The F-functional calculus for unbounded operators

Journal of Geometry and Physics 86 (2014) 392–407

Contents lists available at ScienceDirect

Journal of Geometry and Physics

journal homepage: www.elsevier.com/locate/jgp

The F-functional calculus for unbounded operatorsFabrizio Colombo ∗, Irene SabadiniPolitecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy

a r t i c l e i n f o

Article history:Received 3 December 2013Received in revised form 15 September2014Accepted 17 September 2014Available online 28 September 2014

MSC:47A1047A60

Keywords:F-spectrumF-functional calculus for n-tuples ofunbounded operators

Fueter mapping theorem in integral formSlice monogenic functions vector-valued

a b s t r a c t

In the recent years the theory of slice hyperholomorphic functions has become an impor-tant tool to study two functional calculi for n-tuples of operators and also for its applica-tions to Schur analysis. In particular, using the Cauchy formula for slice hyperholomorphicfunctions, it is possible to give the Fueter–Scemapping theorem an integral representation.With this integral representation it has been defined a monogenic functional calculus forn-tuples of bounded commuting operators, the so called F-functional calculus.

In this paper we show that it is possible to define this calculus also for n-tuples con-taining unbounded operators and we obtain an integral representation formula analogousto the one of the Riesz–Dunford functional calculus for unbounded operators acting on acomplex Banach space. Aswewill see, it is not an easy task to provide the correct definitionof the F-functional calculus in the unbounded case.

This paper is addressed to a double audience, precisely to people with interests inhypercomplex analysis and also to people working in operator theory.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

The theory of slice hyperholomorphic functions is a very important tool in functional analysis because of its applicationsto operator theory. Slice hyperholomorphic functions are called also slicemonogenic,when they are defined on the Euclideanspace Rn+1 and take values in the Clifford algebra Rn, and slice regular in case they are defined on the algebra of quaternionsinto itself. The recent monograph [1] contains the function theory as well as its applications to the so called S-functionalcalculus for n-tuples T = (T1, . . . , Tn) of (non necessarily commuting) operators. Given a slice monogenic function f ,satisfying suitable conditions, the S-functional calculus associates to T the operator function f (T ). This calculus is the naturalextension of the Riesz–Dunford functional calculus for n-tuples of operators, and a suitable variation of it can be applied toquaternionic linear operators.

The theory of slice hyperholomorphic functions has been applied also to someproblems in Schur analysis, see [2,3], whichcan be reformulated and studied in this noncommutative setting, see [4–6].

Slice monogenic functions are related to the classical monogenic functions (i.e. functions in the kernel of the Diracoperator, see [7,8]), by the so-called Fueter mapping theorem, see [9–12]. In [13] it is shown how the Cauchy formula forslice monogenic functions allows to write the Fueter mapping theorem in integral form. Using this result, we have definedthe so called F-functional calculus for n-tuples of bounded operators. This is a monogenic functional calculus in the sensethat, using slice monogenic functions, we definemonogenic functions of n-tuples of operators in the spirit of themonogenicfunctional calculus studied in [14] (see also the literature therein).

The aim of this paper is to extend the F-functional calculus to the case of n-tuples of unbounded operators.

∗ Corresponding author. Tel.: +39 03923994613; fax: +39 0223994626.E-mail addresses: [email protected] (F. Colombo), [email protected] (I. Sabadini).

http://dx.doi.org/10.1016/j.geomphys.2014.09.0020393-0440/© 2014 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.geomphys.2014.09.002

http://www.elsevier.com/locate/jgp

http://www.elsevier.com/locate/jgp

http://crossmark.crossref.org/dialog/?doi=10.1016/j.geomphys.2014.09.002&domain=pdf

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.geomphys.2014.09.002

F. Colombo, I. Sabadini / Journal of Geometry and Physics 86 (2014) 392–407 393

This paper is devoted to people working in hypercomplex analysis as well as in operator theory. For this reason we givesome preliminary results in order to make the paper self contained as much as possible.Monogenic functions. The setting in which wework is the real Clifford algebra Rn over n imaginary units e1, . . . , en satisfyingthe relations eiej + ejei = −2δij. An element in the Clifford algebra will be denoted by

A eAxA where A = i1 . . . ir , iℓ ∈

1, 2, . . . , n, i1 < · · · < ir , is a multi-index, eA = ei1ei2 . . . eir and e∅ = 1. When n = 1, we have that R1 is the algebra ofcomplex numbers C, while when n = 2 we obtain the division algebra of real quaternions H. As it is well known, for n > 2,the Clifford algebras Rn have zero divisors and for n ≥ 2 are noncommutative.

In the Clifford algebraRn, we can identify some specific elementswith the vectors in the Euclidean spaceRn+1: an element(x0, x1, . . . , xn) ∈ Rn+1 will be identified with the element x = x0 + x = x0 +

nj=1 xjej called, in short, paravector. The

norm of x ∈ Rn+1 is defined as |x|2 = x20 + x21 + · · · + x2n. The real part x0 of x will be also denoted by Re(x). Using theabove identification, a function f : U ⊆ Rn+1

→ Rn is seen as a function f (x) of the paravector x. We will denote by S the(n − 1)-dimensional sphere of unit 1-vectors in Rn, i.e.

S = e1x1 + · · · + enxn : x21 + · · · + x2n = 1.

Any I ∈ S is such that I2 = −1.Note that to any nonreal paravector x = x0 + e1x1 + · · · + enxn we can associate a (n − 1)-dimensional sphere defined

as the set, denoted by [x], of elements of the form x0 + I|e1x1 + · · · + enxn| when I varies in S. By CI we denote the set ofelements of the form u + Iv, u, v ∈ R; for any fixed I ∈ S,CI is a complex plane.

The slice monogenic functions (see also [15,16]) are defined as follows.

Definition 1.1. Let U ⊆ Rn+1 be an open set and let f : U → Rn be a real differentiable function. Let I ∈ S and let fI bethe restriction of f to the complex plane CI . We say that f is a (left) slice monogenic function, or s-monogenic function, if forevery I ∈ S, we have

12

∂

∂u+ I

∂

∂v

fI(u + Iv) = 0,

on U ∩ CI . We will denote by SM(U) the set of slice monogenic functions on U . We say that f is a right slice monogenic (forshort right s-monogenic) function if, for every I ∈ S, we have

12

∂

∂ufI(u + Iv)+

∂

∂vfI(u + Iv)I

= 0,

on U ∩ CI .

We define below some domains in which slice monogenic functions are naturally defined.

Definition 1.2. Let U ⊆ Rn+1 be a domain. We say that U is a slice domain (s-domain for short) if U ∩ R is nonempty and ifCI ∩ U is a domain in CI for all I ∈ S.

Let U ⊆ Rn+1. We say that U is axially symmetric if, for all u+ Iv ∈ U , the whole (n−1)-sphere [u+ Iv] is contained in U .

To state the Cauchy formula for slice monogenic functions with values in the Clifford algebra Rn, we need to introducethe Cauchy kernel

S−1C (s, x) := (s − x)(s2 − 2Re(x)s + |x|2)−1,

which is defined for s ∈ [x].

Theorem 1.3. Let U ⊆ Rn+1 be an axially symmetric s-domain and let f : W ⊆ Rn+1→ Rn be a slice monogenic function and

suppose that U ⊂ W. Then, for u ∈ U, we have

f (x) =12π

∂(U∩CI )

S−1C (s, x)dsI f (s), dsI = −Ids (1)

where the integral does not depend neither on U nor on I ∈ S.

