A FUNCTIONAL CALCULUS BASED on the NUMERICAL RANGE. APPLICATIONS the Starting Point of This Work Is the Following Inequality

A FUNCTIONAL CALCULUS BASED ON THE NUMERICAL RANGE. APPLICATIONS

MICHEL CROUZEIX

ABSTRACT. We develop a functional calculus for both bounded and unbounded operators in Hilbert spaces based on a simple inequality related to polynomial functions of a square matrix and involving the numerical range. We present some applications in different areas of mathematics.

1. INTRODUCTION The starting point of this work is the following inequality sup u∗p(A)v ; u, v Cd, u∗u = v∗v = 1 12 sup p(v∗Av) ; v Cd, v∗v = 1 , {| | ∈ }≤ {| | ∈ } which holds for all square matrices A Cd,d, all polynomials p : C C, and all values of d. This inequality may be equivalently written∈ as → p(A) 12 sup p(z) , (1) k k≤ z∈W (A) | | where W (A)= v∗Av ; v Cd, v∗v = 1 is the numerical range of A. We refer to [5] for a proof of (1). { ∈ } Recall that the spectrum σ(A) of a matrix A is contained in its numerical range. Thus if a matrix A is normal we have p(A) = sup p(λ) ; λ σ(A) , and in this particular case inequality (1) is valid with 1 in placek of 12.k The great{| interest| ∈ of (1)} is to be valid for any square matrix. The author conjectures that, in this inequality, 12 may be replaced by 2. An outline of this paper is the following. In Section 2 we show how this inequality allows us to develop a holomorphic calculus for bounded and unbounded operators in a Hilbert space H. In Sections 3 and 4 we describe some applications in different areas of mathematics: second order differential equations, probability, numerical linear algebra, operator theory ...

2. FROMTHEINEQUALITYTOWARDSAFUNCTIONALCALCULUS We assume that the inequality (1) is valid for all square matrices A Cd,d, all polynomials ∈ p : C C, and all values of d. → 2.1. First step : extension to bounded operators in a Hilbert space. From the independence of the dimension we deduce : the inequality p(A) 12 sup p(z) k k≤ z∈W (A) | | holds for any bounded linear operator A (H) on any complex Hilbert space H and for any ∈ L polynomial p.

2000 Mathematics Subject Classiﬁcation. 47A11,47A25. Key words and phrases. functional calculus, numerical range. 1 2 MICHEL CROUZEIX

Proof. Indeed this is clearly true in the case of finite dimensional H. Otherwise let an operator A (H) and a polynomial p of degree n be given ; we consider a vector u H and define the Krylov∈ L subspace ∈ := Span (u, Au, . . . , Anu). K We set A˜ = Π A, where Π denotes the orthogonal projection from H to . Then we have K K K p(A)u = p(A˜)u and, on the finite dimensional space , the linear operator A˜ satisfies W (A˜) K ⊂ W (A). Thus p(A)u = p(A˜)u 12 u sup p(z) 12 u sup p(z) . k k k k≤ k k z∈W (A˜) | |≤ k k z∈W (A) | | The result follows, since u is arbitrary.

2.2. Second step : extension to rational functions. We now claim that the inequality r(A) 12 sup r(z) k k≤ z∈W (A) | | is valid for any bounded linear operator A (H) on a Hilbert space H and for any rational ∈ L function r, bounded on the numerical range W (A).

Proof. Indeed such a rational function r is a uniform limit of polynomials on the bounded set W (A) (as is easily seen by writing the rational function in simple partial fraction form).

2.3. Third step : extension to unbounded operators. We now consider a closed, densely de- ﬁned unbounded operator A : D(A) H H on the Hilbert space H. We assume that its ⊂ → spectrum is contained in the closure of the numerical range: σ(A) W (A). Then we claim that the inequality ⊂ r(A) 12 sup r(z) k k≤ z∈W (A) | | is valid for any rational function r, bounded on the numerical range W (A).

