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A new functional calculus for noncommuting operators Fabrizio Colombo Irene Sabadini 8 Dipartimento di Matematica Dipartimento di Matematica 0 Politecnico di Milano Politecnico di Milano 0 2 Via Bonardi, 9 Via Bonardi, 9 n 20133 Milano, Italy 20133 Milano, Italy a [email protected] [email protected] J 5 Daniele C. Struppa 2 Department of Mathematics ] and Computer Sciences P Chapman University S Orange, CA 92866 USA, . h [email protected] t a m [ Abstract 2 Inthispaperweusethenotionofslicemonogenicfunctions[2]todefineanewfunctional v 4 calculus for an n-tuple T of not necessarily commuting operators. This calculus is different 9 fromtheonediscussedin[5]anditallowstheexplicitconstructionoftheeigenvalueequation 5 for the n-tuple T based on a new notion of spectrum for T. Our functional calculus is 3 consistent with the Riesz-Dunford calculus in the case of a single operator. . 8 0 AMS Classification: 47A10, 47A60, 30G35. 7 Key words: slice monogenic functions, functional calculus, spectral theory, noncommuting op- 0 erators. : v i X 1 Introduction r a In a series of interesting papers, see e.g. [6], [7], [9] as well as in the book [5] and the references therein, the authors have developed a monogenic functional calculus, whose purpose is to deal withn-tuplesofnotnecessarilycommutingoperators. Theinterestinthisproblemcanbetraced to the early works of Taylor, see for example [11], [12], and it is of great physical interest. In this context, one has other approaches such as the Weyl calculus. A complete review of these theories can be found in [5] to which we refer the reader interested in the physical origin of the problem, in a variety of different approaches, and in their mutual relationships. No matter how a functional calculus is developed, one will naturally require that it be consistent with the case in which only one operator is considered, and that it be sufficiently flexibletohandlebothcommutingandnoncommutingoperators. Inthecaseofasingleoperator, one expects to find the usual Riesz-Dunford calculus. When several operators are considered, it is natural to look for functions defined on Rn with values in noncommuting algebras which may allow the formal treatment of non commutativity. This set up is naturally within the scope of monogenic functions [1]. 1 Let R be the real Clifford algebra, i.e. the real algebra generated by the n units e ,...,e n 1 n such that e e +e e = 2δ . An element (x ,x ,...,x ) Rn+1 can be naturally identified i j j i ij 0 1 n − ∈ with the element x +x e +...+x e R . A differentiable function f : U Rn+1 R 0 1 1 n n n n ∈ ⊆ → is said to be monogenic if it is in the kernel of the Dirac operator ∂ +e ∂ + +e ∂ , x0 1 x1 ··· n xn where ∂ is shorthand for ∂/∂ . The theory of such functions is fully developed in [1] (for xi xi the case of several Dirac operators see [3]), and the properties that such functions enjoy closely resemble those of holomorphic functions of a single complex variable. In particular, they can be represented by means of a Cauchy kernel. Specifically, let Σ denote the volume of the unit n n-sphere in Rn+1, let ω, x Rn+1 and ω = x, then if ∈ 6 1 ω¯ x¯ G(ω,x) = − Σ ω xn+1 n | − | where x¯ := x x e ... x e , we have 0 1 1 n n − − − f(x), if x Ω, G(ω,x)n(ω)f(ω)dµ(ω) = ∈ 0, if x Ω, Z∂Ω (cid:26) 6∈ whereΩ Rn+1 isaboundedopensetwithsmoothboundary∂Ωandexteriorunitnormaln(ω), ⊂ and µ is the surface measure of ∂Ω. Note that the kernel G(ω,x) = G (x) can be expanded, see ω [1], as G (x) = W (ω)Vℓ1,...,ℓk(x) ω  ℓ1,...,ℓk  Xk≥0 (ℓ1X,...,ℓk)   in the region x < ω where, for each ω Rn+1 0 , W (ω) = ( 1)k∂ ...