We now recall the Fueter–Sce mapping theorem (often called simply Fueter mapping theorem). Let n be an odd numberand let f : U ⊂ Rn+1

→ Rn be of the form f (x + Iy) = α(x, y)+ Iβ(x, y), where α and β are suitable Rn-valued functionssatisfying the Cauchy–Riemann system and I is a 1-vector in the Clifford algebra Rn such that I2 = −1. This function f isobviously slice monogenic. Then, the function

f (x0, |x|) = ∆(n−1)/2x (α(x0, |x|)+

x|x|β(x0, |x|)),

394 F. Colombo, I. Sabadini / Journal of Geometry and Physics 86 (2014) 392–407

where∆x is the Laplace operator in dimension n + 1, is monogenic, i.e. it is in the kernel of the Dirac operator

∂

∂x0+

ni=1

ei∂

∂xi.

In [13] it is shown that given a slicemonogenic function f , the Cauchy formula (1) allows towrite the Fuetermapping theoremin integral form as

f (x) =12π

∂(U∩CI )

∆n−12

x S−1C (s, x)dsI f (s), dsI = −Ids.

The main advantage of this approach is that there is no need to compute the powers of the Laplacian applied to the slice

monogenic function f in order to obtain f . In fact, it is enough to compute∆n−12

x S−1C (s, x) and to use the above Cauchy formula.

Functional calculus. To better explain our results we quickly recall the Riesz–Dunford functional calculus (for more detailssee [17] or [18]). Let f be a holomorphic function defined on an open set containing Ω , where Ω ⊂ C is an open set thatcontains the spectrum of the bounded operator B defined on a complex Banach space X . We define the functional calculusf (B) as

f (B) =1

2π i

∂Ω

(λI − B)−1f (λ)dλ.

For an unbounded operator A, we assume that the function f is holomorphic at infinity and on the spectrum σ(A) of theclosed linear operator Awhose domain D(A) is contained in X .

Precisely, with the positions Φ(λ) := (λ − α)−1,Φ(∞) = 0 and Φ(α) = ∞, for α ∈ ρ(A) (we assume in this casethat the resolvent set ρ(A) is nonempty) we define f (A) := φ(B) where φ(µ) = f (Φ−1(µ)) and B := (A − αI)−1. For thisfunctional calculus we have the integral representation formula

f (A) = f (∞)I +1

2π i

Γ

(λI − A)−1f (λ)dλ (2)

where Γ is a closed curve, given by the union of a finite number of Jordan arcs, that contain the spectrum σ(A) and the pointat infinity. The function f is holomorphic in an open set that contains Γ in its interior. Note that σ(A) can be unboundedand the representation formula proves also that f (A) is independent of α.

Let us now spend a fewwords on the F-functional calculus for bounded commuting operators. Let us consider an operatorof the form T = T0+e1T1+· · ·+enTn (with commuting components Ti) and its so-called conjugate T = T0−e1T1−· · ·−enTn.To each admissible monogenic function f (we will give the precise definition later) we associate the operator f (T ) using theFueter mapping theorem in integral form.

More precisely, we set

Fn(s, x) := ∆n−12

x S−1C (s, x)

where n is an odd number, and we consider the operator Fn(s, T ) obtained by replacing x by T in Fn(s, x). We obtain theso-called F-resolvent operator

Fn(s, T ) := γn(sI − T )(s2I − s(T + T )+ TT )−n+12 ,

where γn are known constants depending on n. This operator gives rise to a new notion of spectrum for the n-tuples ofoperators T , called F-spectrum of T and denoted by σF (T ). It is the set in which Fn(s, T ) is not defined and it consists of thoses ∈ Rn+1 such that s2I − s(T + T )+ TT is not invertible.

We proved that the F-functional calculus given by

f (T ) =12π

∂(U∩CI )

Fn(s, T )dsI f (s),

is well defined since the integral does not depend neither on the open set U nor on I ∈ S. Here U is a suitable open set thathas smooth boundary and contains the F-spectrum of T .The main results. The goal of this paper is to prove that there is an extension of the F-functional calculus to the case ofcommuting unbounded operators, see Definition 4.11. Indeed, for suitable slicemonogenic functions f , let ψ be the function

ψ(p) := ∆n−12

p (pn−1φ(p))

where ∆p is the Laplace operator in dimension n + 1, n is an odd number and φ(p) := f (Φ−1(p)), p = Φ(s) =

(s − k)−1,Φ(∞) = 0,Φ(k) = ∞. Then, we set A := (T − kI)−1 for some k ∈ ρF (T )∩ R = ∅. The F-functional calculus forunbounded operators in defined byf (T ) := (AA)−

n−12 ψ(A).


The integral representation off (T ) in terms of the F-resolvent of T is given by formula (20) in Section 4,which is the analogueof formula (2) for the Riesz–Dunford functional calculus.

This is a surprising fact because the Riesz–Dunford functional calculus is based on a Cauchy formula, while theF-functional calculus is based on an integral transform, so the fact that there exists an integral representation for theF-functional calculus for unbounded operators was not necessarily expected.The plan of the paper. Section 2 contains some basics on the F-functional calculus for bounded operators. Section 3 containsnew results on slice monogenic functions with values in a Banach module and their Cauchy formula. The main results ofthis paper are in Section 4, where we first collect some technical results that give us the evidences of how the definition ofF-functional calculus for unbounded operators has to be. Definition 4.11 assigns the F-functional calculus for unboundedoperators and Theorem 4.14 shows that the F-functional calculus is well defined and has an integral representation interms of the F-resolvent operator. Finally, in Section 5 we make some concluding comments and show some differencesbetween the F-functional calculus and the SC -functional calculus,which is a slicemonogenic functional calculus based on theF-spectrum.

2. Preliminary results on bounded operators

In this section we collect some definitions and results on the F-functional calculus for n-tuples of bounded commutingoperators, whichwill be useful in the sequel. Moreover, we introduce the definition of slicemonogenic functionswith valuesin a Banach module by providing some properties, among which the Cauchy formula.

By V we denote a Banach space over R with norm ∥ · ∥. We then set Vn = V ⊗ Rn. An element in Vn is of the formA vA ⊗ eA (where A = i1 . . . ir , iℓ ∈ 1, 2, . . . , n, i1 < · · · < ir is a multi-index). The multiplications (right and left) of an

element v ∈ Vn with a scalar a ∈ Rn are defined as

va =

A

vA ⊗ (eAa), and av =

A

vA ⊗ (aeA).

These multiplications make Vn a two-sided Banach module over Rn. For short, we will write

A vAeA instead of

A vA ⊗ eA.Moreover, we define

∥v∥Vn =

A

∥vA∥V .

Let B(V ) be the space of bounded R-homomorphisms of the Banach space V into itself endowed with the natural normdenoted by ∥ · ∥B(V ). If TA ∈ B(V ), we can define the operator T =

A eATA and its action on

v =

B

vBeB

as

T (v) =

A,B

TA(vB)eAeB. (3)

A homomorphism T : Vn → Vn will be said right linear if T (v + u) = T (v) + T (u) and T (va) = T (v)a for allv, u ∈ Vn, a ∈ Rn. Note that an operator T as in (3) is right-linear over Rn. The set B(Vn) of bounded operators T =

A eATA

as in (3) turns out to be a left Rn-module if endowed with the operation of sum T + S =

A(TA + SA)eA, for anyT =

A TAeA, S =

A SAeA ∈ B(Vn) and the operation of left multiplication by a scalar λ =

C λCeC ∈ Rn given by

λT =

C,A λCTAeCeA. We note that it is possible to endow B(Vn) also with an operation of multiplication with a scalar onthe right as Tλ =

A,C TAλCeAeC . The action on a vector v ∈ Vn, v =

B vBeB is

(λT )(v) =

C,A,B

λCTA(vB)eCeAeB, (Tλ)(v) =

A,C,B

λCTA(vB)eAeCeB. (4)

Remark 2.1. From (4) it follows that if an operator T does not contain any imaginary unit, then λT = Tλ.

