An introduction to hyperholomorphic spectral theories and fractional powers of vector operators
aa r X i v : . [ m a t h . SP ] N ov AN INTRODUCTION TO HYPERHOLOMORPHIC SPECTRAL THEORIESAND FRACTIONAL POWERS OF VECTOR OPERATORS
FABRIZIO COLOMBO, JONATHAN GANTNER, AND STEFANO PINTON
Abstract.
The aim of this paper is to give an overview of the spectral theories associated with thenotions of holomorphicity in dimension greater than one. A first natural extension is the theory ofseveral complex variables whose Cauchy formula is used to define the holomorphic functional calcu-lus for n -tuples of operators ( A , ..., A n ). A second way is to consider hyperholomorphic functionsof quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, wehave two different notions of hyperholomorphic functions that are called slice hyperholomorphicfunctions and monogenic functions. Slice hyperholomorphic functions generate the spectral theorybased on the S -spectrum while monogenic functions induce the spectral theory based on the mono-genic spectrum. There is also an interesting relation between the two hyperholomorphic spectraltheories via the F -functional calculus. The two hyperholomorphic spectral theories have differentand complementary applications. Here we also discuss how to define the fractional Fourier’s lawfor nonhomogeneous materials, such definition is based on the spectral theory on the S -spectrum. AMS Classification 47A10, 47A60.Keywords: Spectral theory, S -spectrum, monogenic spectrum, hyperholomorphic spectral theories,fractional powers of vector operators. 1. Introduction
The problem to define functions of an operator A or of an n -tuple of operators ( A , ..., A n ) is veryimportant both in mathematics and in physics and has been investigated with different methodsstarting from the beginning of the last century. The spectral theorem is one of the most importanttools to define functions of normal operators on a Hilbert space and it is of crucial importance inquantum mechanic as well as the Weyl functional calculus.The theory of holomorphic functions plays a central role in operator theory. In fact, the Cauchyformula allows to define the holomorphic functional calculus (often called Riesz-Dunford functionalcalculus) in Banach spaces [14], and this calculus can be extended to unbounded operators. Forsectorial operators the H ∞ -functional calculus, introduced by A. McIntosh in [67], turned out tobe the most important extension.Holomorphic functions of one complex variable f : Ω ⊆ C → C (denoted by O (Ω)) have thefollowing extensions:(E1) Systems of Cauchy-Riemann equations, for functions f : Π ⊆ C n → C , give the theory ofseveral complex variables.(E2) Holomorphicity of vector fields is connected with quaternionic-valued functions or more ingeneral with Clifford algebra-valued functions. There are two different extensions that areobtain by the Fueter-Sce-Qian theorem, also called the Fueter-Sce-Qian construction, andgives two different notions of hyperholomorphic functions, see for more details [30].Consider functions defined on an open set U in the quaternions H or in R n +1 for Clifford algebra-valued functions, then the Fueter-Sce-Qian extension consists of two steps. tep (I) gives the class of slice hyperholomorphic functions (denoted by SH ( U )), thesefunctions are also called slice monogenic for Clifford algebra-valued functions and sliceregular in the quaternionic case.Step (II) gives the monogenic functions (denoted by M ( U )) and Fueter regular functionsin the case of the quaternions.Both classes of hyperholomorphic functions have a Cauchy formula that can be used to definefunctions of quaternionic operators or of n -tuples of operators that do not necessarily commute.(S) The Cauchy formula of slice hyperholomorphic functions generates the S -functional calculusfor quaternionic linear operators or for n -tuples of not necessarily commuting operators, thiscalculus is based on the the notion of S -spectrum. The spectral theorem for quaternionicoperators is also based on the S -spectrum.(M) The Cauchy formula of monogenic functions generates the monogenic functional calculusthat is based on the monogenic spectrum.The hyperholomorphic functional calculi coincide with the Riesz-Dunford functional calculus whenthey are applied to a single complex operator.If we denote by the symbol F SQ the Fueter-Sce-Qian construction then the following diagramillustrates the possible extensions: O (Ω) F SQ − construction −−−−−−−−−−−−→ Hyperholomorphic f unctions y y