∂ G (0), | | | | ∈ \{ } ℓ1,...,ℓk − ωℓ1 ωℓk ω Vℓ1,...,ℓk(x) = 1 z ...z , z = x e x e , and the sum is taken over all different k! j1,...,jk j1 jk j j 0 − 0 j permutation of ℓ ,...,ℓ . 1 k P Consider now an n-tuple T = (T ,...,T ) of bounded linear operators acting on a Banach 1 n space X and let R > (1+√2) n T e . If we formally replace z by T in the Cauchy kernel k j=1 j jk j j series, it can be shown (see [5], Lemma 4.7) that P G (T):= W (ω)Vℓ1,...,ℓk(T) ω  ℓ1,...,ℓk  Xk≥0 (ℓ1X,...,ℓk)   converges uniformly for all ω Rn+1 such that ω R. This fact leads to the following ∈ | | ≥ definition [5]: the monogenic spectrum γ˜(T) of the n-tuple T is the complement of the largest open set U in Rn+1 in which the function G (T) above is the restriction of a monogenic function ω with domain U. It is possible to show that if T is an n-tuple of noncommuting bounded linear operators satisfying suitable reality conditions on their joint spectrum (see [5]), then the function ω G (T)istherestrictiontotheregion ω > (1+√2) n T e ofamonogenicfunctiondefin→ed ω | | k j=1 j jk off Rn. P Denote with the same symbol G (T) its maximal monogenic extension. Under these hy- ω potheses, if Ω Rn+1 is a bounded open neighborhood of γ(T) with smooth boundary and if ⊆ f is a monogenic function defined in an open neighborhood of Ω, then one can show that the expression f(T):= G (T)n(ω)f(ω)dµ(ω) ω Z∂Ω is well defined. 2 In the case of commuting bounded linear operators the spectrum can be determined in an explicit way in view of the following result [5]: Theorem 1.1. Let T = (T ,...,T ) be a n-tuple of commuting bounded linear operator acting 1 n on a Banach space X and suppose that the spectrum of T is real for all j = 1,...,n. Then γ(T) j is the complement in Rn of the set of all λ Rn for which the operator n (λ T )2 is ∈ j=1 jI − j invertible in (X). L P Weobservethattheconditionofinvertibilityof n (λ T )2 givesaneigenvalueequation j=1 jI− j which, at least in some cases, can be easily written. If for example n is odd, it is possible to P write explicitly the Cauchy kernel as (−n−1)/2 n 1 G (T)= ω2 + (ω T )2 (ω T), ω Σ  0I jI − j  I − n j=1 X   whose singularities lie on the set n (0,ω ,...,ω ) Rn+1 0 σ (ω T )2 . 1 n j j  ∈ | ∈  I −   Xj=1    In the case the operators T do not commute the term 1 ω T −n−1(ω T) is not the sum j Σn| I− | I − of the Cauchy kernel series so that it is much more difficult to determine the the spectrum. In this paper we use a different approach to functional calculus. Specifically we replace the use of monogenic functions with the new concept, see [2], of slice-monogenic functions. These functions, whose definition we recall in section 2, have the great advantage that polynomials as well as power series arespecial cases of slice-monogenic functions (this is in contrast to the usual definitionofmonogenicfunctions). Thekeyobservationwhichwillmakeourapproachsuccessful is the fact that the sum of the so-called S-resolvent operator series (see (4)) is a function which can be utilized even in parts of the space where the series does not converge. This