗ = σ(ϕ(T )+ϕ(S))ϕ(R)∗ (k0 )
for all rank one operators R ∈ B (H ). By Lemma 3.3, we conclude that ϕ(T + S) = ϕ(T ) + ϕ(S) for all T, S ∈ B (H ), and ϕ is additive; as desired. Now, let us show that ϕ is homogeneous. Indeed, take a scalar λ ∈ C and an operator T ∈ B (H ), and note that σϕ(λT )ϕ(R)∗ (k0 ) = σ(λT )R∗ (h0 ) = λσT R∗ (h0 ) = λσϕ(T )ϕ(R)∗ (k0 ) = σλϕ(T )ϕ(R)∗ (k0 ) for all rank one operators R ∈ B (H ). By Lemma 3.3, we see that ϕ(λT ) = λϕ(T ) for all T ∈ B (H ) and λ ∈ C. Hence, ϕ is linear. Step 3. ϕ takes the desired form “(2.2)”. By the previous steps, ϕ : F (H ) → F (K ) is a bijective linear map preserving rank one operators in both directions. By [18], either there are two bijective linear operators A, B : H → K such that
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ϕ(x ⊗ y) = Ax ⊗ By,
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(4.6)
or there are two bijective linear operators C, D : H → K such that ϕ(x ⊗ y) = Cy ⊗ Dx.
(4.7)
By Lemma 4.2, ϕ takes only the form (4.6) when it is restricted on F (H ). While Lemma 4.1 tells us that we may and shall assume that A = U and B = V for some unitary operators U and V on B (H , K ), and that Ah0 = U h0 = αk0 for some nonzero scalar α ∈ C. So, for every rank one operator R ∈ B (H ) and every operator T ∈ B (H ), we have σϕ(T )ϕ(R)∗ (k0 ) = σT R∗ (h0 ) 1 = σT R∗ ( h0 ) α = σT R∗ (U ∗ k0 ) = σU T R∗ U ∗ (k0 ) = σU T V ∗ V R∗ U ∗ (k0 ) = σU T V ∗ ϕ(R)∗ (k0 ). Since ϕ bijectively maps F1 (H ) to F1 (K ), Lemma 3.3 shows that ϕ(T ) = U T V ∗ for all T ∈ B (H ), and the proof is therefore complete. 2 5. Proof of Theorem 2.2 Just as for the proof of Theorem 2.1, we need some auxiliary lemmas to prove Theorem 2.2. These lemmas are variant of Lemma 4.1 and Lemma 4.2, and their proofs are somehow similar to the ones of Lemma 4.1 and Lemma 4.2. Lemma 5.1. Let h0 ∈ H and k0 ∈ K be two nonzero vectors of H and A and B be two bijective linear operators from H into K , and ϕ : F1 (H ) → F1 (K ) defined by ϕ(x ⊗ y) := Ax ⊗ By for all x, y ∈ H . If ϕ satisfies σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 )
(5.1)
for all rank one operators T, S ∈ F1 (H ), then there is a unitary operator U ∈ B (H , K ) and a positive scalar μ such that A = μU and B = μ1 U . Moreover, U h0 = αk0 for some nonzero scalar α ∈ C.
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Proof. From (5.1), we get that ∗ ∗ x, uv, yσ(x⊗y) (h0 ) = Ax, AuBv, Byσ(Ax⊗By) (k0 )
(5.2)
for all x, y, u, v ∈ H . As in the proof of Lemma 4.1, one shows that h0 , y = 0 ⇐⇒ k0 , By = 0. This together with (5.2) and Lemma 3.2-(1), we infer that if h0 , y = 0, then x, uv, yx, y = Ax, AuBv, ByAx, By
(5.3)
for all x, y, u, v ∈ H . Now, if h0 , y = 0, (5.3) is true when replacing y by y + λh0 for all nonzero scalar λ, we get an equality between two polynomial functions of λ. Thus, (5.3) is true without any restriction on x, y, u, v ∈ H. Next, let us show that the mappings A and B are continuous. Fix a nonzero vector 2 x,y x ∈ H , and set v = y := B −1 Ax and δx := y By4 . From (5.3), we get that Ax, Au = δx x, u for all u ∈ H . This obviously shows that u → Ax, Au is continuous, and thus, since x is an arbitrary vector in H and A is bijective, the closed graph theorem implies that A itself is continuous. Similarly, we can show that B is continuous, and we therefore omit the details here. Now, let us show that there is a unitary operator U ∈ B (H , K ) and a positive scalar μ such that A = μU and B = μ1 U . To do this, we rewrite (5.3) as follows x, uv, yx, y = A∗ Ax, uB ∗ Bv, yB ∗ Ax, y
(5.4)
for all x, y, u, v ∈ H . We show at first that A∗ Ax and x are linearly dependent for every x ∈ H . Indeed, assume by the way of contradiction that there exists a nonzero vector x1 ∈ H such that A∗ Ax1 and x1 are linearly independent, and let y1 be a nonzero vector in H such that x1 , y1 = 1 and A∗ Ax1 , y1 = 0. From (5.4), we get that 0 = y1 2 = A∗ Ax1 , y1 By1 2 Ax1 , By1 = 0. This contradiction shows that A∗ A = αIH for some positive scalar α. By a similar way, we show that B ∗ B = βIH and B ∗ A = γIH for some positive scalars β and γ. Moreover, observe that (5.4) implies obviously that αβγ = 1. Moreover, such a scalar γ must be 1. Indeed, since A and B are inversibles, we have AB ∗ = γIK , AA∗ = αIK and BB ∗ = βIK and thus αβIH = A∗ AB ∗ B = γ 2 IH .