Several complex variables S − spectrum and monogenic spectrum y y T aylor joint spectrum Hyperholomorphic spectral theories ( HST ) y y Complex spectral theory Connections between ( HST )The first mathematicians who understood the importance of hypercomplex analysis to definefunctions of noncommuting operators on Banach spaces were A. McIntosh and his collaborators,staring from preliminary results in [70]. Using the theory of monogenic functions they developed themonogenic functional calculus and several of its applications, see [69]. The S -functional calculus,and in general the spectral theory on the S -spectrum, started its development only in 2006 whenF. Colombo and I. Sabadini discovered the S -spectrum. The discovery of the S -spectrum and ofthe S -functional calculus is well explained in the introduction of the book [24] with a complete listof the references and it is also described how hypercomplex analysis methods were used to identifythe appropriate notion of quaternionic spectrum whose existence was suggested by quaternionicquantum mechanics.If we denote by B ( V ) the Banach space of all bounded right linear operators acting on a twosided quaternionic Banach space V then the appropriate notion quaternionic spectrum, that iscalled the S -spectrum, is defined in a very counterintuitive way because it involves the square ofthe quaternionic linear operator T and it is define as: σ S ( T ) = { s ∈ H | T − s T + | s | I is not invertible in B ( V ) } . The S -spectrum for quaternionic operators can be naturally defined also for paravector operatorswhen we work in a Clifford algebra, see [43] and the book [46]. he problem of the definition of the quaternionic spectrum for the quaternionic spectral theoremhas been an open problem for long time even though several attempts have been done by severalauthors in the past decades, see e.g. [81, 82], however the correct definition of spectrum was notspecified. Finally in [2] the spectral theorem on the S -spectrum was proved for both bounded andunbounded normal operators on a quaternionic Hilbert space.The main problems with the quaternionic notion of spectrum can be easily described with thefollowing considerations related to bounded linear operators just for the sake of simplicity. Let T : V → V be a right linear bounded quaternionic operator acting on a two sided quaternionicBanach space V . If we readapt the notion of spectrum for a complex linear operator to thequaternionic setting we obtain two different notions of spectra because of the noncommutativity ofthe quaternions. The left spectrum σ L ( T ) of T is defined as σ L ( T ) = { s ∈ H | s I − T is not invertible in B ( V ) } , where the notation s I in B ( V ) means that ( s I )( v ) = sv . The right spectrum σ R ( T ) of T isassociated with the right eigenvalue problem, i.e., the search of those quaternions s such that thereexists a nonzero vector v ∈ V satisfying T ( v ) = vs. In both spectral problems it is unclear how to associate to the spectrum a resolvent operator withthe property of being an hyperholomorphic function operator-valued. In fact, for the left spectrum σ L ( T ) it is not clear what notion of hyperholomorphicity is associated to the map s → ( s I − T ) − ,for s ∈ H \ σ L ( T ) and for the right spectrum it is even more weird because the operator I s − T (where I s means ( I s )( v ) = vs ) is not linear, so it is not clear which operator is the candidate tobe the resolvent operator. Remark . One of the main motivations that suggested the existence of the S -spectrum is thepaper [13] by G. Birkhoff and J. von Neumann, where they showed that quantum mechanics can beformulated also on quaternionic numbers. Since that time, several papers and books treated thistopic, however it is interesting, and somewhat surprising, that an appropriate notion of spectrum forquaternionic linear operators was not present in the literature. Moreover, in quaternionic quantummechanics the right spectrum σ R ( T ) is the most useful notion of spectrum to study the boundedstates of a quantum systems. Before 2006 only in one case the quaternionic spectral theorem wasproved specifying the spectrum and it is the case of quaternionic normal matrices, see [51], wherethe right spectrum σ R ( T ) has been used.Now we recall some research directions and applications of the hyperholomorphic function the-ories and related spectral theories.The first step of FSQ-construction generates slice hyperholomorphic functions and the spectraltheory of the S -spectrum , we have: • The foundation of the quaternionic spectral theory on the S -spectrum are organized in thebooks [23, 24], and for paravector operators see [46]. • The mathematical tools for quaternionic quantum mechanics is the spectral theorem basedon the S -spectrum [2, 55]. • Quaternionic evolution operators, Phillips functional calculus, H ∞ -functional calculus, see[23]. • Quaternionic approximation [54]. • The characteristic operator functions and applications to linear system theory [5]. • Quaternionic spectral operators [56]. • Quaternionic perturbation theory and invariant subspaces [16]. Schur analysis in the slice hyperholomorphic setting [4]. • The theory of function spaces of slice hyperholomorphic functions [6]. • New classes of fractional diffusion problems based on fractional powers of quaternionic linearoperators, see the book [23] and the more recent contributions [17, 18, 19, 28, 29].In the last section of this paper we explain how to treat fractional diffusion problems using thequaternionic spectral theory on the S -spectrum and we show some of the recent results on fractionalFourier’s law for nonhomogeneous materials recently obtained. An example of problems that wecan treat is the following.We warn the reader that in this paper, with an abuse of notations, we use the symbol x for boththe coordinates of a point ( x , x , . . . , x n ) ∈ R n or for the vector part of a quaternion or for theimaginary part of a paravector in a Clifford algebra.Let Ω be a bounded or an unbounded domain in R and let τ > v the temperatureof the material contained in Ω. Let x = ( x , x , x ) ∈ Ω and consider the evolution problem ∂ t v ( x, t ) + div T ( x ) v ( x, t ) = 0 , ( x, t ) ∈ Ω × (0 , τ ] v ( x,
0) = f ( x ) , x ∈ Ω v ( x, t ) = 0 , x ∈ ∂ Ω t ∈ [0 , τ ] , (1)where f is a given datum and the heat flux for the nonhomogeneous material contained in Ω, isgiven by the vector differential operator: T ( x ) = a ( x ) ∂ x e + b ( x ) ∂ x e + c ( x ) ∂ x e , x ∈ Ω . (2)We determine the conditions on the coefficients a , b , c : Ω → R under which the operator T ( x )generates the fractional powers P α ( T ( x )) of T ( x ), for α ∈ (0 , P α ( T ( x )) of T ( x ) is defined to be the nonlocal Fourier’s law associated with T ( x ).The second step of FSQ-construction generates Fueter or monogenic functions and the spectraltheory on the monogenic spectrum . We highlight some references for the research directions in thisarea: • Monogenic spectral theory and applications [69]. Here one can also find the relations of themonogenic functional calculus with the Taylor functional calculus and the Weyl functionalcalculus see also some of the original contributions [64, 65, 66, 68, 75]. • Harmonic analysis in higher dimension, singular integrals and Fourier transform see therecent book [77]. • Algebraic Analysis of Dirac systems [42]. • The theory of spinor valued function [50]. • Boundary value problems treated with quaternionic techniques [60]. • The extension of Schur analysis in the Fueter setting and related topics [8, 9, 10]. • Dirac operator on manifolds and spectral theory [52, 61].We conclude by saying that before the recent works on slice hyperholomorphic functions, thisfunction theory was simply seen as an intermediate step in the Fueter-Sce-Qian’s construction.The literature on hyperholomorphic function theories and related spectral theories is nowadaysvery large. For the function theory of slice hyperholomorphic functions the main books are [6, 44,46, 54, 58], while for the spectral theory on the S -spectrum we mention the books [5, 23, 24, 46]. Forthe Fueter and monogenic function theory and related topics see the books [15, 42, 50, 61, 62, 69, 78].2. Spectral theory in the complex setting
In this section we discuss what is a functional calculus of a single operator on a Banach space andalso for the case of several operators. When we consider a closed linear operator A with domain ( A ) ⊂ X , where X is a Banach space, the resolvent set ρ ( A ) of A is defined as ρ ( A ) = { λ ∈ C | ( λ I − A ) − ∈ B ( X ) } where B ( X ) in the space of all bounded linear operators on X and the spectrum of A is the set σ ( A ) = C \ ρ ( A ) and for λ ∈ ρ ( A ) the map λ → ( λ I − A ) − is called the resolvent operator. Westart with the following intuitive definition of what is a functional calculus. A functional calculus for a closed linear operator A on complex Banach space X is a mathematicaltechnique that allows to construct in a meaningful way an operator f ( A ) for any function f in acertain class of functions F defined on sets that contain the spectrum σ ( A ) of A . The formulation in a meaningful way usually means that the functional calculus is compatiblewith formally plugging A into the function f , whenever this is possible. That is, whenever f ( z )can be expressed by a formula so that formally replacing z by the operator A yields an expressionthat is meaningful, then f ( A ) should correspond to this expression. Some examples should clarifythis idea:(a) For any polynomial p ( z ) = a n z n + . . . + a z + a ∈ F with a ℓ ∈ C , the operator p ( A ) shouldbe given by p ( A ) = a n A n + . . . + a A + a I .(b) For any λ ∈ ρ ( A ) with R λ ( z ) = ( λ − z ) − ∈ F , the functional calculus is compatible withthe resolvent operator at λ . That is, we have R λ ( A ) = ( λ I − A ) − .