Maps on matrices compressing the local spectrum in the spectrum

Linear Algebra Appl. 475 (2015) 176–185

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

Maps on matrices compressing the local spectrum in the spectrum Hassane Benbouziane ∗ , Mustapha Ech-Cherif El Kettani Department of Mathematics, Faculty of Sciences DharMahraz Fes, University Sidi Mohammed BenAbdellah, 1796 Atlas Fes, Morocco

a r t i c l e

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Article history: Received 16 August 2014 Accepted 5 February 2015 Available online 6 March 2015 Submitted by P. Semrl MSC: 47B49 47B10 47A53 47A11

a b s t r a c t Let Mn (C) be the algebra of all complex n × n matrices. Let x0 be a nonzero vector in Cn . We describe maps φ on Mn (C) (not linear or surjective) satisfying σT ±S (x0 ) ⊆ σ(φ(T ) ± φ(S)) for all T, S ∈ Mn (C). We also obtain a similar description by supposing that φ is surjective and σφ(T )±φ(S) (x0 ) ⊆ σ(T ± S) for all T, S ∈ Mn (C). © 2015 Published by Elsevier Inc.

Keywords: Local spectrum Preserver problems Matrix algebras

1. Introduction and statement of the main results Let B(X) be the algebra of all linear bounded operators on a complex Banach space X. The local resolvent set of an operator T ∈ B(X) at a point x ∈ X, denoted by ρT (x), is the union of all open subsets U ⊆ C for which there exists an analytic function f : U → X such that (T − λ)f (λ) = x for all λ ∈ U . The local spectrum of T at x is defined by * Corresponding author. E-mail address: [email protected] (H. Benbouziane). http://dx.doi.org/10.1016/j.laa.2015.02.015 0024-3795/© 2015 Published by Elsevier Inc.

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σT (x) := C \ ρT (x) and is a (possibly empty) closed subset of σ(T ), the usual spectrum of T ∈ B(X). The problem of characterizing spectrum-preserving maps on the algebra Mn(C) of all complex n ×n matrices was considered by a number of authors. In [19], Marcus and Moyls proved that if a linear map φ on Mn (C) preserves eigenvalues (counting multiplicities), then there exists an invertible matrix A ∈ Mn (C) such that φ(T ) = AT A−1 , (T ∈ Mn (C)),

(1)

φ(T ) = AT t A−1 , (T ∈ Mn (C)),

(2)

or

where T t denotes, as usual, the transpose of T ∈ Mn (C). This result has been generalized in different directions; see [4,12,14,20]. In particular, it is shown in [12,14] that for a map φ on Mn (C) with φ(0) = 0, the following statements are equivalent. 1. φ satisfies σ(φ(T ) − φ(S)) ⊆ σ(T − S), (T, S ∈ Mn (C)),

(3)

σ(T − S) ⊆ σ(φ(T ) − φ(S)), (T, S ∈ Mn (C)),

(4)

2. φ satisfies

3. φ takes either the form (1) or (2). This result has been shown before under additional assumptions such as continuity or surjectivity of the map φ. But as stated above, the equivalence between the last two statements is due to Costara [12] and the equivalence between the first and third statements was recently proved by Dolinar, Hou, Kuzma and Qi in [14]. The problem of describing linear or additive maps on B(X) preserving the local spectra has been initiated by Bourhim and Ransford in [8], and continued by several authors; see for instance [5–7,10,11,13,16,17]. In [4], Bendaoud, Douimi and Sarih used Costara’s approach and proved, for a fixed nonzero vector x0 in Cn , that a map φ on Mn (C) with φ(0) = 0 satisfies σφ(T )−φ(S) (x0 ) ⊆ σT −S (x0 ), (T, S ∈ Mn (C))

(5)

if and only if there exists an invertible matrix A ∈ Mn (C) such that Ax0 = x0 and φ(T ) = AT A−1 for all T ∈ Mn (C). They also show that the same conclusion remains valid when the reverse set inclusion in (5) occurs without surjectivity assumption on φ.