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This shows that αβ = γ 2 , and γ 3 = αβγ = 1. Therefore γ = 1 and B ∗ A = IH . √ Now, take U = μ1 A where μ = α and note that U is a unitary operator and U = μB; as desired. Finally, let us show that U h0 and k0 are linearly dependent. Assume by way of contradiction that they are linearly independent, and let u be a vector in H such that h0 , u = 1 and U −1 k0 , u = 0. We have, by Lemma 3.2 and since h0 ⊗ u is idempotent ∗ {1} = σh∗0 ⊗u (h0 ) = σ(h (h0 ) 0 ⊗u)(h0 ⊗u)(h0 ⊗u) ∗ (h0 ) = σ(h ∗ 0 ⊗u)(u⊗h0 ) (h0 ⊗u) ∗ (k0 ) = σϕ(h ∗ 0 ⊗u)ϕ(u⊗h0 ) ϕ(h0 ⊗u) ∗ = σ(U h0 ⊗U u)(U u⊗U h0 )∗ (U h0 ⊗U u) (k0 ) ∗ = σU (h0 ⊗u)(h0 ⊗u)(h0 ⊗u)U ∗ (k0 ) ∗ = σ(h (U −1 k0 ) = {0}. 0 ⊗u)
This arises to a contradiction, and shows that U h0 and k0 are linearly dependent. The lemma is therefore proved. 2 Lemma 5.2. Let h0 ∈ H and k0 ∈ K are two nonzero fixed vectors, and let C and D be two bijective linear operators from H into K , and ϕ : F1 (H ) → F1 (K ) defined by ϕ(x ⊗ y) = Cy ⊗ Dx for all x, y ∈ B (H ). If then there are rank one operators T, S ∈ F1 (H ) such that σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ). Proof. Assume by the way of contradiction that σT S ∗ T (h0 ) = σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) for all rank one operators T, S ∈ F1 (H ) and choose a nonzero vector y1 ∈ K such that k0 , y1 = 0 and x = D−1 y1 . Since x and h0 are nonzero vectors, there exists y ∈ H such that h0 , y = 0 and x, y = 1. We therefore have ∗ ∗ {1} = σx⊗y (h0 ) = σ(x⊗y)(x⊗y)(x⊗y) (h0 ) ∗ ∗ = σ(x⊗y)(y⊗x) ∗ (x⊗y) (h0 ) = σ(Cy⊗Dx)(Cx⊗Dy)∗ (Cy⊗Dx) (k0 ) ∗ (k0 ) = σ(Cy⊗Dx)(Dy⊗Cx)(Cy⊗Dx) ∗ (k0 ) = σCy,CxDy,Dx(Cy⊗Dx) ∗ = σCy,CxDy,Dx(Cy⊗y (k0 ) = {0}. 1)
This leads to a contradiction and the lemma is therefore proved. 2 Let us now establish the proof of Theorem 2.2.
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Proof of Theorem 2.2. Assume that there exists a unitary operator U in B (H , K ) and a nonzero scalar α such that U h0 = αk0 and ϕ(T ) = U T U ∗ for all T ∈ B (H ). We have σϕ(T )ϕ(S)∗ ϕ(T ) (k0 ) = σU T U ∗ U S ∗ U ∗ U T U ∗ (k0 ) = σU T S ∗ T U ∗ (k0 ) = σT S ∗ T (h0 ) for all T, S ∈ B (H ), and (2.3) is established. Conversely, assume that ϕ satisfies (2.3) and let us show that ϕ takes the desired form. The proof breaks down into three steps. Following the same lines as in the proof of Theorem 2.1, we can shows that ϕ is a one-to-one linear map preserving rank one operators in both directions as its range contains F4 (K ). Thus, by [18] and Lemma 5.2, ϕ takes the form (4.6) when it is restricted on F (H ). While Lemma 5.1 tells us that there are a unitary operator U of B (H , K ) and a nonzero scalar α ∈ C such that U h0 = αk0 and ϕ(T ) = U T U ∗ for all T ∈ F (H ). It follows that for every rank one operator R ∈ B (H ) and every operator T ∈ B (H ), we have ∗ ∗ σϕ(R)U T ∗ U ∗ ϕ(R) (k0 ) = σU RU ∗ U T ∗ U ∗ U RU ∗ (k0 ) ∗ = σU RT ∗ RU ∗ (δk0 ) ∗ = σRT ∗ R (h0 ) ∗ = σϕ(R)ϕ(T )∗ ϕ(R) (k0 ).
Since this equality holds for any rank one operator R ∈ B (H ), by Lemma 3.3 we get that ϕ(T )∗ = U T ∗ U ∗ for all T ∈ B (H ), and ϕ has the desired form (2.4). The proof is then complete. 2 Acknowledgement Thanks are due to the referee for his/her careful reading of the manuscript, and some helpful comments. References [1] P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer, Dordrecht, 2004. [2] G. An, J. Hou, Rank-preserving multiplicative maps on B (X), Linear Algebra Appl. 342 (2002) 59–78. [3] R. Bhatia, P. Šemrl, A. Sourour, Maps on matrices that preserve the spectral radius distance, Studia Math. 134 (2) (1999) 99–110. [4] A. Bourhim, J. Mashreghi, Local spectral radius preservers, Integral Equations Operator Theory 76 (1) (2013) 95–104. [5] A. Bourhim, J. Mashreghi, Maps preserving the local spectrum of product of operators, Glasg. Math. J. (2014).
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