Proof. From its boundedness it follows that the rational function r has its poles off W (A) ⊃ σ(A). Thus r(A) is well defined. Note that the numerical range is convex and unbounded for an unbounded operator A. Thus only 3 situations can occur : W (A)= C ; then the inequality is trivially satisfied, since the only bounded rational functions on•C are the constant ones. W (A) is a strip. • There exist θ R and β C such that (0, + ) W (eiθA + β) z C ; Re z 0 . • ∈ ∈ ∞ ⊂ ⊂ { ∈ ≥ } We only look at this last case. Without loss of generality, we can assume that θ = β = 0, and thus (0, + ) W (A) z C ; Re z 0 and A is maximal accretive. Then with ε > 0 we ∞ ⊂ −1 ⊂ { ∈ ≥ } set Aε := A(1+εA) . The operator Aε is bounded, and it is easily verified that W (Aε) W (A). We deduce from the previous step that ⊂

r(Aε) 12 sup r(z) 12 sup r(z) . k k≤ z∈W (Aε) | |≤ z∈W (A) | |

Writing the rational function in simple partial fraction form we see that r(Aε) strongly converges to r(A) as ε 0, which shows the inequality. → A similar proof applies in the strip case, see [5]. A FUNCTIONAL CALCULUS BASED ON THE NUMERICAL RANGE. APPLICATIONS 3

2.4. Last step : extension to holomorphic functions. We now consider the algebra Cb(z) of rational functions, bounded on W (A), and the algebra (W (A)) of continuous and bounded Hb functions in W (A) which are holomorphic in the interior of W (A). We equip these two algebras with the norm f ∞,A := sup f(z) ; z W (A) . The following result shows the existence of a functional calculusk k [12] based{| on the| numerical∈ range.} Theorem 1. For any closed linear unbounded operator A such that σ(A) W (A), the homomor- ⊂ phism r r(A) from the algebra C (z), with norm . , into the algebra (H), is bounded 7→ b k k∞,A L with constant 12. It admits a unique bounded extension from the algebra (W (A)) into (H). Hb L This extension is bounded with constant 12 as well as completely bounded with constant 12. Sketch of proof. As in the previous step we only look at the case of maximal accretive A. First we consider a function f (W (A)) with lim f(z) = 0. Such a function f is a uniform limit ∈Hb z→∞ of rational functions rn on W (A), which allows us to deﬁne f(A) = lim rn(A). For a general function f (W (A)), we associate f and ε > 0 with f (z) := (1+εz)−1f(z) (note that ∈ Hb ε f ( ) = 0) and deﬁne f(A) = (1+εA)f (A). Then we have, if u D(A), ε ∞ ε ∈ f(A)u = (1+εA)f (A)u = f (A)(1+εA)u k k k ε k k ε k 12 f ( u + ε Au ) 12 f ( u + ε Au ). ≤ k εk∞,A k k k k ≤ k k∞,A k k k k Passing to the limit as ε 0 and using the density of D(A) in H we deduce f(A) (H) and → ∈L f(A) 12 sup f(z) . k k≤ z∈W (A) | | From the fact that the inequality (1) is still valid in tensorial form we conclude that the map is completely bounded (see [5] for more details).

3. AN APPLICATION TO SECOND ORDER DIFFERENTIAL EQUATIONS Let us consider two complex Hilbert spaces V H with V continuously and densely embedded ⊂ in H. We also consider a continuous sesquilinear form a( , ) on V V and assume that there · · × exist α> 0, λ, M, N R such that ∈ α v 2 Re a(v, v)+ λ v 2 M v 2 , k kV ≤ k kH ≤ k kV v V. (2) and Im a(v, v) N v v , ∀ ∈ ≤ k kV k kH To this form we associate a closed operator A deﬁned by

D(A) := u V ; there exists K such that a(u, v) K v , v V , { ∈ u | |≤ uk kH ∀ ∈ } and Au, v = a(u, v). h iH It is well known that D(A) is dense in H and σ(A) W (A). The assumption (2) clearly implies ⊂ α W (A) := x+iy ; x + λ y2 . ⊂ P { ≥ N 2 } Note that, if z belongs to the parabolic set , then Im z1/2 ω, with ω := max(λ, N/(2√α)). P | |≤ Thus the function cos(t√z) satisﬁes cos(t√z) eω|t| for all z and all t in R. Also this function is holomorphic in C, since the| cosine function| ≤ is even. Theorem∈ P 1 allows us to deﬁne C(t) := cos(t√A) and we have the following properties : • C(t) (H) and C(t) 12 eω|t| for all t R. ∈L k k≤ ∈ • C(0) = I and C(t+s) 2C(t)C(s)+ C(t s) = 0 for all t,s R. − − ∈ 4 MICHEL CROUZEIX