If we endow B(Vn)with the norm

∥T∥B(Vn) =

A

∥TA∥B(V ),

it turns out to be a two sided Banach module over Rn.In the sequel, we will only consider operators in the so-called paravector form T = T0 +

nj=1 ejTj where Tj ∈ B(V ) for

j = 0, 1, . . . , n. The set of such operators will be denoted by B0,1(Vn) and will be endowed with the norm inherited byB(Vn), namely

∥T∥B0,1(Vn) =

nj=0

∥Tj∥B(V ).

Given T = T0 +n

j=1 ejTj the so-called conjugate of T is T = T0 −n

j=1 ejTj.


The set of n-tuples of bounded commuting operators, in paravector form, is denoted by BC0,1(Vn). If T ∈ BC0,1(Vn), wedefine

T + T = 2T0TT = T 2

0 + T 21 + · · · + T 2

n .

We now recall some results proved in [19,13].

Definition 2.2 (The F-spectrum and The F-resolvent Sets). Let T ∈ BC0,1(Vn). We define the F-spectrum σF (T ) of T as:

σF (T ) = s ∈ Rn+1: s2I − s(T + T )+ TT is not invertible in B(Vn).

The F-resolvent set ρF (T ) is defined by

ρF (T ) = Rn+1\ σF (T ).

Below we state two important properties of the F-spectrum of bounded operators: the first says that it has an axialsymmetry, while the second establishes that it is a compact and nonempty set.

Theorem 2.3 (Structure of The F-spectrum). Let T ∈ BC0,1(Vn) and let p = p0 + p1I ∈ [p0 + p1I] ⊂ Rn+1\ R, such that

p ∈ σF (T ). Then all the elements of the (n − 1)-sphere [p0 + p1I] belong to σF (T ). Thus the F-spectrum consists of real pointsand/or (n − 1)-spheres.

Theorem 2.4 (Compactness of The F-spectrum). Let T ∈ BC0,1(Vn). Then the F-spectrum σF (T ) is a compact nonempty set.Moreover σF (T ) is contained in s ∈ Rn+1

: |s| ≤ ∥T∥ .

The definition of F-resolvent operator is suggested by the Fueter–Sce mapping theorem in integral form.

Definition 2.5 (F-resolvent Operator). Let n be an odd number and let T ∈ BC0,1(Vn). For s ∈ ρF (T ) we define theF-resolvent operator by

Fn(s, T ) := γn(sI − T )(s2I − s(T + T )+ TT )−n+12 , (5)

where the constants γn are given by

γn := (−1)(n−1)/22n−1n − 1

2

!

2.

Now we have to specify which are the functions and which are the suitable open sets containing the F-spectrum forwhich it is possible to define the F-functional calculus for bounded operators.

Definition 2.6. Let T ∈ BC0,1(Vn) and let U ⊂ Rn+1 be an axially symmetric s-domain.(a) We say that U is admissible for T if that contains the F-spectrum σF (T ), and if ∂(U ∩ CI) is union of a finite number of

rectifiable Jordan curves for every I ∈ S.(b) Let W be an open set in Rn+1. A function f ∈ SM(W ) is said to be locally s-monogenic on σF (T ) if there exists an

admissible domain U ⊂ Rn+1 such that U ⊂ W .(c) We will denote by SMσF (T ) the set of locally s-monogenic functions on σF (T ).

The following theorem is a crucial fact for the well posedness of the F-functional calculus.

Theorem 2.7. Let n be an odd number, let T ∈ BC0,1(Vn) and suppose that f ∈ SMσF (T ). Let U be an open set, containingσF (T ), as in Definition 2.6. Then the integral

12π

∂(U∩CI )

Fn(s, T ) dsI f (s), dsI = ds/I (6)

is independent of I ∈ S and of the open set U.

Finally, we are ready to give the following:

Definition 2.8 (The F-functional Calculus for Bounded Operators). Let n be an odd number, T ∈ BC0,1(Vn). Let U be an openset, containing σF (T ), as in Definition 2.6. Suppose that f ∈ SMσF (T ) and let f (x) = ∆

n−12 f (x). We define the F-functional

calculus as

f (T ) =12π

∂(U∩CI )

Fn(s, T ) dsI f (s).

For other properties of the F-functional calculus we refer the reader to [20].


3. Slice monogenic functions with values in a Banach module

In order to extend the results described in Section 2 to the case of unbounded operators, it will be necessary to considerfunctions with values in the Banach Rn-module B(Vn). To this end, we introduce the general notion of slice monogenicfunctions with values in a Banach Rn-module, as well as some basic results which extend the corresponding results in thequaternionic case, see [21].

Since, in general, it is not guaranteed that statements true in the quaternionic case hold also in a higher dimensionalalgebra, we repeat the proofs of the main results where there are differences with respect to the proof given in [21].

We begin by giving the following:

Definition 3.1. Let f : U ⊆ Rn+1→ Rn and let y ∈ U be a nonreal point, y = u0 + Iv0. Let fI be the restriction of f to the

plane CI . Assume that

limx→y, x∈CI

(x − y)−1(fI(x)− fI(y)) (7)

exists. Then we say that f admits left slice derivative in y. If y is real, we assume that

limx→y, x∈CI

(x − y)−1(fI(x)− fI(y)) (8)

exists, equal to the same value, for all I ∈ S. Thenwe say that f admits left slice derivative in y. If f admits left slice derivativefor every y ∈ U then we say that f admits left slice derivative in U or, for short, that f is left slice differentiable in U .

Remark 3.2. We can give an analogous definition by writing the factor (x − y)−1 on the right, thus obtaining the notion offunctions right slice differentiable. When we will refer to right slice differentiable or to right slice monogenic functions wewill write it explicitly. When we will deal with left slice differentiable or to left slice monogenic functions, we will omit tospecify the word ‘‘left’’.

The notion of slice derivability and slice monogenicity are equivalent by virtue of the next result.

Proposition 3.3. Let U ⊆ Rn+1 be an open set and let f : U ⊆ Rn+1→ Rn be a real differentiable function. Then f is slice

monogenic on U if and only if it admits slice derivative on U.

Proof. Let f be a slice monogenic function on U . Then we use the Splitting Lemma, see [1], to write its restriction to thecomplex plane CI . Set I1 = I ∈ S and let I2, . . . , In be a completion to a basis of Rn satisfying the defining relationsIr Is + IsIr = −2δrs. Then there exist 2n−1 holomorphic functions FA : U ∩ CI → CI such that for every z = u+ Iv ∈ U ∩ CI :

fI(z) =

n−1|A|=0

FA(z)IA, IA = Ii1 . . . Iis ,

where A = i1 . . . is is a subset of 2, . . . , n, with i1 < · · · < is, or, when |A| = 0, I∅ = 1. Let y be nonreal and assume thaty ∈ U ∩ CI . Then

limx→y, x∈CI

(x − y)−1(fI(x)− fI(y)) = limx→y, x∈CI

(x − y)−1

n−1|A|=0

(FA(x)− FA(y))IA

=

n−1|A|=0

F ′

A(y)IA (9)

so the limit exists and f admits slice derivative in all nonreal points in U . If y is real then the same computations show thatthe limit in (9) exists on each complex plane CI and since f is slice monogenic at y the limit is equal to ∂

∂u f (y) = ∂uf (y).To show the converse, we assume that f admits slice derivative in U . By (7) and (8) fI admits derivative on U ∩ CI for all

I ∈ S. We can write the restriction fI by splitting its values into components, by fixing a basis I1 = I, I2, . . . , In, Ir Is + IsIr =

−2δrs, obtaining

fI(z) =

n−1|A|=0

FA(z)IA, IA = Ii1 . . . Iis ,

for z ∈ U ∩ CI . Since fI admits derivative on CI , we deduce that all the functions FA admit complex derivative and thus theyare in the kernel of the Cauchy–Riemann operator ∂u + I∂v for all I ∈ S as well as fI . Thus f is slice monogenic.

