Numerical Range and Functional Calculus in Hilbert Space

Numerical range and functional calculus in Hilbert space Michel Crouzeix Abstract We prove an inequality related to polynomial functions of a square matrix, involving the numerical range of the matrix. We also show extensions valid for bounded and also unbounded operators in Hilbert spaces, which allow the development of a functional calculus. Mathematical subject classification : 47A12 ; 47A25 Keywords : functional calculus, numerical range, spectral sets. 1 Introduction In this paper H denotes a complex Hilbert space equipped with the inner product . , . and h i corresponding norm v = v, v 1/2. We denote its unit sphere by Σ := v H; v = 1 . For k k h i H { ∈ k k } a bounded linear operator A (H) we use the operator norm A and denote by W (A) its ∈ L k k numerical range : A := sup A v , W (A) := Av,v ; v ΣH . k k v ΣH k k {h i ∈ } ∈ Recall that the numerical range is a convex subset of C and that its closure contains the spectrum of A. We keep the same notation in the particular case where H = Cd and A is a d d matrix. × The main aim of this article is to prove that the inequality p(A) 56 sup p(z) (1) k k≤ z W (A) | | ∈ holds for any matrix A Cd,d and any polynomial p : C C (independently of d). ∈ → It is remarkable that the completely bounded version of this inequality is also valid. More precisely, we consider now matrix-valued polynomials P : C Cm,n, i.e., for z C, P (z) = → ∈ (p (z)) is a matrix, with each entry p (.) a polynomial C C. Then the inequality ij ij → P (A) 56 sup P (z) , (2) k k≤ z W (A) k k ∈ holds for any matrix A Cd,d, any polynomial P : C Cm,n, and any values of d, m, n. ∈ → We now fix a matrix A Cd,d and consider the algebra C[z] of polynomials p from W (A) C ∈ into , provided with the norm p ,A = supz W (A) p(z) . The map p p(A) is then a k k∞ ∈ | | 7→ homomorphism from the algebra C[z] into the algebra Cd,d. The inequality (1) means that this homomorphism is bounded with constant 56, and the inequality (2) that it is complely bounded with constant 56. These inequalities may be summarized in the following statement. 1 Theorem 1. For any matrix A Cd,d the homomorphism p p(A) from the algebra C[z], with ∈d,d 7→ norm . ,A, into the algebra C is completely bounded, with constant 56. k k∞ It will be clear from the proof that the value 56 is not optimal. Denoting by and the Q Qcb smallest constants such that the inequalities p(A) sup p(z) , P (A) cb sup P (z) , (3) k k ≤ Q z W (A) k k k k ≤ Q z W (A) k k ∈ ∈ hold for any matrix A Cd,d, any polynomial p : C C, P : C Cm,n and any values of ∈ → → d, m, n, we have 2 56. ≤Q≤Qcb ≤ 0 2 The lower bound is attained by the polynomial p(z)= z and the matrix A = , in which 0 0 case W (A) is the unit disk. Using the fact that these constants are independent of d, we will deduce that the inequalities (3) are still valid for any bounded linear operator A (H) on any Hilbert space H. For a ∈ L convex subset E of C we introduce the algebra (E) of continuous and bounded functions in Hb E which are holomorphic in the interior of E. The following statement shows the existence of a functional calculus based on the numerical range. Theorem 2. Let H be a Hilbert space. For any bounded linear operator A (H) the ho- ∈ L momorphism p p(A) from the algebra C[z], with norm . ,A, into the algebra (H), is 7→ k k∞ L bounded with constant . It admits a unique bounded extension from (W (A)) into (H). Q Hb L This extension is bounded with constant and completely bounded with constant . Q Qcb For many applications it is useful to consider unbounded operators. Assume now that A ∈ (D(A),H) is a closed linear operator with domain D(A) densely and continuously embedded in L H. Its numerical range is then defined by W (A) := Av,v ; v Σ D(A) . We assume that {h i ∈ H ∩ } the spectrum σ(A) is included in W (A). The numerical range is still convex but unbounded and we hence cannot apply polynomials. We therefore instead consider the algebra Cb(z) of rational