• For all u H, u(t) := C(t) u satisfies u C(R; H), (strong continuity on H). 0 ∈ 0 ∈ • For all u D(A), u(t) := C(t) u satisfies u C(R; D(A)). 0 ∈ 0 ∈ Proof. Since the map f f(A) is an algebraic homomorphism, the two equalities are direct 7→ consequences of the well-known formulas cos 0 = 1 and cos(a+b) + cos(a b) = 2 cos a cos b. − Since we have C(t) 12 eω|t| and D(A) is dense in H, it suffices to show u(.) C(R; H) k k ≤ ∈ for all u D(A) in order to establish the strong continuity in H. To this end we introduce the 0 ∈ operator B(t, h) := (C(t+h) C(t h))(1+λ+A)−1 and note that, for t + h T , − − | | | |≤ cos((t+h)√z) cos((t h)√z) B(t, h) 12 sup − − k k≤ z∈P 1+ λ + z

sin t√z sin h√z t z 24 h sup ≤ z∈P t√z h√z 1+ λ + z

K(T ) h, for some constant K(T ). ≤ Then, assuming u D(A), 0 ∈ u(t+h) u(t h)= B(t, h) h (1+λ+A)u , − − 0 and u(t+h) u(t h) = K(T ) h (1+λ+A)u , k − − k k 0k which shows that u(.) C(R; H). In the same way we obtain u C(R; D(A)), if u0 D(A). ∈ ∈ ∈

Using the formula cos(t√z) = 1 t(t s)z cos(s√z) ds and the property that we have a − 0 − homomorphism of algebra we deduce, for u D(A), R 0 ∈ t u(t)= u (t s)Au(s) ds. 0 − − Z0 ′ ′′ 2 This shows that u(0) = u0, u (0) = 0, u (t)= A u(t) and thus u C (R; H). In a similar way (see [5] for the details) we can− treat the case where∈ the second initial data is different of 0 and obtain Theorem 2. Assume that u D(A), v V , and set S(t) := t C(s) ds. Then the function 0 ∈ 0 ∈ 0 u(t) := C(t)u + S(t)v is the unique solution of 0 0 R u C2(R; H) C0(R; D(A)), ∈ ∩  u′′(t)+ A u(t) = 0, t R, u(0) = u , u′(0) = v .  ∀ ∈ 0 0 Remark. We have seen that the second order differential equation is well posed as soon as our assumptions (2) are satisﬁed. It is interesting to know that a converse part is also true. More precisely Markus Haase [8] has proved the following result : If A is the generator of a cosine − function on the Hilbert space H (in other words, if this second order problem is well posed), then, with respect to an equivalent scalar product , ◦ , A has its numerical range in a horizontal h· ·i parabola := x+iy ; x + µ ω2y2 for some ω = 0,µ R. P { ≥ } 6 ∈ Using this new scalar product, setting a(u, v) = Au, v ◦ , and denoting by V the closure of h 1i/2 D(A) for the norm v = Re a(v, v) + (µ+1) v, v ◦ , assumptions (2) are satisﬁed with k kV h i α = M = 1, λ = µ+1, and N = ω−1. This shows that the framework used in this section is well suited for a general study of cosine functions in Hilbert spaces. We refer to [10] and [1] for a classical and a modern presentation of the theory of cosine functions. A FUNCTIONAL CALCULUS BASED ON THE NUMERICAL RANGE. APPLICATIONS 5

4. VARIOUS APPLICATIONS 4.1. Application in numerical linear algebra. Let A Cd,d be a square matrix such that 0 / ∈ ∈ W (A). The following quantity δ (A) := min p(A) ; p(0) = 1, p polynomial, deg(p) n . n {k k ≤ } plays an important role in the analysis of convergence of Krylov type algorithms, which are com- monly used for solving large linear systems of equations. A similar term

d (W (A)) := min p ∞ ; p(0) = 1, deg(p) n n {k kL (W (A)) ≤ } has been the object of extensive studies during the last century. Kövari and Pommerenke have obtained the asymptotically optimal estimate 2 2 d (W (A)) , n ≤ φ (0) ≤ φ(0) n 1 | n | | | − where φ is the conformal map from C W (A) onto the exterior of the unit disk such that φ( )= \ ∞ and φ′( ) > 0, and φ the Faber polynomial of degree n defined by ∞ ∞ n (φ(z))n = φ (z)+ O(1/z), as z . n →∞ Note that φ(0) > 1. A direct application of our inequality gives | | 24 24 δ (A) 12 d (W (A)) . n ≤ n ≤ φ (0) ≤ φ(0) n 1 | n | | | − This is a good result, since the estimate is independent of the size of the matrix A. But, noticing that in the proof of inequality (1) some terms vanish for Faber polynomials, Bernhard Beckermann [4] has obtained the improved estimate 2 2 δ (A) . n ≤ φ (0) ≤ φ(0) n 1 | n | | | − 4.2. Application in probability. My interests in developing a functional calculus based on the numerical range started from the reading of the nice paper [7] of Bernard and François Delyon. They have proved the ancestor of the inequality (1) (the difference being that their constant was not universal) and they have used it to prove the Burkholder conjecture : Let T = P1P2 ...Pn be a finite product of conditional expectations with respect to the σ-fields ,..., . Then, for all functions f L2, we have F1 Fn ∈ k lim T f = E[f 1 2 n] almost surely. k→∞ |F ∩ F ∩···∩F