Let Z be a two sided Banach module over Rn. Let us denote by Z∗ the set of bounded, left linear maps from Z to Rn(see [7, 2.3]). We will say that Z∗ is the dual of Z, where Z is considered as a left module over Rn.


Definition 3.4. Let Z be a two sided Banach module over Rn and let Z∗ be its dual. Let U be an open set in Rn+1. A functionf : U → Z is said to be weakly slice monogenic in U ifΛf admits slice derivative for everyΛ ∈ Z∗. A function f : U → Z issaid to be strongly slice monogenic in y if

limx→y,x∈CI

(x − y)−1(fI(x)− fI(y)) (10)

exists in the topology of Z in case y ∈ U is nonreal and y ∈ CI and if

limx→y,x∈CI

(x − y)−1(fI(x)− fI(y)) (11)

exists in the topology of Z for every I ∈ S, and does not depend on I , in case y ∈ U is real.

In order to show that the notions of weakly and strong slice monogenicity coincide it is necessary to prove that theprinciple of uniform boundedness holds also in this setting. Its validity has been proven in the quaternionic setting in [22]but as far as we know this result has never been proved in the framework of Clifford modules, so we insert its proof for lackof reference.

Theorem 3.5. Let Z and Z′ be left (resp. right) Rn-modules. Let Saa∈A, where A is a set, be a family of continuous maps of Zinto Z′ which satisfies the following properties

(a) ∥Sa(u + w)∥ ≤ ∥Sa(u)∥ + ∥Sa(w)∥,∀u, w ∈ Z,(b) ∥Sa(wα)∥ = ∥Sa(w)α∥,∀w ∈ Z, for all α ∈ R, α ≥ 0.

If, for each v ∈ Z, the set Sava∈A is bounded, then limv→0 Sav = 0 uniformly in a ∈ A.

Proof. For ε > 0, a ∈ A and a positive integer k, let us consider the set

Zkdef.=

u ∈ Z :

1k Sa(u)+

1k Sa(−u) ≤

ε

2

which is closed since Sa are continuous. Moreover, by assumption, the sets Sava∈A are bounded, so

Z =

∞k=1

Zk.

Note that the Baire category theorem holds in this setting, indeed its proof is based on arguments that are not affected bythe fact that we are working in a Banach Rn-module (see [17, p. 20]). Thus, there exists a Zk0 containing a ball B(v0, δ)withcenter at v0 and radius δ > 0. Take ∥u∥ < δ so that v0 and v0 + u both belong to B(v0, δ) and to Zk0 . Thus we have 1

k0Sa(v0 + u)

≤ε

2and

1k0

Sa(−v0) ≤

ε

2.

Using assumption (a) we deduce 1k0

Sa(u) ≤

1k0

Sa(v0 + u)+

1k0

Sa(−v0) ,

and using assumption (b) we get 1k0

Sa(u) =

Sa 1k0

u ≤ ε, ∥u∥ < δ, a ∈ A.

The mapping v → v/k0 is a homeomorphism of Z into itself since Z is a topological Rn-module and thus the multiplicationby a real scalar is continuous. Thus limv→0 Sav = 0 uniformly in a ∈ A.

In the particular case of linear maps, Theorem 3.5 gives:

Theorem 3.6 (Principle of UniformBoundedness). Let Taa∈A, whereA is a set, be a family of continuous linearmaps of a BanachRn-module Z into a Banach Rn-module Z′. If, for each u ∈ Vn, the set Tava∈A is bounded, then limv→0 Tav = 0 uniformly ina ∈ A.

By the continuity of linear functional, every strongly slice monogenic function on U is obviously weakly slice monogenicon U . Now we show that also the converse is true.

Theorem 3.7. Let Z be a two sided Banach module over Rn. Every weakly slice monogenic function f : U ⊆ Rn+1→ Z is

strongly slice monogenic on U.


Proof. We will follow the lines of the proof in the complex case given in [23, p. 189]. Let f be a weakly slice monogenicfunction on U . Then, for anyΛ ∈ Z∗ and any I ∈ S, we can use the splitting of the restriction of a function into components,and write (Λf )I(u + Iv) =

A FΛ,A(u + Iv)IA where FΛ,A : U ∩ CI → CI . By hypothesis, for any y ∈ U ∩ CI the limit

limx→y, x∈CI (x − y)−1((Λf )I(x)− (Λf )I(y)) exists, and so the limits

limx→y, x∈CI

(x − y)−1(FΛ,A(x)− FΛ,A(y))

exists for all A. Thus the functions FΛ,A are holomorphic in U ∩ CI and so they admit a Cauchy formula whose integral canbe computed e.g. on a circle γ ⊂ CI , whose interior contains y and is contained in U . When y ∈ R we can pick any complexplane CI . For any increment h in CI we compute (note that (Λf )I = Λ(fI))

Λ(h−1(fI(y + h)− fI(y))− ∂uΛ(fI(y))) =12π

γ

h−1

1

x − (y + h)−

1x − y

−

1(x − y)2

dxIΛ(fI(x)),

where dxI = (du + Idu)/I . Then we observe that Λ(fI(x)) is continuous on γ which is compact, so |Λ(fI(x))| ≤ CΛ for allx ∈ γ . The family of maps f (x) : Z∗

→ Rn is pointwise bounded at each Λ, thus supx∈γ ∥fI(x)∥ ≤ C by the Principle ofuniform boundedness. ThusΛ(h−1(fI(y + h)− fI(y))− ∂uΛ(fI(y)))

≤C2π

∥Λ∥

γ

1x − (y + h)

−1

x − y

−

1(x − y)2

dxI ,so h−1(fI(y+h)− fI(y)) is uniformly Cauchy for ∥Λ∥ ≤ 1 and it converges to a limitw in Z, see Lemma p. 189 in [23] whoseproof adapts to the present case. In case y is real, the above reasoning holds on any complex plane CI and we have to showthat the elementw ∈ Z does not depend on I ∈ S. If the limit depends on I ∈ S we would have:

limx→y,x∈CI

(x − y)−1(fI(x)− fI(y)) = w, limx→y,x∈CJ

(x − y)−1(fJ(x)− fJ(y)) = w′. (12)

Let us applyΛ ∈ Z∗ to both hand sides of (12) and let us take into account the continuity ofΛ. We deduce:

limx→y,x∈CI

(x − y)−1(ΛfI(x)−ΛfI(y)) = Λ(w), limx→y,x∈CJ

(x − y)−1(ΛfJ(x)−ΛfJ(y)) = Λ(w′),

and since Λf is weakly slice monogenic the left hand side coincide so Λ(w − w′) = 0, i.e. w = w′. Thus f admits slicederivative at every y ∈ U and so it is strongly slice hyperholomorphic in U .

