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Polynomials of Almost-Normal Arguments in $ C^* $-Algebras

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Polynomials of Almost-Normal Arguments in $ C^* $-Algebras Polynomials of almost normal arguments in C∗-algebras N. Filonov, I. Kachkovskiy∗ Abstract The functional calculus for normal elements in C∗-algebras is an important tool of analysis. We consider polynomials p(a, a∗) for elements a with small self-commutator norm [a, a ] 6 δ and show that many properties of the functional calculus are retained k ∗ k modulo an error of order δ. 2010 Mathematics Subject Classification: 47A60, 46L05, 11E25. Keywords: C∗-algebras, functional calculus, self-commutator, polynomials, Positivstel- lensatz, pseudospectrum. 1 Introduction Let a be a normal element of a unital C∗-algebra . It is well known that there exists a unique A C∗-algebra homomorphism C(σ(a)) , f f(a) →A 7→ from the algebra of continuous functions on the spectrum σ(a) into such that f(z) = z is mapped into a, σ(f(a)) = f(σ(a)), and A f(a) = max f(z) (1.1) k k z σ(a) | | ∈ (see, for example, [4]). It is called the functional calculus for normal elements and is widely used in analysis. The aim of the present paper is to introduce an analogue of functional calculus for “almost arXiv:1107.6010v2 [math.OA] 10 Feb 2012 normal” elements. More precisely, we shall always be assuming that a 6 1, [a, a∗] 6 δ (1.2) k k k k with a small δ. We restrict the considered class of functions to polynomials in z andz ¯ and show that some important properties of the functional calculus hold up to an error of order δ. If aa∗ = a∗a then the polynomials of a and a∗ are, in general, not uniquely defined. We fix 6 the following definition. For a polynomial k l p(z, z¯)= pklz z¯ (1.3) Xk,l ∗Steklov Institute, St. Petersburg, and King’s College London. The first author was supported by RFBR Grant 11-01-00324-a. The second author was supported by King’s Annual Fund Studentship and King’s Overseas Research Studentship. 1 let k l p(a, a∗)= pkla (a∗) . (1.4) Xk,l It is clear that the map p p(a, a∗) is linear and involutive, that is p(a, a∗)= p(a, a∗)∗ where k l 7→ m m 1 p¯(z, z¯)= p¯ z z¯ . Using the inequality [a, b ] 6 m b − [a, b] and (1.2), one can easily lk k k k k k k show that the map p p(a, a∗) is “almost multiplicative”, P 7→ p(a, a∗)q(a, a∗) (pq)(a, a∗) 6 C(p, q) δ (1.5) k − k where C(p, q)= ls p q . | kl|| st| k,l,s,tX It takes much more effort to obtain an estimate of the norm p(a, a∗) . In the case of an k k k analytic polynomial p(z)= k pkz , according to the von Neumann inequality, P p(a) 6 max p(z) =: pmax k k z 61 | | | | where it is only assumed that a 6 1 (see, for example, [13, I.9]). Our main results are as follows.k k Theorem 1.1. Let p be a polynomial (1.3). There exists a constant C(p) such that the estimate p(a, a∗) 6 p + C(p)δ (1.6) k k max holds for all a satisfying (1.2). Here p(a, a∗) is defined by (1.4), and pmax = max p(z, z¯) . z 61 | | | | If a is normal and f is a continuous function then the functional calculus gives the following more precise estimate, f(a) = max f(z) . (1.7) k k z σ(a) | | ∈ If a and λ σ(a), j =1,...,m 1, then there exists R > 0 such that ∈A j 6∈ − j 1 1 (a λ )− 6 R− , j =1,...,m 1. (1.8) k − j k j − The following theorem gives an analogue of (1.7) for an almost normal a. Theorem 1.2. Let a satisfy (1.2) and (1.8), and let the set ∈A S = z C: z 6 1, z λ > R , j =1,...,m 1 (1.9) { ∈ | | | − j| j − } be nonempty. For each ε > 0 and each polynomial p defined by (1.3) there exists a constant C(p,ε) independent of a such that p(a, a∗) 6 max p(z, z¯) + ε + C(p,ε)δ. z S k k ∈ | | 2 Note that, under the conditions of Theorem 1.2, the set S is a unit disk with m 1 “holes” which contains σ(a). − Finally, assume again that a is normal and µ / f(σ(a)). Then the functional calculus ∈ implies that the element (f(a) µ) is invertible and − 1 