Integr. equ. oper. theory 54 (2006), 1–8 c 2006 Birkh¨ auser Verlag Basel/Switzerland 0378-620X/010001-8, published online October 17, 2005 DOI 10.1007/s00020-005-1375-3
Integral Equations and Operator Theory
Local Spectral Theory of Linear Operators RS and SR C. Benhida and E.H. Zerouali Abstract. In this paper, we study the relation between local spectral properties of the linear operators RS and SR. We show that RS and SR share the same local spectral properties SVEP, (β), (δ) and decomposability. We also show that RS is subscalar if and only if SR is subscalar. We recapture some known results on spectral properties of Aluthge transforms. Mathematics Subject Classification (2000). Primary 47A11; Secondary 47A53. Keywords. Local spectral theory, linear operators, extensions, Aluthge transform.
1. Introduction Throughout this paper, X and Y are Banach spaces and L(X, Y ) denotes the space of all bounded linear operators from X to Y . For a bounded linear operator T ∈ L(X) =: L(X, X), let σ(T ) denote the spectrum, Im(T ) the range and ker(T ) the null space. The local resolvent set ρT (x) of T at x ∈ X is defined as the set of all λ ∈ C for which there exists an analytic X-valued function f on some open neighbourhood U of λ such that (T − µ)f (µ) = x for all µ ∈ U . The local spectrum of T at x is σT (x) = C \ ρT (x), [10]. Note that σT (x) is a closed subset of σ(T ) and it may happen that σT (x) is the empty set. Let D(λ, r) be the open disc centred at λ ∈ C with radius r > 0, and let D(λ, r) be the corresponding closed disc. We say that T has the single valued extension property ( SVEP, for short) at λ ∈ C if there exists r > 0 such that for every open subset U ⊂ D(λ, r), the only analytic solution of the equation (T − µ)f (µ) = 0 is the constant function f ≡ 0. We denote by S(T ), the open set where T fails to have the SVEP. An operator T is said to have the SVEP when T satisfies this property at every complex number (S(T ) = ∅).
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Let U be an open subset of the complex plane and O(U, X) be the Fr´echet algebra of all analytic X-valued functions on U endowed with uniform convergence on compact sets of U. The operator T is said to satisfy Bishop’s property (β) at λ ∈ C if there exists r > 0 such that for every open subset U ⊂ D(λ, r) and for any sequence (fn )n ⊂ O(U, X), if lim (T − µ)fn (µ) = 0 in O(U, X), then n−→∞
lim fn (µ) = 0 in O(U, X). We denote by σβ (T ) by the set where T fails to satisfy
n−→∞
(β) and we say that T satisfies Bishop’s property (β) precisely when σβ (T ) = ∅. The operator T is said to have the decomposition property (δ) if T ∗ satisfies (β). Note that it is known that T is decomposable ”in the sense of Foias” if and only if T satisfies either (δ) and (β). We refer to [10] for more details and further definitions. The property (β) is defined in a similar way as for property (β). To be precise; let E(U, X) be the Fr´echet algebra of all infinitely differentiable X-valued functions on U ⊂ C endowed with the topology of uniform convergence on compact subsets of U of all derivatives. The operator T is said to have property (β) at λ if there exists U a neighbourhood of λ such that for each open set O ⊂ U and for any sequence (fn )n of X-valued functions in E(O, X) the convergence of (T − z)fn (z) to zero in E(O, X) yields to the convergence of fn to zero in E(O, X). Denote by σ(β) (T ) the set where T fails to satisfy (β) . We will say that T satisfies property (β) if σ(β) (T ) = ∅. It is known that property (β) characterizes those operators with some generalized scalar extension (called subscalar). An operator T is said to be generalized scalar if there exists a continuous algebra homomorphism Φ : E(C) → L(X) with Φ(1) = I and Φ(z) = T . See [6], for more details. Our aim is to show (in the light of [4]) that if S : X → Y and R : Y → X are bounded linear operators then SR and RS share many of their local spectral properties. It is proved in section 2 that SR has the single valued extension property (resp. Bishop’s property (β), resp. spectral decomposition property (δ), resp. is subscalar ) if and only if RS has the single valued extension property (resp. Bishop’s property (β), resp. spectral decomposition property (δ), resp. is subscalar). The latter result extends a previous one in [11] given for injective operators. We devote section 3 to show the connection between local spectra of RS and SR and we give in section 4 some applications to Aluthge transforms.