remark will make it possible to compute the spectrum also when the operators do not commute. In fact, the term (T2 2TRe[s]+ s 2 )−1(T s ) is the sum of the series (4) also when the operators − − | | I − I T do not commute, and the singularity of the the Cauchy kernel (the analogue of the maximal j extension G (T)) is given by (T2 2TRe[s]+ s 2 )v = 0. ω − | | I The outline of the paper is the following: in section 2 we introduce the concept of slice- monogenic functions and we recall the properties we need to develop our functional calculus. In section 3 we define the S-resolvent operator and we give the notion of S-spectrum, we show the S-resolvent equation and we develop the functional calculus for bounded operators. Finally in section 4 we treat the case of unbounded operators. Acknowledgements The first and third authors are grateful to Chapman University for the hospitality during the period in which this paper was written. They are also indebted to G.N.S.A.G.A. of INdAM and the Politecnico di Milano for partially supporting their visit. The authors thank the anonymous referee for the valuable suggestions to improve the paper. 2 Slice monogenic functions In this section we collect the basic results on the theory of slice monogenic functions, developed by the authors in [2], to which we refer for the missing proofs in this section. Let R be the real n Clifford algebra over n units e ,...,e such that e e +e e = 2δ . An element in the Clifford 1 n i j j i ij − 3 algebra will be denoted by e x where A = i ...i , i 1,2,...,n , i < ... < i is a A A A 1 r ℓ ∈ { } 1 r multi-index and e = e e ...e . An element x Rn can be identified with a 1-vector in the A i1 i2P ir ∈ Clifford algebra: (x ,x ,...,x ) x = x e +...+x e . A function f : U Rn R is seen 1 2 n 1 1 n n n 7→ ⊆ → as a function f(x) of x. An element in R0 R1 will be written as n⊕ n n x= x +x = x + x e . 0 0 j j j=1 X In the sequel the real part x of x will be also denoted by Re[x]. 0 Let us denote by S the sphere of unit 1-vectors in Rn, i.e. S = x= e x +...+e x x2+...+x2 = 1 . { 1 1 n n | 1 n } The complex line R+IR passing through 1 and I S will be denoted by L . I ∈ An element belonging to L will be denoted by u+Iv, for v, v R. I ∈ Observe that L , for every I S, is a real subspace of Rn+1 isomorphic to the complex plane. I ∈ Definition 2.1. Let U Rn+1 be a domain and let f : U R be a real differentiable function. n ⊆ → Let I S and let f be the restriction of f to the complex line L . We say that f is a (left) I I ∈ slice-monogenic function if for every I S ∈ 1 ∂ ∂ +I f (u+Iv) = 0. I 2 ∂u ∂v (cid:18) (cid:19) Analogously, it is possible to define a notion of right slice-monogenicity which gives a theory equivalent to the one left slice-monogenic functions. In the sequel, unless otherwise stated, we will consider monogenicity on the left and, for simplicity, sometimes we will denote by ∂ I the operator 1 ∂ +I ∂ and we will refer to left slice monogenic functions as s-monogenic 2 ∂u ∂v functions. We will also introduce a notion of I-derivative by means of the operator (cid:0) (cid:1) 1 ∂ ∂ ∂ := I . I 2 ∂u − ∂v (cid:18) (cid:19) Remark 2.2. The s-monogenic functions on U Rn+1 form a right module (U). ⊆ M Remark 2.3. For each a R the monomials x xma are left s-monogenic, while the m n m monomials x a xm are ri∈ght s-monogenic. Thus7→also polynomials N xma are