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Fix a nonzero vector x0 in Cn . In this paper, we give the form of maps φ on Mn (C) satisfying φ(0) = 0 and σT −S (x0 ) ⊆ σ(φ(T ) − φ(S)), (T, S ∈ Mn (C)),

(6)

and show that such a map φ takes either the form (1) or (2). We arrive at the same description by supposing that φ is surjective and satisfies σφ(T )−φ(S) (x0 ) ⊆ σ(T − S), (T, S ∈ Mn (C)).

(7)

Our proofs use Costara’s approach as well but our results cover the main results of [4,12] since a map φ satisfies (7) provided that it verifies either (3) or (5), and φ satisfies (6) provided that it verifies the reverse inclusion of either (3) or (5). The following theorem is our first result. It shows that surjective maps on Mn (C) compressing the local spectrum in the spectrum of the difference of matrices have standard forms. Theorem 1. Fix a nonzero vector x0 in Cn , and let φ be a surjective map on Mn (C) with φ(0) = 0. Then φ satisfies σφ(T )−φ(S) (x0 ) ⊆ σ(T − S), (T, S ∈ Mn (C))

(8)

if and only if φ takes either the form (1) or (2). Next theorem shows that the same conclusion of the above result remains valid without the surjectivity condition but when φ expands the local spectrum in the spectrum of the difference of matrices. Theorem 2. Fix a nonzero vector x0 in Cn , and let φ be a map on Mn (C) with φ(0) = 0. Then φ satisfies σT −S (x0 ) ⊆ σ(φ(T ) − φ(S)), (T, S ∈ Mn (C))

(9)

if and only if φ takes either the form (1) or (2). Let us emphasize once more that it is clear that a map φ on Mn (C) satisfies (8) provided that (3) or (5) holds, and thus Theorem 1 extends [4, Theorem 1.1] and [12, Theorem 1.1]. The same remark applies for reverse inclusions of the previous mentioned ones. With no extra efforts, the same proof as the ones of the above theorems yields the same conclusion by using sums in (8) and (9) instead of subtractions. Theorem 3. Let x0 be a nonzero vector in Cn . A surjective map φ on Mn (C) satisfies σφ(T )+φ(S) (x0 ) ⊆ σ(T + S), (T, S ∈ Mn (C)) if and only if φ takes either the form (1) or (2).

(10)

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The last result is variant of Theorem 2. Theorem 4. Let x0 be a nonzero vector in Cn . A map φ on Mn (C) satisfies σT +S (x0 ) ⊆ σ(φ(T ) + φ(S)), (T, S ∈ Mn (C))

(11)

if and only if φ takes either the form (1) or (2). We close this introduction with the notion of SVEP and a couple of comments about our main results. An operator T ∈ B(X) is said to have the single-valued extension property (abbreviated SVEP) if for every open subset U of C, the equation (λ − T )f (λ) = 0, (λ ∈ U ), has no nontrivial X-valued analytic solution f on U . Note that if T has the SVEP then σT (x) = ∅ for every nonzero vector x ∈ X, and that T has SVEP provided that its point spectrum has an empty interior. In particular, every matrix T ∈ Mn (C) has SVEP and thus σT (x) = ∅ for all T ∈ Mn (C) and all nonzero vectors x ∈ Cn . For further information on the local spectral theory, we refer the reader to the remarkable books [1,18]. We would like to point out here that the condition φ(0) = 0 in Theorem 1 and Theorem 2 is a harmless condition since if a map φ satisfies (8) (resp. (9)) then for any matrix R ∈ Mn (C) the map φ(.) + R satisfies as well (8) (resp. (9)). However, if a map φ on Mn (C) is surjective and satisfies (10) (resp. satisfies (11)), then φ(0) = 0 automatically holds. Indeed, assume for instance φ is a surjective map on Mn (C) satisfying (10), and note that 2σφ(T ) (x0 ) = σφ(T )+φ(T ) (x0 ) ⊂ σ(T + T ) = 2σ(T ) for all T ∈ Mn (C). Hence, σφ(T ) (x0 ) ⊂ σ(T )