(c) For any rational function r ( z ) = p ( z ) /q ( z ) ∈ F with polynomials p ( z ) = a n z n + . . . + a z + a and q ( z ) = b n z n + . . . + b z + b with a ℓ , b ℓ ∈ C , the operator r ( A ) should be given by r ( A ) = p ( A ) q ( A ) − , where p ( A ) = a n A n + . . . + a A + a I and q ( A ) = b m A m + . . . + b A + b I .(d) If A is the infinitesimal generator of a strongly continuous group U A ( t ) , t ≥ tz ) ∈ F , then exp( tA ) = U A ( t ).Of course in the case of unbounded operators one has to pay attention to the domain of the opera-tors. Usually the class F constitutes an algebra, often even a Banach algebra, and the meaningful-ness of the functional calculus as described above follows from the compatibility of the functionalcalculus with the algebraic operation. Precisely, a functional calculus usually satisfies several (orall) of the following statements:(I) The functional calculus is an algebra homomorphism that is ( af + g )( T ) = af ( T ) + g ( T )for all f, g ∈ F and all a ∈ C .(II) For f and g ∈ F such that f g ∈ F we expect ( f g )( A ) = f ( A ) g ( A ).(III) For f ( z ) = 1, we have f ( A ) = 1( A ) = I .(VI) For the identity f ( z ) = z , we have f ( A ) = z ( A ) = A .(V) If X is a Hilbert space and both f and f ( z ) = f ( z ) belong to F , then f ( A ) = f ( A ) ∗ .(VI) If F is normed, then the functional calculus defines a continuous mapping into the space ofbounded linear operators B ( X ) on X , that is k f ( A ) k B ( X ) ≤ C k f k F .We restrict ourselves to the case F consists of functions that are at least continuous and to the casethat the topology on F is coarses than the topology of locally uniform convergence. In particular,convergence in F implies pointwise convergence.For the measurable functional calculus the main statements of this section hold true, but it needsdifferent arguments to show them.There are two main methods for defining a functional calculus, often both of them can be appliedin order to construct a specific functional calculus. Method 2.1.
One considers a subalgebra F of F that is dense in F such that f ( A ) can be definedeasily for any f ∈ F (for instance the set of polynomials or the set of rational functions in F ). If f ∈ F is arbitrary, one chooses an approximating sequence ( f n ) n ∈ N in F for f and defines f ( A ) := lim n → + ∞ f n ( A ) . ethod 2.2. If any f ∈ F admits an integral representation f ( z ) = Z K ( ξ, z ) f ( ξ ) dµ ( ξ ) , and formally replacing z by A in K ( ξ, z ) yields a meaningful operator K ( ξ, A ) , then one may define f ( A ) := Z K ( ξ, A ) f ( ξ ) dµ ( ξ ) . An example for method 2.1 is the continuous functional calculus. With method 2.2 we define forexample the Riesz-Dunford-functional calculus or the Philips functional calculus.There is also another concept behind the notion of functional calculus: the operator f ( A ) shouldbe defined by letting f act on the spectral values of A . In particular, this means that Ax = λx = ⇒ f ( A ) x = f ( λ ) x, for x ∈ X. (3)This idea is usually not to much emphasized in the complex setting when one introduces andexplains the concept of a functional calculus and we shall see here in the sequel the reason waythis happens. However, it is this relation that explains why functional calculi are the fundamentaltechniques for investigating linear operators. If it doesn’t hold, then a functional calculus does notprovide any information about the operator even though it generates functions of operators.Interestingly enough a deep difference between the theory of complex and the theory of quater-nionic linear operators (or more in general for hyperholomorphic spectral theories) is revealed hereso that in the latter, the relation (3) needs to be addressed explicitly.Let us start our considerations by justifying the importance of the relation (3). We thereforerecall the easiest result that is shown by an application of a functional calculus. Theorem 2.3.
Let A ∈ C m × m . Then A has an eigenvalue and for any polynomial p ∈ C [ n ] with p ( A ) = 0 , the set of eigenvalues of A is contained in the set of roots of p . It is obvious that even for the above, very easy, and fundamental result, the fact that thepolynomial functional calculus satisfies the relation (3) is crucial.The fact that the relation (3) trivially holds true for any known functional calculus in thecomplex setting is shown in the next two results, and this is the reason for which it is not explicitlymentioned.
Theorem 2.4.
Let
Φ : F → B ( X ) be a functional calculus for an operator A defined via method2.1. If (3) holds true for any function in F , then the functional calculus Φ is compatible with (3) .This is in particular the case if F consists of polynomials or rational functions. Theorem 2.5.