left s- monogenic and7→anympower series +∞ xma is left s-monogenic in its domm=a0in of cmonvergence. m=0 m P The function R(x) = (x y )−m, m N is s-monogenic (left and right) if and only if y R. 0 0 − P ∈ ∈ Definition 2.4. Let U be a domain in Rn+1 and let f : U R be an s-monogenic function. n → Its s-derivative ∂ is defined as s ∂ (f)(x), x= u+Iv, v = 0, ∂s(f)= ∂If(u), x= u R. 6 (1) u (cid:26) ∈ Note that the definition of derivative is well posed because it is applied only to s-monogenic functions. Furthermore, any holomorphic function f : ∆(0,R) C can be extended (uniquely, → up to a choice of an order for the elements in the basis of R ) to an s-monogenic function n f˜:B(0,R) R . n → A key fact is that any s-monogenic function can be developed into power series and also that it admits a Cauchy integral representation. 4 Proposition 2.5. If B = B(x ,R) Rn+1 is a ball centered in a real point x with radius 0 0 ⊆ R > 0, then f : B R is s-monogenic if and only if it has a series expansion of the form n → 1 ∂mf f(x)= xm (x ) (2) m!∂um 0 m≥0 X converging on B. Given an element x = x +x Rn+1 let us set 0 ∈ x if x = 0, I = x 6 x ( a|n|y element of S otherwise. We have the following: Theorem 2.6. Let B = B(0,R) Rn+1 be a ball with center in 0 and radius R > 0 and let ⊆ f : B R be an s-monogenic function. If x B then n → ∈ 1 f(x) = (ζ x)−1dζ f(ζ) 2π − Ix Z∂∆x(0,r) where ζ L B, dζ = dζI and r > 0 is such that ∈ Ix Ix − x T ∆ (0,r) = u+I v u2+v2 r2 x x { | ≤ } contains x and is contained in B. Remark 2.7. An analogue statement holds for regular functions in an open ball centered in a real point x . 0 Remark 2.8. If f is an s-monogenic function on a domain U and γ : [a,b] Rn+1 is a curve, → then the integral f(ξ)dξ is defined as bf(γ(t)) γ′(t) dt. In particular, the curve γ can have γ a values on a complex plane L . I R R The key ingredient to define a functional calculus is what we call noncommutative Cauchy kernel series. Definition 2.9. Let x = Re[x]+x, s = Re[s]+s be such that sx = xs. We will call noncom- 6 mutative Cauchy kernel series the following expansion S−1(s,x) := xns−1−n (3) n≥0 X defined for x < s . | | | | Theorem 2.10. (See [2]) Let x= Re[x]+x, s = Re[s]+s be such that xs= sx. Then 6 xns−1−n = (x2 2xRe[s]+ s 2)−1(x s) − − | | − n≥0 X for x < s . | | | | 5 We will call the expression (x2 2xRe[s]+ s 2)−1(x s), defined for x2 2xRe[s]+ s 2 = 0, − | | − − | | 6 noncommutative Cauchy kernel. Therefore note that the noncommutative Cauchy kernel is defined on a set which is larger then the set (x,s) : x < s where the noncommutative { | | | |} Cauchykernelseriesisdefined. Sincex= sisasolutionofs2 2sRe[s]+ s 2 = 0,onemaywonder − | | if the factor (x s¯) can be simplified from the expression of the noncommutative Cauchy kernel. − However, as shown in the next result, this is not possible and the noncommutative Cauchy kernel cannot be extended to a continuous function in x = s. With an abuse of notation, we will denote the noncommutative Cauchy kernel series and the noncommutative Cauchy kernel with the same symbol S−1(s,x). Theorem 2.11. Let S−1(s,x) be the noncommutative Cauchy kernel with xs = sx. Then 6 S−1(s,x) is irreducible and lim S−1(s,x) does not exist. x→s Proof. We prove that we cannot find a degree one polynomial Q(x) such that x2 2xRe[s]+ s 2 = (s+x 2 Re[s])Q(x). − | | − The existence of Q(x) would allow