(12)

for all T ∈ Mn (C). On the other hand, if φ(R) = 0 for some R ∈ Mn (C), then σφ(T ) (x0 ) = σφ(T )+φ(R) (x0 ) ⊂ σ(T + R)

(13)

for all T ∈ Mn (C). From (12) and (13), we see that ∅ = σφ(T ) (x0 ) ⊆ σ(T + R) ∩ σ(T ) for all T ∈ Mn (C). Now, Lemma 4 tells us that R = 0 and thus φ(0) = 0; as claimed. 2. Preliminaries In this section, we introduce some preliminary results which will be used to prove our main results in the next section. We start with the following result quoted from [9]. For an element a in a complex unital Banach algebra A, we denote by σ(a) = {λ ∈ C : a − λ is not invertible} the spectrum of a.

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Lemma 1. Let A be a complex semisimple unital Banach algebra, and a, b ∈ A. If σ(a +x) and σ(b + x) are finite and equal for all x in some nonempty open subset O of A, then A is finite dimensional and a = b. Proof. See [9, Corollary 2.4].

2

Let K(C) denote the set of all nonempty compact subsets of C, endowed with the Hausdorff metric. The following lemma was proved by Aupetit [2]. Lemma 2. Let X be a finite dimensional normed space. Then the map σ : T ∈ B(X) → σ(T ) ∈ K(C) is continuous. Proof. It is a consequence of [2, Corollary 3.4.5].

2

In the case of the local spectrum, Dollinger and Oberai in [15], showed that the local spectrum function is lower semi-continuous on Mn (C), in the following sense: let x0 be a nonzero vector in Cn and (Tk )k a sequence in Mn (C) such that Tk −→ T ∈ Mn (C), then σT (x0 ) ⊆ lim inf σTk (x0 ). k−→∞

The following result was used by several authors, and its proof is contained in [3]. Recall that the spectral radius of an operator T ∈ B(X) is given by r(T ) = max{|λ| : λ ∈ σ(T )}, 1

and coincides with the limit of the convergent sequence ( T k k )k . Lemma 3. Let (Ak )k be a sequence of matrices in Mn (C). If (r(Ak + B))k is bounded for all B in a neighborhood of 0 in Mn (C), then (Ak )k is bounded. Proof. See [3, The proof of Lemma 2.2].

2

The following lemma gives a spectral characterization of the zero matrix of Mn (C). Lemma 4. Let A be a matrix in Mn (C). The following conditions are equivalent: (1) A = 0. (2) σ(A + T ) ∩ σ(T ) is nonempty for every T ∈ Mn (C).

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Proof. See [21, Proposition 5.2].

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2

For every matrix T ∈ Mn (C), we denote by tr(T ) the usual trace of T and by Sk (T ) the kth symmetric function on the eigenvalues of T . Note that S1 (T ) = tr(T ), Sn (T ) = det(T ) and (tr(T ))2 = tr(T 2 ) + 2S2 (T ).

(14)

for all T ∈ Mn (C). Next, we introduce a subset of Mn (C) which plays an important role in the rest of the paper. Let x0 be a nonzero fixed vector in Cn , we denote by Mx0 the set given by Mx0 := {T ∈ Mn (C) : {T x0 , T 2 x0 , . . . , T n x0 } is a basis of Cn and |σ(T )| = n}, where |σ(T )| is the number of elements of σ(T ). We end this section with the following lemma that summarizes some important properties of Mx0 . Lemma 5. Let x0 be a nonzero fixed vector in Cn . Then the following statements hold. (1) Mx0 is an open dense subset of Mn (C). (2) If T ∈ Mx0 then σ(T ) = σT (x0 ). Proof. See [7] and [17].