Let
Φ : F → B ( X ) be a functional calculus for an operator A defined via method2.2. If (3) holds true for K ( λ, · ) for any λ , then the functional calculus Φ is compatible with (3) . We now recalling that the spectral theorem works as a functional calculus. In the finite dimen-sional case when we pick an n × n matrix A = ( a i,j ) for i, j = 1 , . . . n of complex numbers suchthat a i,j = a j,i for all i, j = 1 , . . . , n . The spectrum σ ( A ) consists of eigenvalues of A , that is,complex numbers λ for which the equation Av = λv has a nonzero vector v ∈ C n as a solution.The hermitian matrix A has a unique decomposition as a finite sum A = n X j =1 λ j E λ j ,A where E λ j ,A is the orthogonal projection onto the eigenspace of the eigenvalue λ j . In the case ofbounded selfadjoint (or more in general normal operators) operators A acting in Hilbert space the pectral theorem is the most important tool for the complete description of such operators, in factwe have A = Z σ ( A ) λdE λ ; A with respect to a spectral measure E λ ; A associated with A . From the spectral theorem we candefine f ( A ) by f ( A ) = Z σ ( A ) f ( λ ) dE λ ; A for any continuous (but also bounded Borel measurable) function f : σ ( A ) → C . The mapping f → f ( A ) is an algebra homomorphism into the space of bounded linear operators. The spectraltheorem can be generalized to the case of n -tuple of commuting bounded selfadjoint operators( A , . . . , A n ), as f ( A , . . . , A n ) = Z σ ( A ,...,A n ) f ( λ ) dE λ ; A ,...,A n is valid for the joint spectral measure E λ ; A ,...,A n associated with A , . . . , A n . The joint spectrumof A , . . . , A n in R n is the support of E λ ; A ,...,A n and f : σ ( A , . . . , A n ) → C is any bounded Borelmeasurable function. The theorem holds more in general for unbounded normal operators but onehas to pay attention to the definitions of commutativity in this case, see the book [80].In the case we work in a Banach space the most natural way to define functions of bounded (andalso of unbounded) operators is the Riesz-Dunford functional calculus f ( A ) = 12 πi Z C ( λ I − A ) − f ( λ ) dλ which holds for all holomorphic functions f defined in a neighborhood of σ ( A ) in the complexplane. The simple closed contour C surrounds σ ( A ) and is contained in the domain of the function f . There is a natural generalization of Riesz-Dunford functional calculus for n -tuples of boundedoperators A , . . . , A n as f ( A , . . . , A n ) = 1(2 πi ) n Z C . . . Z C n ( λ I − A − . . . ( λ n I − A n ) − f ( λ , . . . , λ n ) dλ where dλ = dλ · · · dλ n and f is any holomorphic function in a neighborhood of σ ( A ) × . . . × σ ( A n )in C n . For each j = 1 , . . . , n the simple closed contour C j surrounds σ ( A j ) and C × . . . × C n iscontained in the domain of f in C n . Also when the operators A , . . . , A n do not commute witheach other, the functional calculus makes sense with any change in the operator ordering of thefunction ( λ , . . . , λ n ) ( λ I − A − . . . ( λ n I − A n ) − in C n \ ( σ ( A ) × . . . σ ( A n )) . For non commuting operators things are in general more complicated and we will not enter intothe details here.The material discussed in this section can be found is several classical books.