2. Local spectral theory for RS and SR Let S : X → Y and R : Y → X be bounded linear operators. We start with the following result. Proposition 2.1. Let λ ∈ C be a complex number, then RS has the SVEP (resp. (β)) at λ if and only if SR has the SVEP (resp. (β)) at λ. In particular RS has the SVEP (resp. (β)) if and only if SR has the SVEP (resp. (β)). Proof. We only give the proof for (β), the case of the SVEP is clearly similar. Let λ ∈ C \ σβ (RS) and let (fn )n be a sequence of Y -valued analytic functions in a
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neighbourhood of λ such that lim (SR − µ)fn (µ) = 0 in O(V (λ), Y ).
(1)
n→+∞
Then lim (RS − µ)Rfn (µ) = 0 in O(V (λ), X). It follows that lim Rfn (µ) = 0 n→+∞
in O(V (λ), X) and then,
n→+∞
lim SRfn (µ) = 0. In Equation (1), we deduce that
n→+∞
(µfn (µ))n converges to 0 on compact sets. Now since fn are analytic, the maximum modulus principle implies (fn )n converges to 0 on compact sets. Thus λ ∈ C \ σβ (SR). The previous result extends [12, Theorem 5] by removing the injectivity of R and S. By passing to adjoints in Proposition 2.1, and by using the definition of decomposable operators, we obtain Corollary 2.1. RS has the decomposition property (δ) (resp. is decomposable) if and only if SR has the decomposition property (δ) (resp. is decomposable). For (β) property, we have Theorem 2.1. σ(β) (RS) = σ(β) (SR). In particular RS is subscalar if and only if SR is subscalar We need a lemma of independent interest. Lemma 2.1. Let O be an open set and (fn )n be a sequence in E(O, X) such that (zfn (z))n converges to zero in E(O, X). Then (fn )n converges to zero in E(O, X). Proof. Consider the operator Tz : f ∈ E(O, X) → g ∈ E(O, X) given by g(z) = zf (z). From the division properties in E(O, X), it follows that Tz has a closed range ([13, p. 72]). Since Tz is clearly one-to-one it follows that Tz is left invertible. The lemma is hence proved. Proof of Theorem 2.1. Suppose λ ∈ / σ(β) (RS) and let O be a neighbourhood of λ such that O ∩ σ(β) (RS) = ∅. If (fn )n is any sequence in E(O, X) such that (RS − z)fn (z) converges to zero in E(O, X), then lim (SR − z)Sfn (z) = 0 and n→∞
hence lim Sfn (z) = 0. It follows that (zfn (z))n converges to zero in E(O, X). By n→∞
/ σ(β) (SR). lemma 2.1, we have (fn )n converges to zero in E(O, X). Finally λ ∈ The reverse implication is obtained by symmetry. The previous theorem is given by Lin Chen et al. [11, Theorem A] under the extra assumption that R and S are one-to-one.