2

3. Proof of the main theorems The main ingredients are collected in the previous section, and we therefore are able to prove the main results of this paper. In fact, in what follows, we shall only prove Theorem 1 and Theorem 2 as Theorem 3 and Theorem 4 have identical proofs to those of Theorem 1 and Theorem 2. Our arguments are influenced by the ones given in [12]. We start with the proof of Theorem 2. But before that, we add few more comments. Assume that φ is a map on Mn (C) satisfying (9) with φ(0) = 0, and note that, in view of Lemma 5 and the fact that the cardinality of the spectrum of each matrix in Mx0 is n, we always have σ(S − T ) = σ(φ(T ) − φ(S))

(15)

for all T, S ∈ Mn (C) such that S − T ∈ Mx0 . Since φ(0) = 0, we particularly have σ(T ) = σ(φ(T )) for all T ∈ Mx0 .

(16)

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Proof of Theorem 2. The “if” part is obvious. Indeed, assume that φ takes the form (1) and note that σAT A−1 (Ax0 ) = σT (x0 ) for all T ∈ Mn (C). This shows that σT −S (x0 ) = σA(T −S)A−1 (Ax0 ) = σφ(T )−φ(S) (Ax0 ) ⊆ σ(φ(T ) − φ(S)) for all T, S ∈ Mn (C), and thus (9) is satisfied. In a similar way, one shows that this inclusion remains valid if φ takes the form (2) instead of (1). To prove the “only if” part, suppose that φ satisfies (9) with φ(0) = 0. We aim to show that (15) holds for all T, S ∈ Mn (C) and thus [12, Theorem 2] applies to conclude that φ takes the desired forms. For that, let us first show that φ is continuous. To do so, it is enough to show that φ is continuous at 0 since for any R ∈ Mn (C) the map φR (T ) := φ(T + R) − φ(R), (T ∈ Mn (C)), satisfies (9) with φR (0) = 0, and the continuity of φR at 0 entails the continuity of φ at R. So, let A ∈ Mx0 and let us first show that there are two positive constants , δ such that {S ∈ Mn (C) : S − φ(A) < } ⊆ φ ({T ∈ Mn (C) : T − A < δ}) .

(17)

Since Mx0 is an open set, there is δ > 0 such that T and T − S are in Mx0 for all T, S ∈ Mn (C) such that A − T , S < δ. By (15), we see that σ(T − S) = σ(φ(T ) − φ(S)), ( A − T , S < δ).

(18)

In particular, tr(φ(T ) − φ(S)) = tr(T − S) and S2 (φ(T ) − φ(S)) = S2 (T − S) for all T, S ∈ Mn (C) such that A − T , S < δ. By (14), we have tr((φ(T ) − φ(S))2 ) = tr((T − S)2 ), ( A − T , S < δ).

(19)

Taking S = 0 in (19), we obtain that tr((φ(T ))2 ) = tr(T 2 )

(20)

for all T ∈ Mn (C) such that A − T < δ. On the other hand, (14) and (16) applied to any S ∈ Mx0 such that S < δ give tr((φ(S))2 ) = tr(S 2 )

(21)

for all S ∈ Mx0 such that S < δ. The linearity of the trace together with (19), (20) and (21) entail that tr(φ(T )φ(S)) = tr(T S) for all T, S ∈ Mn (C) such that A − T , S < δ and S ∈ Mx0 .

