and Michael White 1. Introduction

Transcription

1 SPECTRAL CHARACTERIZATION OF ALGEBRAIC ELEMENTS Thomas Ransford 1 and Michael White 1. Introduction Let A be a unital Banach algebra. An element a A is algebraic if there exists a nonconstant polynomial p such that p(a) = 0. In these circumstances, the spectrum σ(a) is a subset of the roots of p, and so is a finite set. Moreover, the following result of Aupetit and Zemánek [4, Theorem 2.7] shows that the spectrum σ(a + x) satisfies a Lipschitz condition for small perturbations a+x of a. Here and in what follows, denotes the usual Hausdorff distance between sets. Theorem 1.1. Let A be a unital Banach algebra and let a A. Suppose that p(a) = 0 for some non-constant polynomial p, and let m be the maximum of the multiplicities of the roots of p. Then there exist constants C, δ > 0 such that (1) ( σ(a + x), σ(a) ) C x 1/m (x A, x < δ). Another proof of this result is sketched in Johnson s review [5] of [4], and a further can be found in [6], the latter yielding better estimates for the constants C, δ. The purpose of this article is to establish the converse to Theorem 1.1. Note that Theorem 1.1 remains true if a is merely algebraic modulo the radical, and this is thus the most one could hope to prove in any converse. If we want to conclude that a is genuinely algebraic, we need to assume in addition that the radical is zero, i.e. that A is semisimple. Theorem 1.2. Let A be a semisimple unital Banach algebra and let a A. Suppose that σ(a) is the finite set {λ 1,..., λ k }, and that there exist an integer m 1 and constants C, δ > 0 such that (1) holds. Then p(a) = 0, where p(z) = k j=1 (z λ j) m Mathematics Subject Classification : Primary 46H05; Secondary 47A10 Research supported by grants from NSERC (Canada) and the Fonds FCAR (Québec) 1

2 Theorem 1.2 was motivated in part by the following result of Aupetit [3, Theorem 1.1], which now becomes a special case. Corollary 1.3. Let A be a semisimple unital Banach algebra and let a A. Suppose that σ(a) {0, 1} and that there exist constants C, δ > 0 such that ( σ(a + x), σ(a) ) C x (x A, x < δ). Then a is an idempotent. Thus idempotents can be characterized purely in terms of the spectrum. This result was used by Aupetit in [3] to show that if T is a spectrum-preserving linear isomorphism between semisimple Banach algebras, then T maps idempotents to idempotents (leading ultimately to a proof of a conjecture of Kaplansky in the case of von Neumann algebras). Evidently, Theorem 1.2 now permits us to conclude that such a T also sends algebraic elements to algebraic elements, nilpotents to nilpotents, etc. Another case of Theorem 1.2 worthy of note corresponds to taking {λ 1,..., λ k } = {0}. In this case, the hypothesis can be expressed simply in terms of the spectral radius ρ. Corollary 1.4. Let A be a semisimple unital Banach algebra and let a A. Suppose that there exists an integer m 1 and constants C, δ > 0 such that ρ(a + x) C x 1/m (x A, x < δ). Then a m = Proof of a special case The proof of Theorem 1.2 actually goes via the special case cited in Corollary 1.4. Our aim in this section is to prove the following slight extension of Corollary 1.4, which can also be viewed as a generalization of a version of Zemánek s characterization of the radical [1, Theorem (v)]. 2

3 Theorem 2.1. Let A be a unital Banach algebra and let a, b A. Suppose that there exist an integer m 1 and constants C, δ > 0 such that (2) ρ(a + bx) C x 1/m (x A, x < δ). Then a m b rad(a). Evidently Corollary 1.4 follows upon taking b = 1. However, the proof based on an induction for which it is convenient to work with a general b A. It uses some ideas borrowed from [2], together with the following reduction lemma. Lemma 2.2. Let A be a unital Banach algebra, let a, b A and let α > 1. Suppose that there exist constants C, δ > 0 such that (3) ρ(a + bx) C x 1/α (x A, x < δ). Then there exist constants C, δ > 0 such that (4) ρ(a + abx) C x 1/(α 1) (x A, x < δ ). have Proof of Lemma 2.2. Assume that (3) holds. Setting C 1 = max(c, a δ 1/α ), we (5) ρ(a + bx) C 1 x 1/α + b x (x A). Indeed, if x < δ, then this follows immediately from (3), while if x δ then it is an easy consequence of the fact that ρ(a + bx) a + bx a + b x. Without loss of generality b 0. Set δ = 1/(3 b ). We shall show that (4) holds for an appropriate value of C. Let x A with x < δ. Then bx b δ = 1/3, so 1 bx is invertible in A. Set y = (1 bx) 1. Note that 1 y = bx(1 + bx + (bx) 2 + ) = bxz, where z 3/2. Let λ C \ {0}. Then ρ(1 y λa) = ρ(bxz + λa) = λ ρ(a + bxz/λ) λ ( C 1 xz/λ 1/α + b xz/λ ) C 1 λ 1 1/α (3 x /2) 1/α + 1/2. 3