We now prove the following result:

Proposition 3.8. Let Z be a two sided Banachmodule and let U ⊆ Rn+1 be an open set. A real differentiable function f : U → Zis weakly slice monogenic if and only if

(∂u + I∂v)fI(u + Iv) = 0,

for all I ∈ S.Proof. Assume that f is weakly slice hyperholomorphic. Then for every nonreal y ∈ U, y ∈ CI , we can compute the limit(10) for the function ΛfI choosing first an increment x = y + h with h ∈ R and then x = y + Ih with h ∈ R. We obtain,respectively, ∂ufIΛ(y) and −I∂vΛfI(y)which must coincide. Thus we get (∂u + I∂v)ΛfI(u+ Iv) = Λ(∂u + I∂v)fI(u+ Iv) = 0for anyΛ ∈ X∗ and the statement follows by the Hahn–Banach theorem. If y is real, then the statement follows analogously,since the limit (11) exists for all I ∈ S.

Conversely, let us assume that fI satisfies the Cauchy–Riemann equation on U ∩ CI . Then Λ((∂u + I∂v)fI(u + Iv)) = 0for allΛ ∈ X∗ and all I ∈ S. SinceΛ is linear and continuous we can write (∂u + I∂v)ΛfI(u + Iv) = 0 and thusΛfI(u + Iv)belongs to the kernel of ∂u + I∂v for allΛ ∈ X∗ or, equivalently by Proposition 3.3, it admits slice derivative. Thus at everyy ∈ U ∩ CI we have

limx→y,q∈CI

(x − y)−1(ΛfI(x)−ΛfI(y)) = limx→y,x∈CI

Λ((x − y)−1(fI(x)− fI(y))),

for allΛ ∈ X∗. We deduce that f is weakly slice hyperholomorphic.

With all the above preliminary results on slice monogenic functions with values in a Banach module we deduce theCauchy formula whose proof is as in the quaternionic setting, see [21].

Theorem 3.9 (Cauchy Formulas). Let Z be a two sided Banach module and let W be an open set in Rn. Let U ⊂ W be an axiallysymmetric s-domain, and let ∂(U ∩CI) be the union of a finite number of rectifiable Jordan curves for every I ∈ S. Set dsI = ds/I .If f : W → Z is slice monogenic, then, for x ∈ U, we have

f (x) =12π

∂(U∩CI )

S−1C (s, x)dsI f (s), (13)

and the integral (13) does not depend on the choice of the imaginary unit I ∈ S and on U ⊂ W.


Proof. We have proved that weakly slice hyperholomorphic functions are strongly slice hyperholomorphic functions so, inparticular, they are continuous functions. The validity of the formula (13) follows as in point (b) p. 80 in [18].

The following theorem will be crucial in the proof of the main result in the next section.

Theorem 3.10. Let g(s) := ((s2I − s(T + T )+ TT )−1)r , for s ∈ ρF (T ) and r ∈ N. Suppose that f ∈ SM(U). Then the productgf is slice monogenic with values in B(Vn).

Proof. Here we have to note that T + T , and TT do not contain the imaginary units ei. If we set s = u + Iv, we get

(∂u + I∂v)(gf )(s) = [(∂u + I∂v)g(s)]f (s)+ g(s)(∂u + I∂v)f (s) = 0.

So the statement follows from Proposition 3.8.

Theorem 3.11. Let hr(s) := (sI − T )((s2I − s(T + T ) + TT )−1)r , for r ∈ N and s ∈ ρF (T ). Then hr is right slice monogenicwith values in B(Vn). In particular for r =

n+12 , where n is an odd number, we have that the F-resolvent operator is right slice

monogenic.

Proof. Using the notion of right slice monogenicity, we get the statement with direct computations.

4. The F -functional calculus: unbounded operators

We will define the F-functional calculus for unbounded operators for the following class of operators that we will calladmissible.

Definition 4.1 (Admissible Operators). Let V be a real Banach space and Vn = V ⊗ Rn. Let Tj : D(Tj) ⊂ V → V be linearclosed operators for j = 0, 1, . . . , n, such that TjTi = TiTj, for all i, j = 0, 1, . . . , n. Let D(T ) =

nj=0 D(Tj) be the domain of

the operator T = T0 +n

j=1 ejTj : D(T ) ⊂ Vn → Vn.We say that T is an admissible operator if

(1)n

j=0 D(Tj) is dense in Vn,(2) sI − T is densely defined in Vn, where s ∈ Rn+1,(3) D(TT ) ⊂ D(T ) is dense in Vn.

The definition of extended F-spectrum, F-resolvent set and of extended F-spectrum are as follows.

Definition 4.2. Consider the operator s2I − s(T + T ) + TT and let its domain be D := ∩nj=0 D(T 2

j ). The F-resolvent setρF (T ) of T is defined as the set of s ∈ Rn+1 such that

(i) s2I − s(T + T )+ TT : D → Vn is bijective;(ii) (s2I − s(T + T )+ TT )−1

: Vn → Vn belongs to B(Vn).

The F-spectrum σF (T ) of T is defined as

σF (T ) = Rn+1\ ρF (T )

and the extended F-spectrum is defined as:

σ F (T ) := σF (T ) ∪ ∞.

Remark 4.3. In the case of unbounded operators the F-spectrum is not necessarily bounded, it can be also the whole Rn+1

or it can also be empty. For this reason we will always assume that the resolvent set ρF (T ) is nonempty.

We can give the definitions of the pseudo F-resolvent operator and of the F-resolvent operators in the unbounded case.

Definition 4.4 (The F-resolvent Operators in TheUnbounded Case). Let s ∈ ρF (T )wedefine the pseudo F-resolvent operator as

Qs(T ) := (s2I − s(T + T )+ TT )−1, (14)

and the F-resolvent operator as

Fn(s, T ) := γn(sI − T )Qs(T )n+12 ,

where γn := (−1)(n−1)/22n−1

n−12

!

2.

Definition 4.5. We say that f is an s-monogenic function at ∞ if f is an s-monogenic function in a set D′(∞, r) = x ∈

Rn+1: |x| > r, for some r > 0, and limx→∞ f (x) exists and it is finite. We define f (∞) to be the value of this limit.


Definition 4.6. Let T : D(T ) → Vn be an admissible operator, assume ρF (T ) = ∅. Let U ⊂ Rn+1 be an axially symmetrics-domain that contains the F-spectrum σF (T ) of T and such that ∂(U ∩ CI) is union of a finite number of rectifiable Jordancurves for every I ∈ S. Assume that U and ∞ are contained in an open set in which f is s-monogenic.

(a) A function f is said to be locally s-monogenic on σF (T ) if there exists an open set U as above such that f is s-monogenicon U and at infinity.

(b) We will denote by SMσF (T ) the set of locally s-monogenic functions on σF (T ).

As we have pointed out in [24], the open set U associated with f ∈ SMσF (T ) need not to be connected. Moreover, as inthe classical functional calculus, U can depend on f and can be unbounded.

The following results are important since they will lead us to the definition of F-functional calculus for unboundedoperators.

Proposition 4.7. Let k ∈ ρF (T ) ∩ R = ∅ and let n be an odd number and let p = (s − k)−1. Set A := (T − kI)−1 for Tadmissible. Then we have the following relations for the pseudo F-resolvent operators Qs(T ) and Qp(A):

Qp(A) = (AA)−1Qs(T )p−2

= (k2I − k(T + T )+ TT )Qs(T )p−2

and

Qp(A)n+12 = (AA)−

n+12 Qs(T )

n+12 p−(n+1)

= (k2I − k(T + T )+ TT )n+12 Qs(T )

n+12 p−(n+1).