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3. Local spectra for RS and SR Proposition 3.1. Let R and S be as above. For every x ∈ X, we have i) σSR (Sx) ⊂ σRS (x) ⊂ σSR (Sx) ∪ {0} (resp. σRS (Ry) ⊂ σSR (y) ⊂ σRS (Ry) ∪ {0} for every y ∈ Y ). ii) If moreover S is one-to-one, σRS (x) = σSR (Sx) (resp. if R is one-to-one, σRS (Ry) = σSR (y) for every y ∈ Y ). Proof. Let λ ∈ / σRS (x) and x(µ) be an X-valued analytic function in a neighbourhood O of λ such that x = (RS−µ)x(µ) for every µ ∈ O. Then Sx = (SR−µ)Sx(µ) and hence λ ∈ / σSR (Sx). The first inclusion in i) is proved. To show the second inclusion, let λ ∈ / σSR (Sx) ∪ {0} and y(µ) be an Y -valued analytic function on an open neighbourhood O of λ, such that Sx = (SR−µ)y(µ)(µ ∈ O). For z(µ) =: µ1 (Ry(µ) − x), it is easy to check that x = (RS − µ)z(µ) and thus λ∈ / σRS (x). To prove ii), it suffices to consider the case λ = 0. Suppose 0 ∈ / σSR (Sx) and let y(µ) be an Y -valued analytic function in a neighbourhood of 0 such that Sx = (SR − µ)y(µ). Note first that from the injectivity of S, it follows that x = Ry(0). Moreover, we have µy(µ) = SRy(µ) − Sx = S(Ry(µ) − x) and so y(µ) = S[ µ1 (Ry(µ) − x)]. Set 1 if µ = 0 µ (Ry(µ) − x) z(µ) = if µ = 0, Ry (0) then, S[x − (RS − µ)z(µ)] = 0 and since S is one-to-one we get x = (RS − µ)z(µ) and hence 0 ∈ / σRS (x). If S is not one-to-one, it may happen that σRS (x) = σSR (Sx). To this aim, let S be the shift operator defined on the usual Hardy space and let S ∗ be its adjoint operator. Then, S ∗ S is the identity operator, while SS ∗ is the projection operator on Im(S). In particular σS ∗ S (x) = {1}, for every nonzero x ∈ H, σSS ∗ (x) = {1} if x ∈ Im(S), σSS ∗ (x) = {0} if x ∈ ker(S ∗ ) and σSS ∗ (x) = {0, 1} otherwise. The inclusion σS ∗ S (Sx) ⊂ σSS ∗ (x) is then strict in the last case. It is known that σ(RS) \ {0} = σ(SR) \ {0} and the previous example shows that the equality σ(RS) = σ(SR) fails to be true in general. The later equality has been proved in the case where R (or S) is normal, see [5]. As an immediate application of Proposition 3.1, we obtain Corollary 3.1. Let R and S be as above, then σ(RS) = σ(SR) in the following cases: 1. S and R are injective. 2. S or R is injective with dense range. Proof. For an arbitrary operator T , we have σ(T ) = σT (x) ∪ S(T ), see [3]. Now x if S is injective, then σ(RS) = σRS (x) ∪ S(RS) = σSR (Sx) ∪ S(SR) ⊂ x∈X
x∈X
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σSR (y) ∪ S(SR) = σ(SR), (1) is then obtained by symmetry.
(2) by passing to duals.