4.3. Applications in operator theory. We have seen that the map uA from the algebra of rational functions bounded on W (A) into the C∗-algebra (H), deﬁned by u (r)= r(A), is completely L A bounded with a complete norm uA cb 12. A direct application of Paulsen’s theorem, see [13], gives : k k ≤ There exists an invertible operator S (H) such that S S−1 12 and ∈L k k k k≤ r(S−1AS) sup r(z) , for any rational function r bounded in W (A), k k≤ z∈W (A) | | or, equivalently, W (A) is a spectral set for the operator S−1AS. Note that we have a reduction of the numerical range : W (S−1AS) W (A). ⊂ 6 MICHEL CROUZEIX

Then we deduce from a result due to Arveson (see [2], [13]) that there exist a larger Hilbert space , containing H as a subspace, and a normal operator N acting on , with spectrum K K σ(N) ∂W (A), such that ⊂ r(A)= S P r(N) S−1, for any rational function r bounded in W (A), H |H where P denotes the orthogonal projection from onto H. In other words, A is similar to an H K operator having a normal ∂W (A) dilation. − Let us now consider a sector S = 0 = z C ; arg(z) < α , α (0, π/2]. We will say that α { 6 ∈ | | } ∈ the closed operator A is α-accretive iff

σ(A) W (A) S . ⊂ ⊂ α According to a well-known result of Kato if A is α-accretive and if 0 < β 1, then Aβ is αβ-accretive. ≤ Using the similarity S given by Paulsen’s theorem Christian Le Merdy [11] has remarked the converse statement if A is α-accretive and if 1 < β π , then S−1AβS is αβ-accretive . ≤ 2α A classical consequence of the von Neumann inequality for the half-plane is if A is α-accretive, then Ais eα|s| for all s R. Christian Le Merdy has alsok obtainedk≤ the converse∈ part if Ais eα|s| for all s R, then the operator S−1AS is α-accretive. k k≤ ∈ Another application can be found in [8], where the inequality is used for simplifying the proof of a result due to Boyadzhiev and de Laubenfels (Theorem 7.2.1).

4.4. Application in numerical analysis. It is clear that inequality (1) and its extensions have many applications in numerical analysis, since polynomial or rational functions of a matrix often occur in this ﬁeld. We consider for instance a simple parabolic problem

u′(t)+ A u(t) = 0, t> 0, (P ) 0 ( u(0) = u . Here A is a maximal accretive operator on a Hilbert space H corresponding to a sesquilinear form a( , ), which is continuous and elliptic on a dense Hilbert subspace V of H. The numerical approximation· · of this problem is splitted in two steps :

Space discretization. If we use for instance a ﬁnite element method, the spaces V and H are • approximated by a ﬁnite dimensional space V V and the initial data by an element u0 V . h ⊂ h ∈ h The continuous problem (P ) is then replaced by

u′ (t)+ A u (t) = 0, t> 0, 1 h h h (Ph ) 0 ( uh(0) = uh. Here A (V ) is the operator deﬁned by h ∈L h A u , v = a(u , v ), u , v V . h h h hiH h h ∀ h h ∈ h Note that we have the interesting property W (A ) W (A). h ⊂ A FUNCTIONAL CALCULUS BASED ON THE NUMERICAL RANGE. APPLICATIONS 7