(22)

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Given a matrix X = (xij ) ∈ Mn (C), let us introduce its n2 row vector RX := (x11 , x12 , . . . , x1n , x21 , . . . , x2n , xn1 , . . . , xnn ) and its n2 column vector CX := (x11 , x21 , . . . , xn1 , x12 , . . . , xn2 , x1n , . . . , xnn )t . Since tr(XY ) = RX CY for all X, Y ∈ Mn (C), we rewrite (22) as follows: Rφ(T ) Cφ(S) = RT CS

(23)

for all T, S ∈ Mn (C) such that A − T , S < δ and S ∈ Mx0 . Let {B1 , . . . ., Bn2 } be a basis of Mn (C) such that Bj ∈ Mx0 and Bj < δ for all j = 1, . . . , n2 . Using Bj in place of S in (23), we obtain a system of n2 linear equations Rφ(T ) Cφ(Bj ) = RT CBj , j = 1, . . . , n2 . Introduce two n2 × n2 matrices: Q = [Cφ(B1 ) |Cφ(B2 ) |...|Cφ(Bn2 ) ] and C = [CB1 |CB2 |... |CBn2 ]. This system can be rewritten into Rφ(T ) Q = RT C,

( A − T < δ).

(24)

To show that Q is invertible, consider a basis {A1 , . . . , An2 } of Mn (C) such that

Aj − A < δ for all j = 1, . . . , n2 . Using Aj in place of T in (24), the identity (24) can be rewritten into a matrix equation PQ = RC, where R ∈ Mn2 (C) has rows {RA1 , . . . , RAn2 } and P ∈ Mn2 (C) has rows {Rφ(A1 ) , . . . , Rφ(An2 ) }. Since {A1 , . . . , An2 } is a basis, the matrix R is invertible. Likewise, since {B1 , . . . , Bn2 } is a basis, C is invertible. This implies that Q is also invertible. In particular, (24) becomes Rφ(T ) = RT W for all T ∈ Mn (C) such that A − T < δ, where W = CQ−1 . The invertibility of W implies that there exists a scalar  > 0 such that (17) holds. Now, let us prove that φ is continuous at 0. Let (Tk )k ⊆ Mn (C) be a sequence converging to 0, and let us show that (φ(Tk ))k is bounded. By (18), we have r(φ(Tk ) − φ(T )) = r(Tk − T ) ≤ T + Tk for all T ∈ Mn (C) such that A − T < δ, and k large enough. This and (17) entail that the sequence (r(B + φ(A) − φ(Tk )))k is also bounded for all B ∈ Mn (C) such that B < . By Lemma 3, the sequence (φ(Tk ) − φ(A))k is bounded and so is the sequence (φ(Tk ))k . Let (φ(Tkj ))j be a subsequence of (φ(Tk ))k

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converging to a matrix R ∈ Mn (C), and let us show that R = 0. By (18), we have σ(φ(T ) − φ(Tkj )) = σ(T − Tkj ) for all matrices T ∈ Mn (C) such that A − T < δ and j large enough. The continuity of the spectrum implies that σ(φ(T ) − R) = σ(T )

(25)

for all matrices T ∈ Mn (C) such that A − T < δ. The inclusion (17) implies that, for every matrix S in the open ball O of Mn (C) centered at φ(A) and of radius , there is T ∈ Mn (C) such that S = φ(T ) and A − T < δ, and thus σ(S − R) = σ(φ(T ) − R) = σ(T ) = σ(φ(T )) = σ(S) for all S ∈ O; see (16) and (25). By Lemma 1, we see that R = 0 and thus φ is continuous at 0. Hence, φ is in fact continuous on Mn (C); as desired. Finally, we show that φ takes the desired forms. Let T, S ∈ Mn (C), and note that, since Mx0 is dense in Mn (C), there exists a sequence (Tk )k ⊆ Mx0 converging to T − S. By (15), we have σ(Tk ) = σ(φ(Tk + S) − φ(S)) for all k. The continuity of φ and the continuity of the spectrum on Mn (C) (Lemma 2) imply that σ(T − S) = σ(φ(T ) − φ(S), (T, S ∈ Mn (C)).