the simplification S−1(s,x) = Q−1(x)(s+x 2 Re[s])−1(s+x 2 Re[s])= Q−1(x). − − We proceed as follows: first of all note that Q(x) has to be a monic polynomial of degree one, so we set Q(x)= x r − where r = r + n r e . The equality 0 j=1 j j P (s+x 2 Re[s])(x r)= x2 2xRe[s]+ s 2 − − − | | gives sx sr xr+2r Re[s] s 2 = 0. − − −| | Solving for r, we get r = (s+x 2 Re[s])−1(sx s 2), − −| | which depends on x. Let us now prove that the limit does not exists. Let ε = ε + n ε e , 0 j=1 j j and consider P S−1(s,s+ε) = ((s+ε)2 2(s+ε)Re[s]+ s 2)−1ε = ((s+ε)2 2(s+ε)Re[s]+ s 2)−1ε − | | − | | = (sε+εs+ε2 2εRe[s])−1ε − = (ε−1(sε+εs+ε2 2εRe[s]))−1 = (ε−1sε+s+ε 2Re[s]))−1. − − If we now let ε 0, we obtain that the term ε−1sε does not have a limit because → ε ε−1sε = sε ε2 | | ε ε s i j ℓ contains scalar addends of type with i,j,ℓ 0,1,2,3 that do not have limit. ε2 ∈ { } | | The following result will be useful in the sequel and its proof follows by a simple computation. Proposition 2.12. Let s= Re[s]+s. Then the following identity holds s2 2sRe[s]+ s 2 = 0. − | | 6 3 Slice-monogenic functional calculus for bounded operators In the sequel, we will consider a Banach space V over R (the case of complex Banach spaces can be discussed in a similar fashion) with norm . It is possible to endow V with an operation k·k of multiplication by elements of R which gives a two-sided module over R . We recall that a n n two-sided moduleV over R is called a Banach moduleover R , if there exists a constant C 1 n n ≥ such that va C v a and av C a v for all v V and a R . n k | ≤ k k| | k | ≤ | |k k ∈ ∈ In the sequel, we will make use of the following notations. By V we denote a Banach space over R with norm . • k·k By V we denote the two-sided Banach module over R corresponding to V R . An n n n • ⊗ elementinV isofthetype v e (whereA = i ...i ,i 1,2,...,n ,i < ... < i n A A⊗ A 1 r ℓ ∈ { } 1 r is a multi-index). The multiplications of an element v V with a scalar a R are n n P ∈ ∈ defined as va= v (e a) and av = v (ae ). We will write v e instead A A⊗ A A A⊗ A A A A of v e . We define v 2 = v 2 . A A⊗ A P k kVn Ak PAkV P (VP) is the space of bounded R-homPomorphisms of the Banach space V to itself endowed • B with the natural norm denoted by . B(V) k·k Let T (V). We define an operator T = T e and its action on v = v e V • A ∈ B A A A B B ∈ n as T(v) = T (v )e e . The operator T is a right-module homomorphism which is A,B A B A B P P a bounded linear map on V . The set of all such bounded operators is denoted by (V ). n n n P B We define T 2 = T 2 . k kBn(Vn) Ak AkB(V) P 3.1 The S-resolvent operator for bounded operators Throughout the rest of this section, and unless otherwise specified, we will only consider operators of the form T = T + n e T where T (V) for µ = 0,1,...,n. The set of such 0 j=1 j j µ ∈ B 0,1 operators in (V ) will be denoted by (V ). Bn n P Bn n Definition 3.1. Let T 0,1(V ) and s= Re[s]+s. We define the S-resolvent operator series n n ∈ B as S−1(s,T):= Tns−1−n (4) n≥0 X for T < s . k k | | Theorem 3.2. Let T 0,1(V ) and s = Re[s]+s. Then n n ∈ B Tns−1−n = (T2 2TRe[s]+ s 2 )−1(T s ), (5) − − | | I − I n≥0 X for T < s . k k | | Proof. InTheorem2.10thecomponentsofxandsarerealnumbersandthereforetheyobviously commute. When we formally replace x by operator T we cannot assume that T T = T T and µ ν ν µ so we need to verify independently