Some remarks in view of the hyperholomorphic spectral theories. (I) In operator theory on the S -spectrum, the statements of Theorems 2.4 and 2.5 still hold truebut just for a subclass of functions. The conditions that the functions in the dense subspace F resp.the kernel K ( λ, A ) satisfy (3) is not true in this setting. It is neither satisfied by the S -resolventoperator, nor by the F -resolvent operator, nor by slice hyperholomorphic rational functions withnon-real coefficients. In particular it is not satisfied, whenever the left-linear structure of the spacehas a prominent role.(II) In general the definitions of hyperholomorphic functional calculi ( S -functional calculus, the F -functional calculus, the monogenic functional calculus) the product rule, the composition rule or he spectral mapping theorem do not hold. One needs additional arguments to show that functionalcalculi based on the S -spectrum satisfy (3) at least for a subclass of functions namely, the class ofintrinsic functions and for these functions, the problems mentioned before do not occur.3. Spectral theories in the hyperholomorphic setting
At the beginning of the last century several authors started the study of hyperholomorphicfunctions and the most popular class of functions are nowadays called Fueter (or Cauchy-Fueter)regular functions in the case of the quaternions and monogenic functions (or functions in the kernelof the Dirac) for Clifford algebra setting. The second class of hyperholomorphic functions havebeen developed more recently, just at the beginning of this century, and different definitions arepossible even though they are not totally equivalent.In the following we will discuss mainly the implications of the Fueter-Sce-Qian construction inthe Clifford setting, the quaternionic setting is similar and we will use it for the fractional powersof vector operators in the last section of this paper.Let R n be the real Clifford algebra over n imaginary units e , . . . , e n satisfying the relations e ℓ e m + e m e ℓ = 0, ℓ = m , e ℓ = − . An element in the Clifford algebra will be denoted by P A e A x A where A = { ℓ . . . ℓ r } ∈ P{ , , . . . , n } , ℓ < . . . < ℓ r is a multi-index and e A = e ℓ e ℓ . . . e ℓ r , e ∅ = 1.A point ( x , x , . . . , x n ) ∈ R n +1 will be identified with the element x = x + x = x + P nj =1 x j e j ∈ R n called paravector and the real part x of x will also be denoted by Re( x ). The imaginary part of x is defined by Im( x ) = x e + . . . + x n e n and for the sake of simplicity we also use the notation x forIm( x ). The conjugate of x is denoted by x = x − Im( x ) and the Euclidean modulus of x is givenby | x | = x + . . . + x n . The sphere of purely imaginary paravectors with modulus 1, is defined by S = { x = e x + . . . + e n x n | x + . . . + x n = 1 } . The element I ∈ S are such that I = − I by C I . For this reason the elements of S are also called imaginary units. Given a non-realparavector x = x + Im( x ) = x + J x | Im( x ) | , J x := Im( x ) / | Im( x ) | ∈ S , we can associate to it thesphere defined by [ x ] = { x + J | Im( x ) | | J ∈ S } . The set of quaternions will be denoted by H and the above definitions adapts in this setting in anatural way. Definition 3.1.
Let U ⊆ R n +1 (or U ⊆ H ). We say that U is axially symmetric if, for every u + Iv ∈ U , all the elements u + J v for J ∈ S are contained in U .For operator theory the most appropriate definition of slice hyperholomorphic functions is theone that comes from the Fueter-Sce-Qian mapping theorem because it allows to define functionson axially symmetric open sets. Definition 3.2.
Let U ⊆ R n +1 be an axially symmetric open set and let U ⊆ R × R be such that x = u + J v ∈ U for all ( u, v ) ∈ U . We say that a function f : U → R n of the form f ( x ) = f ( u, v ) + J f ( u, v )is left slice hyperholomorphic if f , f are R n -valued differentiable functions such that f ( u, v ) = f ( u, − v ) , f ( u, v ) = − f ( u, − v ) for all ( u, v ) ∈ U and if f and f satisfy the Cauchy-Riemann system ∂ u f − ∂ v f = 0 , ∂ v f + ∂ u f = 0 . he above definition adapt naturally to the quaternionic setting. Since we will restrict just toleft slice hyperholomorphic function on U we introduce the symbol SH L ( U ) to denote them. Thesubset of intrinsic functions consist of those slice hyperholomophic functions such that f , f arereal-valued and is denoted by N ( U ). We recall that right slice hyperholomorphic functions are ofthe form f ( x ) = f ( u, v ) + f ( u, v ) J where f , f