(26)

By [12, Theorem 2], we conclude that φ takes either the form (1) or (2). The proof is therefore complete. 2 We close this paper with the proof of Theorem 1. Proof of Theorem 1. Just as the beginning of the above proof, one sees that the “if” part is obvious. So, we only need to prove the “only if” part. Suppose that φ satisfies (8), and let us show that φ is injective. Indeed, let A, B ∈ Mn (C) such that φ(A) = φ(B) and let us prove that A = B. By (8), we have σφ(T +A)−φ(A) (x0 ) ⊆ σ(T ) and σφ(T +A)−φ(A) (x0 ) = σφ(T +A)−φ(B) (x0 ) ⊆ σ(T + A − B) for all T ∈ Mn (C). Hence, ∅ = σφ(T +A)−φ(A) (x0 ) ⊆ σ(T + A − B) ∩ σ(T )

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for all T ∈ Mn (C). This and Lemma 4 imply that A − B = 0, and thus φ is injective. It is in fact a bijective map since it is supposed to be surjective. As φ−1 satisfies (9) and φ−1 (0) = 0, Theorem 2 applied to φ−1 shows that φ takes the desired forms. The proof is therefore complete. 2 Acknowledgement The authors wish to express their thanks to the referee for carefully reading the paper and for giving valuable suggestions. References [1] P. Aiena, Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer Acad. Publishers, 2004. [2] B. Aupetit, A Primer on Spectral Theory, Springer, New York, 1991. ˘ [3] R. Bhatia, P. Semrl, A.R. Sourour, Maps on matrices that preserve the spectral radius distance, Studia Math. 134 (1999) 99–110. [4] M. Bendaoud, M. Douimi, M. Sarih, Maps on matrices preserving local spectra, Linear Multilinear Algebra 61 (7) (2013) 871–880. [5] A. Bourhim, J. Mashreghi, Local spectral radius preservers, Integral Equations Operator Theory 76 (2013) 95–104. [6] A. Bourhim, Surjective linear maps preserving local spectra, Linear Algebra Appl. 432 (2010) 383–393. [7] A. Bourhim, V.G. Miller, Linear maps on Mn (C) preserving the local spectral radius, Studia Math. 188 (1) (2008) 67–75. [8] A. Bourhim, T.J. Ransford, Additive maps preserving local spectrum, Integral Equations Operator Theory 55 (2006) 377–385. [9] G. Braatvedt, R.M. Gareth, Uniqueness and spectral variation in Banach algebras, Quaest. Math. 36 (2) (2013) 155–165. [10] J. Bračič, V. Müller, Local spectrum and local spectral radius of an operator at a fixed vector, Studia Math. 194 (2009) 155–162. [11] C. Costara, Surjective maps on matrices preserving the local spectral radius distance, Linear Multilinear Algebra 62 (7) (2014) 988–994. [12] C. Costara, Maps on matrices that preserve the spectrum, Linear Algebra Appl. 435 (2011) 2674–2680. [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] G. Dolinar, J. Hou, B. Kuzma, X. Qi, Spectrum nonincreasing maps on matrices, Linear Algebra Appl. 438 (8) (2013) 3504–3510. [15] M. Dollinger, K. Oberai, Variation of local spectra, J. Math. Anal. Appl. 39 (1972) 324–337. [16] M.E. El Kettani, H. Benbouziane, Additive maps preserving operators of inner local spectral radius zero, Rend. Circ. Mat. Palermo 63 (2) (2014) 311–316. [17] M. González, M. Mbekhta, Linear maps on Mn (C) preserving the local spectrum, Linear Algebra Appl. 427 (2007) 176–182. [18] K.B. Laursen, M.M. Neumann, An Introduction to Local Spectral Theory, London Math. Soc. Monogr. (N.S.), vol. 20, The Clarendon Press, Oxford University Press, New York, 2000. [19] M. Marcus, B.N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math. 11 (1959) 61–66. [20] J. Mrčun, Lipschitz spectrum preserving mappings on algebras of matrices, Linear Algebra Appl. 215 (1995) 113–120. [21] A.R. Sourour, Inversibility preserving linear maps on L(X), Trans. Amer. Math. Soc. 348 (1996) 13–30.