that (5) still holds. To this aim, we check that (T − − s )−1(T2 2TRe[s]+ s 2 ) is the inverse of Tns−1−n. In what follows, we assume the I − | | I n≥0 convergence of the series to be in the norm of (V ): n n BP (T s )−1(T2 2TRe[s]+ s 2 ) Tns−1−n = − − I − | | I I n≥0 X 7 so then we get ( s 2 T2+2TRe[s]) Tns−1−n = T +(s 2 Re[s]) . −| | I − − I n≥0 X Observing that s 2 T2+2TRe[s] commutes with Tn we can write −| | I − Tn( s 2 T2+2TRe[s])s−1−n = T +(s 2 Re[s]) . −| | − − I n≥0 X Now expand the series as Tn( s 2 T2+2Re[s]T)s−1−n = ( s 2 T2+2TRe[s])s−1 −| | I − −| | I − n≥0 X +T1( s 2 T2+2TRe[s])s−2+T2( s 2 T2+2TRe[s])s−3+... −| | I − −| | I − = s 2s−1+T( 2sRe[s]+ s 2)s−2+T2(s2 2sRe[s]+ s 2)s−3+... − | | − | | − | | and using Propo(cid:16)sition 2.12, we get (cid:17) Tn( s 2 T2+2TRe[s])s−1−n = s 2s−1 +Ts2s−2 = s 2s−1 +T −| | − −| | I −| | I n≥0 X = sss−1 +T = s +T = (s 2 Re[s]) +T. − I − I − I Proposition 3.3. When Ts = sT, the operator S−1(s,T) equals (s T)−1 when the series I I − (4) converges. Proof. It follows by direct computation. Definition 3.4. (The S-spectrum and the S-resolvent set) Let T 0,1(V ) and s = Re[s]+s. n n ∈B We define the S-spectrum σ (T) of T as: S σ (T) = s Rn+1 : T2 2 Re[s]T + s 2 is not invertible . S { ∈ − | | I } An element in σ (T) will be called an S-eigenvalue. S The S-resolvent set ρ (T) is defined by S ρ (T) = Rn+1 σ (T). S S \ Definition 3.5. (The S-resolvent operator) Let T 0,1(V ) and s = Re[s]+s ρ (T). We n n S ∈ B ∈ define the S-resolvent operator as S−1(s,T) := (T2 2Re[s]T + s 2 )−1(T s ). (6) − − | | I − I Example 3.6. (Pauli matrices) As an example, we compute the S-spectrum of two Pauli matrices σ , σ (compare with example 4.10 in [5]): 3 1 1 0 0 1 σ = σ = . 3 0 1 1 1 0 (cid:20) − (cid:21) (cid:20) (cid:21) Let us consider the matrix T = σ e +σ e and let us compute T2 2Re[s]T + s 2 . We obtain 3 1 1 2 − | | I the matrix s 2 2 2Re[s]e 2(e Re[s])e 1 1 2 | | − − − 2(e +Re[s])e s 2 2+2Re[s]e 1 2 1 (cid:20) − | | − (cid:21) whose S-spectrum is σ (T) = 0 s R3 : Re[s]= 0, s = 2 . S { }∪{ ∈ | | } 8 Theorem 3.7. Let T 0,1(V ) and s = Re[s]+s ρ (T). Let S−1(s,T) be the S-resolvent n n S ∈ B ∈ operator defined in (6). Then S−1(s,T) satisfies the (S-resolvent) equation S−1(s,T)s TS−1(s,T) = . (7) − I Proof. Replacing (6) in the above equation we have (T2 2Re[s]T + s 2 )−1(T s )s+T(T2 2Re[s]T + s 2 )−1(T s )= (8) − − | | I − I − | | I − I I and applying (T2 2Re[s]T + s 2 ) to both hands sides of (8), we get − | | I (T s )s+(T2 2Re[s]T + s 2 )T(T2 2Re[s]T + s 2 )−1(T s ) − − I − | | I − | | I − I = T2 2Re[s]T + s 2 . − | | I Since T and T2 2Re[s]T + s 2 commute, we obtain the identity − | | I (T s )s+T(T s )= T2 2Re[s]T + s 2 − − I − I − | | I which proves the statement. 3.2 Properties of the spectrum and the functional calculus Theorem 3.8. (Structure of the S-spectrum) 0,1 Let T (V ) and let p = Re[p]+p be an S-eigenvalue of T with p = 0. Then all the elements n n ∈ B 6 of the sphere s = Re[s]+s with s = p and s = p are S-eigenvalues of T. 0 0 | | | | Proof. It is immediate and is left to the reader. Definition 3.9. Let Sn = x Rn+1 : x = 1 denote the unit sphere of Rn+1. For any set { ∈ | | } A Rn+1, let us define the circularization of A as set ⊆ circ(A) := u+vSn. u+vI∈A [ Definition 3.10. Let T =T + n e T 0,1(V ). Let U Rn+1 be an open set such that 0 j=1 j j ∈ Bn n ⊂ (i) ∂(U L ) is