satisfy the above conditions. Definition 3.3 (Monogenic functions) . Let f : U → R n be a continuously differentiable functiondefined on an open subset U ⊆ R n +1 . We say that f is (left) monogenic on U , if Df ( x ) = 0where D is the Dirac operator defined by D = ∂ x + n X j =1 e j ∂ x j . The definition of slice hyperholomorphic functions and of monogenic functions can be seen asto two steps in the Fueter-Sce- Qian constructions to extend holomorphic functions to dimensiongreater than one for the vector-valued functions (quaternionic or Clifford valued-functions).In fact, starting from holomorphic functions, R. Fueter in 1935, see [53], showed an interestingway to generate Cauchy-Fueter regular functions. More then 20 years later in 1957 M. Sce, see[79], extended this result in a very pioneering and general way that includes Clifford algebras, seethe English translation of his works in hypercomplex analysis with commentaries collected in therecent book [30].In the original construction of R. Fueter the holomorphic functions are defined on open sets ofthe upper half complex plane. This condition can be relaxed by taking function g ( z ) = g ( u, v ) + ig ( u, v ) , z = x + iy defined in a set D ⊆ C , symmetric with respect to the real axis such that g ( u, − v ) = g ( u, v ) and g ( u, − v ) = − g ( u, v )namely if g and g are, respectively, even and odd functions in the variable v . Additionally the pair( g , g ) satisfies the Cauchy-Riemann system. The above remark holds also for M. Sce’s theoremthat we state in the following for Clifford algebras. Theorem 3.4 (Sce [79]) . Consider the Euclidean space R n +1 whose elements are identified withparavectors x = x + x . Let ˜ f ( z ) = f ( u, v ) + if ( u, v ) be a holomorphic function defined in adomain (open and connected) D in the upper-half complex plane and let Ω D = { x = x + x | ( x , | x | ) ∈ D } be the open set induced by D in R n +1 . The following map f ( x ) = T F S ( ˜ f ) := f ( x , | x | ) + x | x | f ( x , | x | ) takes the holomorphic functions ˜ f ( z ) and induces the Clifford-valued function f ( x ) . Then thefunction ˘ f ( x ) := T F S (cid:16) f ( x , | x | ) + x | x | f ( x , | x | ) (cid:17) , where T F S := ∆ n − n +1 and ∆ n +1 is the laplacian in n + 1 dimensions, is in the kernel of the Diracoperator, i.e., D ˘ f ( x ) = 0 on Ω D . he case in which the operator ∆ n − n +1 has a fractional index has been treated by T. Qian in[74]. Observe that for the Fueter’s theorem the operator T F S is equal to the laplacian ∆ in4 dimensions. Further developments can be found in [71, 72, 73] see also the survey [76]. Wecan summarize the Fueter-Sce contractions as follows. Denoting by O ( D ) the set of holomorphicfunctions on D , by N (Ω D ) the set of induced functions on Ω D (which turn out to be intrinsic slicehyperholomorphic functions) and by AM (Ω D ) the set of axially monogenic functions on Ω D theFueter-Sce construction can be visualized by the diagram: O ( D ) T F S −−−−→ N (Ω D ) T F S =∆ ( n − / −−−−−−−−−−−−→ AM (Ω D ) , where T F S denotes the first linear operator of the Fueter-Sce construction and T F S the secondone. The Fueter-Sce mapping theorem induces two spectral theories according to the two classesof hyperholomorphic functions it generate.Recently also the problem of construction the inversion of the maps that appear in the Fueter-Sce-Qian extension has been treated, we mention the papers [11, 12, 39, 40, 41, 31], while a differentmethod to connect slise monogenic and monogenic functions is via the Radon and dual Radontransform, see [27]. Remark . The theory of slice hyperholomorphic functions was somewhat abandoned until 2006when G. Gentili and D. C. Struppa (inspired by C. G. Cullen [49]) introduced in [57] the notionof slice regular functions for the quaternions. Further developments of the theory of slice regularfunctions were discussed also in [25] and the above definition was extended by F. Colombo, I.Sabadini and D.C. Struppa, in [45], (see also [47, 48, 32]) to the Clifford algebra setting. Sliceregular functions as defined in [57] and their generalization to the Clifford algebra as in [45], calledslice monogenic functions, possess good properties on specific open sets that are called axiallysymmetric slice domains. When it is not necessary to distinguish between the quaternionic caseand the Clifford algebra case we call these functions slice hyperholomorphic. The extension slicehyperholomorphic functions on real alternative algebras can be found in [59].