4. Applications to Aluthge transforms 4.1. Aluthge transforms Let T ∈ L(H) be a bounded operator on some Hilbert space H and U |T | be 1 its polar decomposition, where |T | = (T ∗ T ) 2 and U is the appropriate partial isometry. The generalized Aluthge transform associated with T and s, t ≥ 0, is 1 1 defined by T (s, t) = |T |s U |T |t . In the case s = t = 12 , the operator T˜ = |T | 2 U |T | 2 is called the Aluthge transform of T and was first considered by A. Aluthge to extend some inequalities related to hyponormality. More precisely, an operator A ∈ L(H) is said to be p-hyponormal (p > 0) if (A∗ A)p ≥ (AA∗ )p (i.e. |A|2p ≥ |A∗ |2p ). A 12 −hyponormal operator is called semi-hyponormal ([1], [16]). The L¨ owner-Heinz inequality implies that if A is q-hyponormal then it is p-hyponormal for any 0 < p ≤ q. An invertible operator A is said to be log-hyponormal ( [15]) if log(A∗ A) ≥ log(AA∗ ). It is known that if T is p-hyponormal (p > 0), then T˜ is semihyponormal ˜ and T˜ is hyponormal. If T is log-hyponormal, then T˜˜ is semihyponormal. T is said to be w-hyponormal ([2]) if |T˜| ≥ |T | ≥ |T˜ ∗ |, so T˜ is semi-hyponormal if T is whyponormal. It is also known ([2]) that p-hyponormal (p > 0) and log-hyponormal operators are w-hyponormal. Since then this concept received a lot of interest by numerous mathematicians. In series of papers [7, 8, 9], I. B. Jung, E. Ko and C. Pearcy have investigated common spectral properties of T˜ and T . They showed that T˜ and T share most of their spectral properties. Let r ≤ t, R = |T |r and S = |T |s U |T |t−r . Then SR = T (s, t) and RS = T (s + r, t − r). It follows that T (s, t) and T (s + r, t − r) (and in particular T˜ and T ) almost have the same local spectral properties. Proposition 4.1. Let T ∈ L(H), s ≥ 0 and 0 ≤ r ≤ t, then T (s, t) has the property (β) (resp. (δ) or is subscalar) if and only if T (s + r, t − r) has the property (β) (resp. (δ) or is subscalar). In particular, T =: T (0, 1) has the property (β) (resp. (δ) or is subscalar) if and only if T (r, 1 − r) (0 < r < 1) has the property (β) (resp. (δ) or is subscalar) if and only if T˜ has the property (β) (resp. (δ) or is subscalar). The previous result stands in [11] under the restrictive condition ker(T ) ⊂ ker(T ∗ ). We also deduce Corollary 4.1. If T ∈ L(H) is p-hyponormal, log-hyponormal or w-hyonormal, then T is subscalar.
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4.2. Spectral picture Recall that an operator T ∈ L(H) is a Fredholm operator if Im(T ) is closed and if max(dim(ker(T )), codim(Im(T ))) is finite. Left Fredholm operators (resp. Right Fredholm operators) are given by operators T with closed range and such that dim(ker(T )) is finite (resp. codim(Im(T )) is finite). An operator is said to be semiFredholm if it is either left Fredholm or right Fredholm. Thus T is semi-Fredholm if Im(T ) is closed and if min(dim(ker(T )), codim(Im(T ))) is finite. The index of a semi-Fredholm operator T is defined to be ind(T ) = dim(ker(T )) − codim(Im(T )). Denote by F (resp. SF , LF and RF ) the family all Fredholm (resp. semiFredholm, left Fredholm and right Fredholm) operators. The essential spectrum / F }. The left essential spectrum σle (T ) and of T is σe (T ) = {λ ∈ C : T − λ ∈ the right essential spectrum σre (T ) are defined similarly. Weyl spectrum, σw (T ), is the set of all λ ∈ C such that T − λ is not Fredholm with index zero. It is well known that the mapping ind : SF → Z ∪ {−∞, +∞} is continuous. We shall call a hole in σe (T ) any bounded component of C\σe (T ) and a pseudo hole a component of σe (T ) \ σle (T ) or σe (T ) \ σre (T ) The spectral picture SP (T ) of an operator was introduced by C. Pearcy in [14] as the collection of holes and pseudo holes and the associated