1 (f(a) µ)− = . (1.10) − dist (µ, f(σ(a))) The equality (1.10) also admits the following approximate analogue with σ(a) replaced by S and f(σ(a)) by p(S), where p(S) is the image of S under p considered as a map from C to C. Theorem 1.3. Let S be defined by (1.9), and let p be a polynomial (1.3). Then for each ε> 0 and κ > 0 there exist constants C(p, κ,ε), δ0(p, κ,ε) such that for all δ < δ0(p, κ,ε) and for all µ C satisfying dist(µ,p(S)) > κ the estimate ∈ 1 1 (p(a, a∗) µ1)− 6 κ− + ε + C(p, κ,ε)δ k − k holds for all a satisfying (1.2) and (1.8). ∈A The authors’ interest to the subject was drawn by its relation with Huaxin Lin’s theorem (see [6, 5]). It says that if a is an n n-matrix satisfying (1.2), then the distance from a to the × set of normal matrices is estimated by a function F (δ) such that F (δ) 0 as δ 0 uniformly in n. This result implies Theorems 1.1–1.3 with δ replaced by F (δ) in→ the right→ hand side. By homogenuity reasons, F (δ) can not decay faster than Cδ1/2 as δ 0. Therefore this approach → gives weaker results in terms of power of δ. Also, our results hold in any unital C∗-algebra, while the infinite-dimensional versions of Lin’s theorem require additional index type assumptions on a (see, for example, [5]). Our proofs are based on certain representation theorems for positive polynomials. If a real polynomial of x , x is non-negative on the unit disk x : x2 + x2 < 1 then, by a result of [11], 1 2 { 1 2 } it admits a representation r (x)2 + 1 x2 x2 s (x)2 (1.11) j − 1 − 2 j j j X X with real polynomials rj and sj (see Proposition 3.2 below). Representations similar to (1.11) are usually referred to as Positivstellensatz. We also make use of Positivstellensatz for poly- nomials positive on the sets (1.9). The corresponding results for sets bounded by arbitrary algebraic curves were obtained in [2, 9, 10, 11]. In order to prove Theorem 1.3, we need uniform with respect to µ estimates for polynomials appearing in Positivstellensatz-type representations. In order to obtain the estimates, we use the scheme introduced in [12, 7]. The authors thank Dr. A. Pushnitski and the referee for valuable comments. 2 Proofs of the main results The proofs of all three theorems consist of two parts. This section is devoted to the “operator- theoretic” part, which is essentially based on Lemma 2.2. The “algebraic” part is the existence of representations (2.2) for the polynomials (2.3), (2.4), (2.7) which is discussed in Section 3. 3 2.1 Positive elements of C∗-algebras Recall that a Hermitian element b is called positive (b > 0) if one of the following two ∈ A equivalent conditions holds (see, for example, [4, 1.6]): § 1. σ(b) [0, + ). ⊂ ∞ 2. b = h∗h for some h . ∈A The set of all positive elements in is a cone: if a, b > 0, then αa + βb > 0 for all real α, β > 0. There exists a partial ordering onA the set of Hermitian elements of : a 6 b iff b a > 0. For A − a Hermitian b, b 1 6 b 6 b 1 (2.1) −k k k k and, moreover, if 0 6 b 6 β1, β R, then b 6 β. The following fact is also well known. ∈ k k 2 Proposition 2.1. Let h , ρ> 0. Then h∗h > ρ 1 if and only if the element h is invertible 1 1 ∈A and h− 6 ρ− . k k Our proofs use the following simple lemma. Lemma 2.2. Let a satisfy (1.2), and let ∈A N m 1 N − 2 2 q = rj + rij gi, (2.2) j=0 i=0 j=0 ! X X X where rj, rij, gi are real-valued polynomials of the form (1.3). Assume that gi(a, a∗) > 0, i =0,...,m 1. Then − q(a, a∗) > Cδ1 − with some non-negative constant C depending on rj, rij, gj. Proof. Note that q is real-valued, so q(a, a∗) is self-adjoint. Since gi(a, a∗) > 0, we have g (a, a∗)= b∗b for some b . Then i i i i ∈A rij(a, a∗)gi(a, a∗)rij(a, a∗)=(birij(a, a∗))∗(birij(a, a∗)) > 0. 2 We also have rj(a, a∗) > 0. From (1.5), we have 2 q(a, a∗) r (a, a∗) r (a, a∗)g (a, a∗)r (a, a∗) 6 C′δ, k − j − ij i ij k j i,j X X and now the proof is completed by using (2.1).
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