Proof. Observe that

p2I − p(A + A)+ AA = (p2(AA)−1− p(A + A)(AA)−1

+ I)(AA)= (p2(AA)−1

− p(A−1+ (A)−1)+ I)(AA)

where we have used the fact that (AA)−1= A−1(A)−1

= (A)−1A−1. Let k ∈ ρF (T ) ∩ R. Recalling that A := (T − kI)−1, A :=

(T − kI)−1 we obtain

(A)−1A−1= k2I − k(T + T )+ TT ,

A−1+ (A)−1

= T + T − 2kI,

and replacing these expressions in Qp(A)we get

Qp(A) := (p2I − p(A + A)+ AA)−1

= (AA)−1(p2(AA)−1− p(A−1

+ (A)−1)+ I)−1

= (k2I − k(T + T )+ TT )(p2(k2I − k(T + T )+ TT )− p(T + T − 2kI)+ I)−1.

Observe now that T + T and TT are operators that do not contain any imaginary unit ei, thus p commutes with them (seeRemark 2.1) and so

Qp(A) = (k2I − k(T + T )+ TT )× ((k2I − k(T + T )+ TT )− p−1(T + T − 2kI)+ p−2I)−1p−2.

Finally

k2 − k(T + T )+ TT − p−1(T + T − 2kI)+ p−2I = k2I − k(T + T )+ TT − p−1(T + T )+ 2kp−1I + p−2I

= TT − (p−1+ k)(T + T )+ (p−2

+ k2 + 2kp−1)I

= TT − s(T + T )+ s2I

so

Qp(A) = (k2I − k(T + T )+ TT )(Qs(T )p−2)

but since k ∈ R, sp = s(s − k)−1= pswe have that (k2I − k(T + T )+ TT ),Qs(T ) and p−2 commute among themselves so

Qp(A)n+12 = (k2I − k(T + T )+ TT )

n+12 Qs(T )

n+12 p−(n+1).

Theorem 4.8 (First Relation Between Fn(p, A) and Fn(s, T )). Let k ∈ ρF (T )∩ R = ∅, n be an odd number and let p = (s− k)−1.Let us put A := (T − kI)−1 for T admissible and suppose that p ∈ ρF (A) and p = 0. Then we have

Fn(s, T ) = −(A)n−12 A

n+12 Fn(p, A)pn. (15)


Proof. We recall that Fn(p, A) := γn(pI − A)Qp(A)n+12 and thanks to Proposition 4.7 we have

Fn(p, A) = γn(pI − A)(A)−n+12 A−

n+12 Qs(T )

n+12 p−(n+1).

Since s = p−1+ k commutes with p we have

Fn(p, A) = γn(pI − A)(A)−n+12 A−

n+12 p−1Qs(T )

n+12 p−n

and also

Fn(p, A) = γn(pI − A)(A)−n+12 A−

n+12 p−1(sI − T )−1(sI − T )Qs(T )

n+12 p−n.

Observe that Fn(s, T ) = γn(sI−T )Qs(T )n+12 and since (A)−

n+12 A−

n+12 does not depend on the imaginary units ei, it commutes

with p (see Remark 2.1) and thus with (pI − A), so we get

Fn(p, A) = (A)−n+12 A−

n+12 (pI − A)p−1(sI − T )−1Fn(s, T )p−n.

Replace now s = p−1+ k to get

Fn(p, A) = (A)−n+12 A−

n+12 (pI − A)p−1(p−1I + kI − T )−1Fn(s, T )p−n

= (A)−n+12 A−

n+12 (pI − A)p−1(p−1I − (A)−1)−1Fn(s, T )p−n

so we obtain

Fn(p, A) = −(A)−n+12 A−

n+12 (pI − A)p−1(p(pI − A)−1A)Fn(s, T )p−n

that becomes

Fn(p, A) = −(A)−n+12 A−

n+12 AFn(s, T )p−n

= −A(A)−n+12 A−

n+12 Fn(s, T )p−n

= −(A)−n−12 A−

n+12 Fn(s, T )p−n,

and so we get the statement.

Theorem 4.9 (Second Relation Between Fn(p, A) and Fn(s, T )). Let k ∈ ρF (T ) ∩ R = ∅ and let n be an odd number and letp = (s − k)−1. Recall that A := (T − kI)−1 for T admissible. Let s ∈ ρF (T ) and p = 0. Then we have

(AA)−n−12 Fn(p, A)pn+1

= γnp(s2I − s(T + T )+ TT )−n−12 − Fn(s, T ). (16)

Proof. Recall that

AA = (k2I − k(T + T )+ TT )−1: D(TT ) → Vn,

and

A + A = (T + T − 2kI)AA : D(TT ) → D(T ).

Using the relation s = p−1+ k we get

p2I − p(A + A)+ AA = p2s2I − s(T + T )+ TT

(T − kI)−1(T − kI)−1 (17)

where the right hand side of (17) is the composition of the maps (T − kI)−1(T − kI)−1: Vn → D(TT ) and of

s2I −

s(T + T )+ TT

: D(TT ) → Vn. As prescribed by the left hand side of (17), the operator p2I − p(A + A)+ AAmaps Vn intoitself. We now consider Fn(p, A) in terms of the above positions to get

Fn(p, A) = γn[(pI − (T − kI)−1)(T − kI)(T − kI)]

× (T − kI)n−12 (T − kI)

n−12

s2I − s(T + T )+ TT

−n+12p−2(n+1),

and now observe that the term [(pI − (T − kI)−1)(T − kI)(T − kI)], using s = p−1+ k, becomes

[p(k2I − k(T + T )+ TT )+ kI − T ] = p(s2I − s(T + T )+ TT )− (sI − T )


from which we obtain

Fn(p, A) = γn(T − kI)n−12 (T − kI)

n−12 [p(s2I − s(T + T )+ TT )− (sI − T )]

×

s2I − s(T + T )+ TT

−n+12p−(n+1),

which gives


= γnp(s2I − s(T + T )+ TT )−n−12 − Fn(s, T ).

Definition 4.10. Let k ∈ R and define the homeomorphismΦ : Rn+1→ Rn+1

,

p = Φ(s) = (s − k)−1, Φ(∞) = 0, Φ(k) = ∞. (18)

Definition 4.11 (The F-functional Calculus for Unbounded Operators). Let n be an odd number and let T : D(T ) → Vn be anadmissible operator with ρF (T ) ∩ R = ∅ and suppose that f ∈ SMσF (T ). Let us consider the functions

φ(p) := f (Φ−1(p)),

ψ(p) := ∆n−12

p (pn−1φ(p)),

∆p is the Laplace operator in dimension n + 1, and the operator

A := (T − kI)−1, for some k ∈ ρF (T ) ∩ R.

With the notations above we definef (T ) := (AA)−n−12 ψ(A), (19)

for those functions f such that f (k) = 0.

Remark 4.12. We point out that the function φ(p) := f (Φ−1(p)) given in Definition 4.11 is well defined since it is thecomposition of a slice monogenic function f with the function Φ−1(p) = p−1

+ k which is a monogenic function with realcoefficients. The function (pn−1φ(p)) remains slice monogenic, in its domain of definition, because it is the product of amonogenic function with real coefficients multiplied by a slice monogenic functions. See the book [1, Proposition 4.11.5].