Time discretization. We now introduce a time step ∆t> 0 and are looking for an approxima- • n tion uh of the solution at the levels tn := n∆t. Classical methods are un+1 un un+1 un h − h + A un = 0, or h − h + A un+1 = 0, ∆t h h ∆t h h un+1 un un+1 +un or h − h + A h h = 0. ∆t h 2 These methods are respectively called (forward) Euler scheme, backward Euler scheme, and Crank-Nicolson scheme. They can also be written as un+1 = (I ∆t A )un, resp. un+1 = (I +∆t A )−1un, h − h h h h h resp. un+1 = (I ∆t A )(I ∆t A )−1un. h − 2 h − 2 h h Therefore, each of these methods may be expressed in the form n+1 n n n 0 uh = r(∆t Ah)uh, and thus uh = r(∆t Ah) uh, with r(z) = 1 z for Euler, r(z)= 1 for backward Euler, and r(z)= 1−z/2 for Crank-Nicolson. − 1+z 1+z/2

More generally the main part of one-step methods for the discretization of differential equations (and in particular all the famous Runge-Kutta methods) provide an approximation of the form n+1 n n n 0 uh = r(∆t Ah)uh, and thus uh = r(∆t Ah) uh, where r(z) is a rational approximation of e−z. A crucial point for the numerical analysis of a method is its stability. In our framework, the method is called stable if there exists a constant C, independent of h and ∆t, such that n 0, r(∆t A ) n C. ∀ ≥ k h k≤ 0 Remark. This a very natural requirement. Indeed, if we replace the initial data uh by a perturbed 0 0 n n n 0 one uh +εh, the numerical approximation uh is replaced by uh + r(∆t Ah) εh. The stability is just the guarantee that the effect of the initial perturbation cannot be amplified by a factor larger than C. An interesting class of schemes is the subset of A-stable methods, i.e., by definition the methods such that r(z) < 1 for all z belonging to the right half-plane Re z > 0. (This is the case if r(z) | | is a diagonal or a subdiagonal Padéapproximation of exp( z), in particular for backward Euler and Crank-Nicolson). Then we deduce from the von Neumann− inequality for the half-plane n n r(∆t Ah) sup r(z) 1, k k≤ Re z>0 | |≤ which shows that these methods are stable (independently of ∆t). But it may be useful to work with non A-stable methods. (An A-stable method has for instance the drawback to be necessarily implicit.) Our inequality shows that, under the assumption ∆tW (A ) := z C ; r(z) 1 , we have r(∆t A ) n 12. h ⊂ Dr { ∈ | |≤ } k h k≤ The set r is called the stability domain associated to r and has been the object of numerous studies.D Up to now such a sufficient condition implying the stability of the method was only known for self-adjoint operators. Concerning, respectively, the discretization of parabolic equations and the stability properties of numerical schemes, the standard reference books are [14] and [9]. 8 MICHEL CROUZEIX

REFERENCES 1. W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, 2002. 2. W. Arveson, Subalgebras of C∗-algebras. II, Acta Math. 118 (1972), 271–308. 3. C. Badea, M. Crouzeix, B. Delyon, Convex domains and K-spectral sets, Math. Z. 252, no. 2, 345–365, 2006. 4. B. Beckermann, Image numérique, GMRES et polynômes de Faber, CRAS, série 1, 340 (2006), 855–860. 5. M. Crouzeix, Numerical range and functional calculus in Hilbert space, preprint, available from http://perso.univ-rennes1.fr/michel.crouzeix/ 6. M. Crouzeix, Bounds for analytic functions of matrices, Int. Equ. Op. Th. 48 (2004), 461–477. 7. B. & F. Delyon, Generalization of Von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. France, 1, (1999), 25–42. 8. M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, vol. 169, Birkhäuzer Verlag, Basel, Boston, Berlin, 2006. 9. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential Algebraic Problems, 2nd edition, Springer Series in Computational Mathematics 14, Springer-Verlag, Berlin, 1996. 10. H.O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland, Amsterdam, 1985. 11. C. Le Merdy, Similarities of ω-accretive operators, International Conference on Harmonic Analysis and Related Topics (Sydney, 2002), 84–95, Proc. Centre Math. Appl. Austral. Nat. Univ., 41, Austral. Nat. Univ., Canberra, 2003. 12. A. McIntosh, Operators which have an H∞ functional calculus. Miniconference on operator theory and partial differential equations (North Ryde, 1986), Austral. Nat. Univ. Canberra, 1986, 210–231. 13. V.Paulsen, Completely bounded maps and operator algebras, Cambridge Univ. Press, 2002. 14. V.Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathe- matics 25, second edition, Springer-Verlag, Berlin, 2006.

INSTITUTDE RECHERCHE MATHEMATIQUE´ DE RENNES, UMR 6625, UNIVERSITEDE´ RENNES 1, CAMPUS DE BEAULIEU, 35042 RENNES CEDEX,FRANCE E-mail address: [email protected]