functions which are bounded on W (A), and provide it with the norm r ,A = supz W (A) r(z) . k k∞ ∈ k k We then have the following statement similar to Theorem 2 Theorem 3. For any closed linear unbounded operator A such that σ(A) W (A), the ho- ⊂ momorphism r r(A) from the algebra Cb(z), with norm . ,A, into the algebra (H), is 7→ k k∞ L bounded with constant . It admits a unique bounded extension from the algebra (W (A)) into Q Hb (H). This extension is bounded with constant and completely bounded with constant . L Q Qcb An outline of this paper is the following : in Section 2 we deduce Theorems 2 and 3 from the inequality (3), and illustrate them by application to the theory of Cosine functions. Section 3 is devoted to an integral representation formula for P (A), which is the key to proving (2). In Section 4 we prove some preliminary bounds which allow us to treat most matrices. The remaining case, which corresponds to very thin numerical ranges, is more complicated and is considered in Section 5. Section 6 is devoted to the proof of a geometrical inequality for convex domains in the plane, which is needed in Section 4. Some concluding remarks are given in Section 7. The author conjectures that, in fact, = = 2. Q Qcb 2 2 The extension to infinite dimension Proof of Theorem 2. We assume that (3) is valid if H = Cd, for all integers d. This implies that (3) holds for any finite dimensional Hilbert space. Let now H be infinite-dimensional and consider an operator A (H) and a polynomial p of degree n. For u H given, we introduce ∈L ∈ the Krylov space = Span u, Au, . , Anu H, denote by Π the orthogonal projection Kn { } ⊂ n onto n, and set An := ΠnA n . We clearly have p(A) u = p(An) u and W (An) W (A), and K |K ⊂ A acts on the n+1 dimensional space . We then deduce that n Kn p(A) u = p(An) u sup p(z) u sup p(z) u , k k k k ≤ Q z W (An) | | k k ≤ Q z W (A) | | k k ∈ ∈ which implies p(A) sup p(z) . The same proof works also for polynomials with matrix k k ≤ Q z W (A) | | ∈ values, thus the inequalities (3) are still valid. The Mergelyan theorem states that C[z] is dense in (W (A)), which completes the proof. Hb Proof of Theorem 3. We consider now a closed linear unbounded operator A such that σ(A) ⊂ W (A) and a rational function r bounded on W (A). It is clear that r(A) is well defined and belongs to (H) (write r in simple partial fraction form). In the case W (A) = C, the space L (W (A)) is reduced to constant functions and Theorem 2 follows immediately (but has no Hb interest). In the other cases, changing, if needed, A to α + eiθA, α C, θ R, we have only ∈ ∈ two possibilities to consider : a) The numerical range is a strip : W (A)= z C ; 0 Re z a , a> 0. { ∈ ≤ ≤ } 1 1 We then set Aε = (1+εA∗)− A(1+εA)− , for ε> 0. b) x ; x> 0 W (A) z C ; Re z 0 . { }⊂ ⊂ { ∈ ≥ } 1 We then set Aε = A(1+εA)− , for ε> 0. In both cases it is easy to verify that A is a bounded operator and that W (A ) W (A). Let ε ε ⊂ r be a rational function bounded in W (A), clearly r (W (A )). From Theorem 2 we have ∈Hb ε r(Aε) sup r(z) sup r(z) . (4) k k ≤ Q z W (Aε) | | ≤ Q z W (A) | | ∈ ∈ We now note that limε 0 r(Aε)= r(A), strongly on H. Indeed, it suffices to verify that → 1 1 lim(λ Aε)− u = (λ A)− u for all λ / W (A) and all u D(A). ε 0 − − ∈ ∈ → We consider the case of a strip, the other situation being simpler. We have by a simple calculation 1 1 1 1 1 1 (λ Aε)− u (λ A)− u = ε(λ Aε)− (1+εA∗)− (A+A∗)(1+εA)− (λ A)− Au − − − − − 1 1 1 − 1 ε(λ A )− (1+εA∗)− (εA∗)(1+εA)− A(λ A)− Au. − − ε − The convergence to 0 of the second member follows from the bounds 1 1 A + A∗ 2 a, (1 + εA∗)− 1, (1 + εA∗)− (εA∗) 1, k k≤ k k≤ k k≤ 1 1 1 1 (λ A )− , (λ A)− , k − ε k≤ d(λ, W (A)) k − k≤ d(λ, W (A)) 1 λ A (λ A)− | | . k − k≤ d(λ, W (A)) 3 Taking the limit as ε 0 in (4), we obtain that the map r r(A) is bounded with constant → 7→ . The space C (z) is dense in the subset (W (A)) of functions of (W (A)) which have a Q b Hb,0 Hb limit at .

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