Theorem 4.13. Let n be an odd number. If k ∈ ρF (T )∩R = ∅ andΦ, φ are as above, thenΦ(σF (T )) = σF (A) and the relationφ(p) := f (Φ−1(p)) determines a one-to-one correspondence between f ∈ MσF (T ) and φ ∈ MσF (A).

Proof. Suppose that s ∈ ρF (T ) and p = 0. Then by relation (16), that is


= γnp(s2I − s(T + T )+ TT )−n−12 − Fn(s, T )

we have that p ∈ ρF (A). Let us prove the converse. If p ∈ ρF (A) and p = 0 then the relation (15) in Theorem 4.8, that is

Fn(s, T ) = −(A)n−12 A

n+12 Fn(p, A)pn,

shows that s ∈ ρF (T ). The point p = 0 is in σS(A) since A−1= T − kI is unbounded. The fact the φ(p) := f (Φ−1(p))

determines a one-to-one correspondence between f ∈ MσF (T ) and φ ∈ MσF (A) follows from the definition.

Theorem 4.14. Let n be an odd number and let T be admissible with ρF (T ) ∩ R = ∅ and suppose that f ∈ Mσ F (T ). Let W bean open set as in Definition 4.6 and let f be a s-monogenic function such that its domain of slice monogenicity contains W. SetdsI = −dsI for I ∈ S. If f (k) = 0, then the operatorf (T ) := (AA)−

n−12 ψ(A), defined in (19), does not depend on k ∈ ρF (T )∩ R.

Moreover, we have the integral formula

f (T ) =

∂(W∩CI )

Fn(s, T )dsI f (s). (20)

Proof. The first part of the statement follows from the validity of formula (20) since the integral is independent of k.Given k ∈ ρF (T )∩ R and the setW we can assume that k ∈ W ∩ CI ,∀I ∈ S since otherwise, by the Cauchy theorem, we

can replaceW byW ′, on which f is s-monogenic, such that k ∈ W′∩ CI , without altering the value of the integral (20), this

is true because Fn(s, T ) is slice monogenic thanks to Theorem 3.11. Moreover, the integral (22) is independent of the choiceof I ∈ S, thanks to the structure of the spectrum (see Theorem 2.3) which holds also in the case of unbounded operators.

We have that

V ∩ CI := Φ−1(W ∩ CI)


is an open set that contains σF (T ) and its boundary

∂(V ∩ CI) = Φ−1(∂(W ∩ CI))

is positively oriented and consists of a finite number of continuously differentiable Jordan curves. Using the second relationbetween Fn(p, A) and Fn(s, T ), see formula (16), we have

∂(W∩CI )

(γnp(s2I − s(T + T )+ TT )−n−12 − Fn(s, T ))dsI f (s)

= (AA)−n−12

∂(V∩CI )

Fn(p, A)dpIpn−1φ(p).

Now we work on the left hand side∂(W∩CI )

(γnp(s2I − s(T + T )+ TT )−n−12 − Fn(s, T ))dsI f (s)

=

∂(W∩CI )

(γnp(s2I − s(T + T )+ TT )−n−12 dsI f (s)−

∂(W∩CI )

Fn(s, T ))dsI f (s)

=

∂(W∩CI )

γn (s − k)−1 dsI(s2I − s(T + T )+ TT )−n−12 f (s)−

∂(W∩CI )

Fn(s, T )dsI f (s)

= (k2I − k(T + T )+ TT )−n−12 f (k)−

∂(W∩CI )

Fn(s, T )dsI f (s)

where we have used the fact that

(s2I − s(T + T )+ TT )−n−12 dsI = dsI(s2I − s(T + T )+ TT )−

n−12

since T + T and TT do not depend on the imaginary units ei. The integral∂(W∩CI )

γn (s − k)−1 dsI(s2I − s(T + T )+ TT )−n−12 f (s)

can be written as∂(W∩CI )

γn S−1C (s, k) dsI(s2I − s(T + T )+ TT )−

n−12 f (s).

Since k is a real number we have that

(s − k)−1= S−1

C (s, k)

and by Theorem 3.10 the function

(s2I − s(T + T )+ TT )−n−12 f (s)

is operator valued slice monogenic. By the Cauchy formula (13) we have∂(W∩CI )

(s − k)−1 dsI(s2I − s(T + T )+ TT )−n−12 f (s) = (k2I − k(T + T )+ TT )−

n−12 f (k).

So from the second F-resolvent relation and the above considerations we have

γn(AA)−n−12 f (k)−

∂(U∩CI )

Fn(s, T )dsI f (s) = −

∂(V∩CI )

(AA)−n−12 Fn(p, A)dpIpn−1φ(p).

By the assumption f (k) = 0 and so (AA)−n−12 does not depend on p and the function pn−1φ(p) is slice monogenic, we

conclude that∂(U∩CI )

Fn(s, T )dsI f (s) = (AA)−n−12

∂(V∩CI )

Fn(p, A)dpIpn−1φ(p)

= (AA)−n−12 ψ(A).

So we get the statement.


5. Some concluding comments

In this section we point out some facts that appeared during the investigation that led the definition of the F-functionalcalculus for n-tuples of unbounded commuting operators. We also discuss the differences and the analogies with respect tothe SC -functional calculus for unbounded operators. We quickly recall the SC -functional calculus for unbounded operators,for more details see [19], which is a slice monogenic functional calculus based on the F-spectrum.

Definition 5.1. Let T : D(T ) → Vn be an admissible operator with ρF (T ) ∩ R = ∅ and suppose that f ∈ SMσF (T ). Let usconsider

φ(p) := f (Φ−1(p))

and the operator A := (T − kI)−1, for some k ∈ ρS(T ) ∩ R. We define

f (T ) = φ(A). (21)

The fact that the SC -functional calculus for n-tuples of commuting unbounded operators is well defined is based on thefact that f (T ) = φ((T − kI)−1) does not depend on k thanks to the following theorem.

Theorem 5.2. Let T : D(T ) → Vn be an admissible operator with ρF (T ) ∩ R = ∅ and suppose that f ∈ MσF (T ). Then theoperator f (T ) defined in (21) is independent of k ∈ ρF (T ) ∩ R. Let U be as in Definition 4.6. Set dsI = ds/I for I ∈ S, then wehave

f (T ) = f (∞)I +12π

∂(U∩CI )

S−1C (s, T )dsI f (s), (22)

where

S−1C (s, T ) := (sI − T )Qs(T ) (23)

is the SC -resolvent operator.

To prove Theorem 5.2 it is of crucial importance the relation between the resolvents S−1C (s, T ) and S−1

C (p, A) given by thefollowing result, see [19]:

Theorem 5.3. If k ∈ ρF (T ) ∩ R = ∅ and Φ, φ are as above, then Φ(σF (T )) = σF (A) and the relation φ(p) := f (Φ−1(p))determines a one-to-one correspondence between f ∈ MσF (T ) and φ ∈ MσF (A). Moreover the following identity holds

S−1C (s, T ) = pI − S−1

C (p, A)p2. (24)

We are now in the position to conclude with the following comments.

(I) In [13] it is shown that∆hxS

−1C (p, x) is a right slice monogenic function in p for every h, while it is a monogenic function

in x only for h = (n − 1)/2 where n is an odd number. It is also immediate to observe that when h = 0 we reobtainthe function S−1

C (p, x). In this case, we have the relation (24) of the SC -functional calculus, indeed for p ∈ ρF (A) theSC -resolvent operators are given by:

S−1C (p, A) := (pI − A)(p2I − p(A + A)+ AA)−1,

and since, by (5), F1(p, A) = S−1C (p, A) and γ1 = 1, we have

S−1C (p, A)p2 = pI − S−1

C (s, T ).