union of a finPite number of rectifiable Jordan curves for every I S, I ∩ ∈ (ii) U contains the circularization of the S-spectrum σ (T). S A function f is said to be locally s-monogenic on σ (T) if there exists an open set U Rn+1, S ⊂ as above, on which f is s-monogenic. We will denote by the set of locally s-monogenic functions on σ (T). MσS(T) S Remark 3.11. Note that any open set U containing the circularization of the S-spectrum contains open balls with center in x for all x σ (T) R. Moreover, by Theorem 3.8, if the 0 0 S ∈ ∩ (n 1)-sphere σ = s Rn+1 : Re[s] = s , s = r belongs to σ (T), then U must contain an 0 S − { ∈ | | } open annular domain with center in s R. In fact, set m = min dist(s,∂U). Then for 0 s∈circ(σ) ∈ any R < m the annular domain x Rn+1 r R < x s < r+R is contained in U. 0 { ∈ | − | − | } 9 Theorem 3.12. Let T 0,1(V ) and f . Let U Rn+1 be an open set as in ∈ Bn n ∈ MσS(T) ⊂ Definition 3.10 and let U = U L for I S. Then the integral I I ∩ ∈ 1 S−1(s,T) ds f(s) (9) I 2π Z∂UI does not depend on the choice of the imaginary unit I and on the open set U. Proof. We first note that the integral (9) does not depend on the choice of U by the Cauchy theorem applied on the plane L , see [2]. We now show the independence of the choice of I I S. Note that since the S-spectrum is bounded (because it is contained in the ball s ∈ { ∈ Rn+1 : s T ) we can choose a finite number of open balls B ,...,B and of open annular 1 ν | |≤ k k} domains A ,...,A , ν,µ N, containing the S-spectrum of T. We observe that thanks to the 1 µ ∈ the Cauchy theorem we can write: 1 S−1(s,T)ds f(s) I 2π Z∂UI ν µ 1 1 = S−1(s,T)ds f(s)+ S−1(s,T)ds f(s), (10) I I 2π 2π i=1Z∂(Bi∩LI) i=1Z∂(Ai∩LI) X X where the right hand side does not depend on the choice of the B ’s and A ’s. Since f admits i i series expansion on the B ’s for Taylor theorem and on A ’s by theLaurent theorem (see [2]), we i i can integrate term by term. Let us now choose another imaginary unit I′ S, I = I′ and let us ∈ 6 write the analogue of (3.2) on LI′. The S-spectrum contains either real points or, by Theorem 3.8,(n 1)-spheresofthetype s Rn+1 : Re[s]= s , s = r . EverycomplexlineL = R+IR 0 I − { ∈ | | } contains all the real points belonging to the S-spectrum. Let s Rn+1 : s = s +It, t = r 0 { ∈ | | } be an (n 1)-sphere in the S-spectrum. The two points of the sphere lying on the complex − line L are s rI so, on the plane, they have coordinates (s , r). The coordinates of the two I 0 0 ± ± intersection points on a different complex line LI′ are still (s0, r), so the right hand side of ± (3.2) does not depend on the choice of I S. Thus ∈ 1 1 S−1(s,T) dsIf(s)= S−1(s,T) dsI′f(s). 2π 2π Z∂UI Z∂UI′ We give a result that motivates the functional calculus. Theorem 3.13. Let x = Re[x]+x, a = Re[a]+a Rn+1, m N and consider the monomial xma. Consider T 0,1(V ), let U Rn+1 be a∈n open set∈as in Definition 3.10, and set n n ∈ B ⊂ U = U L for I S. Then I I ∩ ∈ 1 Tma= S−1(s,T) ds sm a. (11) I 2π Z∂UI Proof. Let us consider the power series expansion for the operator S−1(s,T) and a circle C r centered in the origin and of radius r > T . We have: k k 1 1 S−1(s,T) ds sm a = Tn s−1−n+m ds a= Tm a, (12) I I 2π 2π Z∂UI n≥0 ZCr X since ds s−n−1+m = 0 if n = m, ds s−n−1+m = 2π if n = m. (13) I I 6 ZCr ZCr TheCauchy theoremshowsthattheaboveintegrals arenotaffected ifwereplaceC by∂U . r I 10