4 This will be < 1 provided that C 1 λ 1 1/α (3 x /2) 1/α < 1/2. Suppose that λ satisfies this, i.e. 1/λ > 2(3C1 α x ) 1/(α 1). Then ρ(1 y λa) < 1, which implies that y + λa is invertible, and hence so is (y + λa)(1 bx). But (y + λa)(1 bx) = 1 λ(a abx), so 1/λ / σ(a abx). Therefore ρ(a abx) 2(3C1 α x ) 1/(α 1). This proves that (4) holds with C = 2(3C1 α ) 1/(α 1). Proof of Theorem 2.1. Suppose that (2) holds. Without loss of generality, δ 1, so (3) holds with α = m We now apply the lemma repeatedly, first with α = m + 1 2, then with α = m 1 2 and b replaced by ab etc. After m such applications, we deduce that there exist constants C, δ > 0 such that (6) ρ(a + a m bx) C x 2 (x A, x < δ ). Now fix x A, and consider the function λ ρ(λa + a m bx). If λ > x /δ, then using (6) we have ρ(λa + a m bx) = λ ρ(a + a m b(x/λ)) λ C x/λ 2 = C x 2 / λ. But also, by Vesentini s theorem [1, Theorem 3.4.7], the function λ ρ(λa + a m bx) is subharmonic on C. Hence, by the maximum principle, for each R > x /δ we have Letting R, we obtain ρ(a m bx) = 0. ρ(a m bx) max λ =R ρ(λa + am bx) C x 2 /R. Finally, since x is an arbitrary element of A, the standard characterization for the radical (see e.g. [1, p.36]) allows us to conclude that a m b rad(a). 3. Proof of the General Case To go from the special case σ(a) = {0} to the general case σ(a) = {λ 1,..., λ k }, we employ a localization technique similar to that used in [3]. It is based upon the following lemma. (We write ρ B to denote the spectral radius measured with respect to an algebra B.) 4

5 Then: Lemma 3.1. Let A be a unital Banach algebra, and let e A be an idempotent. (i) eae is a unital Banach algebra with identity e; (ii) ρ eae (x) = ρ A (x) for all x eae; (iii) rad(eae) = rad(a) eae. Proof. (i) It is clear that eae is an algebra with identity e. Also it is closed in A, for if (x n ) is a sequence in eae and x n x in A, then also x n = ex n e eye, so y = eye eae. It may happen that e > 1, but it is always possible to re-norm eae with an equivalent norm such that e = 1. (ii) For x eae, the spectral radius formula applied both in eae and in A gives ρ eae (x) = lim n xn 1/n = lim n xn 1/n = ρ A (x). (iii) Let x eae. Then, given y A, ρ A (xy) = ρ A (exey) = ρ A (xeye) = ρ eae (x(eye)), the last equality using (ii). Hence, by the standard characterization of the radical, x rad(a) ρ A (xy) = 0 (y A) ρ eae (xeye) = 0 (y A) x rad(eae). This completes the proof. We shall also need a result about the spectrum of the sum of orthogonal elements. Lemma 3.2. Let A be a unital Banach algebra and let a, b A. Suppose that ab = ba = 0. Then σ(a + b) {0} = σ(a) σ(b) {0}. Proof. Given a subset S of A, we write S c for the set of elements in A which commute with every element of S. It is readily checked that S c is a closed (unital) subalgebra of A, which is full, in the sense that each element of S c which is invertible in A is also invertible 5