In the case n = 3 (which can be related to the quaternionic case, see [13]) we have, since γ3 = −4:

(AA)−1F3(p, A)p4 = −4p(s2I − s(T + T )+ TT )−1− F3(s, T ).

(II) Observe that in the definition of the SC -functional calculus for unbounded operators we have used the transformationp = Φ(s) in formula (18), the composition φ(p) := f (Φ−1(p)) and the bounded operator A := (T − kI)−1, for somek ∈ ρF (T ) ∩ R. Then the definition of the SC -functional calculus in (21) is inspired by the analogue definition of theRiesz–Dunford functional calculus for unbounded operators, see [17]. In the definition of the F-functional calculus forunbounded operators there is a substantial difference because the SC -functional calculus is defined by a Cauchy formula,while the F-functional calculus is defined by the Fueter mapping in integral form which is an integral transform thatmaps slicemonogenic functions intomonogenic functions. This integral transform is obtained applying suitable powersof the Laplace operator to the Cauchy kernel in the Cauchy formula for slice monogenic functions. So natural attemptsfor the definition would be:


(a) For f = f (x), set φ(p) := f (Φ−1(p)) and define

f1(T ) := (∆n−12

x f )(φ(A));

(b) or to define

f2(T ) := (∆n−12

p φ)(A).

But it seems that for both the operatorsfj(T ), j = 1, 2 it is not possible to show a relation between the F-resolventoperators of T and ofA analogous to (24). Formula (24)makes possible to prove (22)which says that for the SC -functionalcalculus f (T ) := φ(A) is independent on k and that φ(A) can be written in terms of the SC -resolvent of T .

(III) The reason for which we have defined the F-functional calculus as in (19), that isf (T ) := (AA)−n−12 ψ(A) is essentially

due to relation (16) in Theorem 4.9 namely


= γnp(s2I − s(T + T )+ TT )−n−12 − Fn(s, T ).

Thanks to this relation we can prove thatf (T ) is independent of k and admits the integral representation (20).(IV) We observe that in the proof of Theorem 4.14 we have used the F-functional calculus for bounded operators, but it

is necessary also to use the Cauchy formula for vector valued functions, see Theorem 3.9, because in the relation (16)there is the term

γnp(s2I − s(T + T )+ TT )−n−12 .

While in the proof of Theorem 5.2 we only use the SC -functional calculus for bounded operators, and not the Cauchyformula for vector valued functions, since the operator

(s2I − s(T + T )+ TT )−n−12

is replaced by the identity operator.(V) The SC -functional calculus is the commutative version of the so called S-functional calculus which is a slice monogenic

functional calculus and is defined for non necessarily commuting operators (bounded but also for unbounded) and isbased on the notion of S-spectrum, see the book [1] or the original papers [24,25]. In the case the operators commute,the S-spectrum turns out to be equal to the F-spectrum. The advantage of the F-spectrum is that, in the concrete cases,it is easier to compute than the S-spectrum, because it takes into account the commutativity of the operators.

Acknowledgment

The authors are grateful to the referee for the useful comments on the manuscript.

References

[1] F. Colombo, I. Sabadini, D.C. Struppa, Noncommutative Functional Calculus. Theory andApplications of Slice Hyperholomorphic Functions, in: Progressin Mathematics, Birkhäuser, Basel, 2011.

[2] D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory, AmericanMathematical Society, Providence, RI, 2001, Translated fromthe 1998 French original by Stephen S. Wilson, Panoramas et Synthèses.

[3] D. Alpay, A. Dijksma, J. Rovnyak, H. de Snoo, Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces, in: Operator Theory:Advances and Applications, vol. 96, Birkhäuser Verlag, Basel, 1997.

[4] D. Alpay, F. Colombo, I. Sabadini, Schur functions and their realizations in the slice hyperholomorphic setting, Integral Equations Operator Theory 72(2012) 253–289.

[5] D. Alpay, F. Colombo, I. Sabadini, Pontryagin De Branges Rovnyak spaces of slice hyperholomorphic functions, J. Anal. Math. 121 (2013) 87–125.[6] D. Alpay, F. Colombo, I. Sabadini, Krein-Langer factorization and related topics in the slice hyperholomorphic setting, J. Geom. Anal. 24 (2014) 843–872.[7] F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, in: Research Notes in Mathematics, vol. 76, Pitman, Advanced Publishing Program, Boston, MA,

1982.[8] F. Colombo, I. Sabadini, F. Sommen, D.C. Struppa, Analysis of Dirac Systems and Computational Algebra, in: Progress in Mathematical Physics, vol. 39,

Birkhäuser, Boston, 2004.[9] R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 4 (1932) 9–20.

[10] K.I. Kou, T. Qian, F. Sommen, Generalizations of Fueter’s theorem, Methods Appl. Anal. 9 (2002) 273–290.[11] T. Qian, Generalization of Fueter’s result to Rn+1 , Rend. Mat. Acc. Lincei 8 (1997) 111–117.[12] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Acc. Lincei Rend. Fisica 23 (1957) 220–225.[13] F. Colombo, I. Sabadini, F. Sommen, The Fueter mapping theorem in integral form and the F -functional calculus, Math. Methods Appl. Sci. 33 (2010)

2050–2066.[14] B. Jefferies, Spectral Properties of Noncommuting Operators, in: Lecture Notes in Mathematics, vol. 1843, Springer-Verlag, Berlin, 2004.[15] F. Colombo, I. Sabadini, D.C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009) 385–403.[16] F. Colombo, I. Sabadini, D.C. Struppa, Extension properties for slice monogenic functions, Israel J. Math. 177 (2010) 369–389.[17] N. Dunford, J. Schwartz, Linear Operators, Part I: General Theory, J. Wiley and Sons, 1988.[18] W. Rudin, Functional Analysis, in: McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, Düsseldorf, Johannesburg, 1973.[19] F. Colombo, I. Sabadini, The F -spectrum and the SC-functional calculus, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012) 479–500.[20] D. Alpay, F. Colombo, I. Sabadini, On some notions of convergence for n-tuples of operators, Math. Methods Appl. Sci. (2014)

http://dx.doi.org/10.1002/mma.2982. in press.[21] D. Alpay, F. Colombo, I. Lewkowicz, I. Sabadini, Realizations of slice hyperholomorphic generalized contractive and positive functions, preprint 2013.

http://refhub.elsevier.com/S0393-0440(14)00205-8/sbref1



















http://dx.doi.org/doi:10.1002/mma.2982


[22] D. Alpay, F. Colombo, I. Sabadini, Inner product spaces and Krein spaces in the quaternionic setting, in: Recent Advances in Inverse Scattering,Schur Analysis and Stochastic Processes. A Collection of Papers Dedicated to Lev Sakhnovich, in: Operator Theory Advances and Applications. LinearOperators and Linear Systems, 2014.

[23] M. Reed, B. Simon, Methods of Modern Mathematical Physics. I, Academic Press Inc., Harcourt Brace Jovanovich Publishers, New York, 1980.[24] F. Colombo, I. Sabadini, The Cauchy formula with s-monogenic kernel and a functional calculus for noncommuting operators, J. Math. Anal. Appl. 373

(2011) 655–679.[25] F. Colombo, I. Sabadini, D.C. Struppa, A new functional calculus for noncommuting operators, J. Funct. Anal. 254 (2008) 2255–2274.





Documents

The F-functional calculus for unbounded operators