Fredholm indices. Since for every λ = 0, Im(RS − λ) is closed if and only if Im(SR−λ) is closed and dim(ker(RS−λ)) = dim(ker(SR−λ)), one conclude easily that RS and SR have the same spectral picture. Thus in section 4.1, T and T˜ have the same spectral picture as proved in [8]. For T a bounded operator, we write iso σ(T ) for the set of all isolated points of σ(T ), and we define Π00 (T ) = {λ ∈ iso σ(T ) : 0 < dim(ker(T − λ)) < ∞}. An operator is said to satisfy Weyl’s theorem if σw (T ) = σ(T ) \ Π00 (T ) From the discussion above, it follows that Π00 (RS) \ {0} = Π00 (SR) \ {0} and that σw (RS) \ {0} = σw (SR) \ {0}. Thus / Π00 (SR) ∪ Π00 (RS), then RS Proposition 4.2. If 0 ∈ Π00 (SR) ∩ Π00 (RS) or 0 ∈ satisfies Weyl’s theorem if and only if SR satisfies Weyl’s theorem. The preceding Proposition and Corollary 3.1 allow to get Corollary 4.2. If R and S are injective, then RS satisfies Weyl’s theorem if and only if SR satisfies Weyl’s theorem. Since σp (T ) = σp (T˜ ), we have Corollary 4.3. T satisfies Weyl’s theorem if and only if T˜ satisfies Weyl’s theorem. We finally note that the equivalence in Proposition 4.2 does not hold in general. Indeed, let S be the unilateral shift on the Hardy space and R = S ∗
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its adjoint operator. Then RS is the identity operator and hence satisfies clearly Weyl’s theorem, while for SR, we have σw (SR) = {0, 1} = σ(SR) \ Π00 (SR) = {1}.
References [1] A. Aluthge, On p-hyponormal operators for 0 < p < 1. Integral Equations Operator Theory 13 (1990), 307–315. [2] A. Aluthge and D. Wang, w-hyponormal operators II. Integral Equations Operator Theory 37 (2000), 324–331. [3] J. K. Finch, The single valued extension property on a Banach space. Pacific J. Math. 58 (1975), 61–69. [4] B. Barnes, Common operator properties of the linear operators RS and SR. Proc. Amer. Math. Soc. 126 (1998), 1055–1061. [5] M. Cho, On spectra of AB and BA, Proc. KOTAC 3 (2000), 15–19. [6] J. Eschmeier and M. Putinar, Bishop’s condition (β) and rich extensions of linear operators. Indiana Univ. Math. J. 37 (1998), 325–348. [7] I. B. Jung, E. Ko and C. Pearcy, Aluthge transforms of operators. Integral Equations Operator Theory 37 (2000), 437–448. [8] I. B. Jung, E. Ko and C. Pearcy, Spectral picture of Aluthge transforms of operators. Integral Equations Operator Theory 40 (2001), 52–60. [9] I. B. Jung, E. Ko and C. Pearcy, The iterated Aluthge transforms. Integral Equations Operator Theory 45 (2003), 375–387. [10] K. B. Laursen, M.M. Neumann, An Introduction to Local Spectral Theory. London Mathematical Society Monograph, New series 20, 2000. [11] Lin Chen, Yan Zikun and Ruan Yingbin, p-Hyponormal operators are subscalar. Proc. Amer. Math. Soc. 131 (2003), 2753–2759. [12] Lin Chen, Yan Zikun and Ruan Yingbin, Common operator properties of operators RS and SR and p−Hyponormal operators. Integral Equations Operator Theory 43 (2002), 313–325. [13] B. Malgrange, Ideals of differentiable functions. Oxford University Press, London, 1967. [14] C. Pearcy, Some recent developments in operator theory. CBMS Regional Conference Serie in Mathematics 36, Amer. Math. Soc., Providence, 1978. [15] Kˆ otarˆ o Tanahashi, On log-hyponormal operators, Integral Equations Operator Theory 34 (1999), 364–372. [16] D. Xia, Spectral theory of hyponormal operators, Birkh¨ auser Verlag, Basel, 1983.
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C. Benhida Universit´e de Lille 1 UFR de Math´ematiques CNRS-UMR 8524, Bˆ at M2 59655 Villeuneuve cedex France e-mail:
[email protected] E.H. Zerouali Facult´e des Sciences de Rabat BP 1014 Rabat Morocco e-mail:
[email protected] Submitted: January 30, 2004 Revised: October 30, 2004
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