6 in S c. Also, if the elements of S commute with one another, then, although S c need not be commutative, S cc always is. Indeed, S S c implies S c S cc, which in turn implies S cc S ccc, as claimed. Armed with these preliminary observations, we can easily prove the lemma. B = {a, b} cc. Then B is a full commutative subalgebra of A, so for each x B we have σ A (x) = σ B (x) = {χ(x) : χ Φ B }, where Φ B denotes the character space of B. Moreover, given χ Φ B, we have χ(a)χ(b) = χ(ab) = χ(0) = 0, so either χ(a) = 0 or χ(b) = 0. Thus, given λ C \ {0}, we have λ σ(a + b) iff λ = χ(a) for some χ Φ B with χ(b) = 0, or λ = χ(b) for some χ Φ B with χ(a) = 0, and this in turn is true iff λ σ(a) σ(b). This proves the lemma. Set We are now ready to prove the main result. Proof of Theorem 1.2. Let e 1,..., e k be the spectral projections of a corresponding to λ 1,..., λ k respectively. The e j are orthogonal projections whose sum is 1, and they all commute with a. Let x e 1 Ae 1. Then a λ x = e 1 (a λ x)e 1 + By Lemma 3.2, applied repeatedly, we have k e j (a λ 1 1)e j. 2 σ(a λ x) {0} = σ ( ) k e 1 (a λ x)e 1 σ ( ) e j (a λ 1 1)e j {0}. Using the spectral mapping theorem for the functional calculus, we have σ(e j (a λ 1 1)e j ) = {0, λ j λ 1 }. Hence σ(a λ x) {0} = σ ( ) e 1 (a λ x)e 1 {0, λ2 λ 1,..., λ k λ 1 }. Set η = min 2 j k λ j λ 1 (or η = 1 if k = 1). By upper semicontinuity of the spectrum, if x is sufficiently small, then σ(e 1 (a λ 1 + x)e 1 ) is contained in an η/2-neighbourhood of σ(e 1 (a λ 1 1)e 1 ) = {0}. Hence, for such x, ( σ(e 1 (a λ x)e 1 ), σ(e 1 (a λ 1 1)e 1 ) ) = ( σ(a λ x), σ(a λ 1 1) ) = ( σ(a + x), σ(a) ). 6 2

7 Now the left-hand side is just ρ(e 1 (a λ x)e 1 ), and, if x < δ, then by (1) the right-hand side is majorized by C x 1/m. Hence, there exists δ such that, ρ(e 1 (a λ 1 1)e 1 + x) C x 1/m (x e 1 Ae 1, x < δ ). By Lemma 3.1, this implies that ρ e1 Ae 1 (e 1 (a λ 1 1)e 1 + x) C x 1/m (x e 1 Ae 1, x < δ ). Applying Theorem 2.1 to the algebra e 1 Ae 1, we deduce that (e 1 (a λ 1 1)e 1 ) m rad(e 1 Ae 1 ). Now rad(e 1 Ae 1 ) = rad(a) e 1 Ae 1 = {0}, since A is semisimple. Thus e 1 (a λ 1 1) m e 1 = 0. Similarly we have e j (a λ j 1) m e j = 0 for j = 1, 2,..., k. Hence, finally, (a λ 1 1) m (a λ k 1) m = This completes the proof. k e j (a λ 1 1) m e j e j (a λ k 1) m e j = 0. j=1 Acknowledgement. while this paper was being written. The second author thanks Université Laval for its kind hospitality References [1] B. Aupetit, A Primer on Spectral Theory (Springer-Verlag, New York, 1991). [2] B. Aupetit, Spectral characterization of the radical in Banach and Jordan Banach algebras, Math. Proc. Camb. Phil. Soc. 114 (1993), [3] B. Aupetit, Spectrum-preserving linear mappings between Banach algebras or Banach Jordan algebras, preprint. [4] B. Aupetit and J. Zemánek, Local behaviour of the spectrum near algebraic elements, Linear Algebra Appl. 52/53 (1983), [5] B. E. Johnson, review of [4] above, Mathematical Reviews, 84h: [6] A. Nokrane and T. J. Ransford, Estimates for the spectrum near algebraic elements, preprint. 7

8 Département de mathématiques et de statistique, Université Laval, Québec (QC), Canada G1K 7P4 Department of Mathematics, University of Newcastle, Newcastle upon Tyne NE1 7RU United Kingdom 8