Remark . It is also possible to define slice hyperholomorphic functions, as functions in the kernelof the first order linear differential operator (introduced in [26]) Gf = (cid:16) | x | ∂∂x + x n X j =1 x j ∂∂x j (cid:17) f = 0 , where x = x e + . . . + x n e n . While, a forth way to introduce slice hyperholomorphicity, done in1998 by G. Laville and I. Ramadanoff in the paper [63], is inspired by the Fueter-Sce-Qian mappingtheorem. They introduce the so called Holomorphic Cliffordian functions defined by the differentialequation D ∆ m f = 0 over R m +1 , where D is the Dirac operator. Observe that the definition viathe global operator G requires less regularity of the functions with respect to the definition in [63].We now recall the hyperholomorphic Cauchy formulas that are the heart of the hyperholomorphicspectral theories. It is important to remark that the hypotheses of the following Cauchy formulaare related to the Definition 3.2 of slice hyperholomorphic functions. Theorem 3.7 (Cauchy formula for slice hyperholomorphic functions) . Let U ⊆ R n +1 be an axiallysymmetric open set such that ∂ ( U ∩ C I ) is union of a finite number of continuously differentiableJordan curves, for every I ∈ S . Let f be an R n -valued slice hyperholomorphic function on an openset containing U and, for any I ∈ S , we set ds I = − Ids . Then, for every x ∈ U , we have: f ( x ) = 12 π Z ∂ ( U ∩ C I ) S − L ( s, x ) ds I f ( s ) , (4) here the slice hyperholomorphic Cauchy kernel is given by S − L ( s, x ) = − ( x − s ) x + | s | ) − ( x − s ) (5) and the value of the integral (4) depends neither on U nor on the imaginary unit I ∈ S .Remark . In the paper [57] was introduced the notion of slice regularity, besides the defini-tion, the authors treated power series centered at the origin and some consequences. Withoutany tools the Cauchy formula with slice hyperholomorphic kernel and the representation formulawere originally determine with the following elementary considerations. To determine the slicehyperholomorphic Cauchy kernel S − L ( s, q ) we observe that from the definition of slice regularityits expansion S − L ( s, q ) := ∞ X m =0 q m s − − m , | q | < | s | is true when q and s belong to the same complex plane C I , for I ∈ S . Then we ask ourself whatis the closed form of the series in the case q and s do not belong to the same complex plane C I observing that (cid:16) ∞ X m =0 q m s − − m (cid:17) s − q (cid:16) ∞ X m =0 q m s − − m (cid:17) = 1 (6)is true also when s and q do not belong to the same complex plane C I . In the quaternionic case itwas observed that the inverse S of S − L ( s, q ) is the non trivial solution of the quaternionic equation S + Sq − sS = 0 , which easily follows from (6). The unknown S was determined using the Niven’s Algorithm as itis shown in the historical Note 4.18.3 in the book [46] and it gives S ( s, q ) = ( q − s ) − s ( q − s ) − q. Taking the inverse of S ( s, q ) we have the Cauchy kernel defined in (5) for the quaternions. Thisstrategy to determine the Cauchy kernel shows that in the Clifford setting the Cauchy kernelremains the same if we consider S − L ( s, x ) := ∞ X m =0 x m s − − m , | x | < | s | where x = x + x e + ... + x n e n and s = x + s e + ... + s n e n are paravectors. Moreover, observethat a direct computation of the integral12 π Z ∂ ( U ∩ C I ) S − L ( s, x ) ds I f ( s )by computing the residues of the singularities of the kernel S − L ( s, x ) in the complex plane C I , gives:12 π Z ∂ ( U ∩ C I ) S − L ( s, x ) ds I f ( s ) = 12 h f ( u + J v ) + f ( u − J v ) i + I h J [ f ( u − J v ) − f ( u + J v )] i , choosing any J ∈ S for all x = u + Iv ∈ U . From here one can clearly see the existence of thestructure formula (or representation formula) for slice monogenic functions, see for example [32](or [33]): f ( u + Iv ) = 12 h f ( u + J v ) + f ( u − J v ) i + I h J [ f ( u − J v ) − f ( u + J v )] i . The quaternionic setting is just a particular case and from these observations started a full devel-opment of the theory of slice hyperholomorphic functions. he second ingredient for our discussion in the following is the Cauchy formula for monogenicfunctions. Theorem 3.9 (Cauchy formula for monogenic functions) . Let U ⊂ R n +1 be an open set withsmooth boundary ∂U and let η ( ω ) be the outer unit normal to ∂U and dS ( ω ) be the scalar elementof surface area on ∂U . Let f be a monogenic function on an open set that contains U then f ( x ) = Z ∂U G ω ( x ) η ( ω ) f ( ω ) dS ( ω ) for every x in U , where the monogenic Cauchy kernel is given by G ω ( x ) := 1 σ n ω − x | ω − x | n +1 , x, ω ∈ R n +1 , x = ω and σ n := 2 π n +12 / Γ (cid:16) n +12 (cid:17) is the volume of unit n-sphere in R n +1 . Before to introduce the basic fact on the hyperholomorphic spectral theories we need someimportant considerations.(I) Holomorphic functions of one complex variable and harmonic analysis are strongly connectedsince the Cauchy-Riemann operator factorizes the Laplace operator. The holomorphic functionalcalculus and the spectral theorem are based on the same notion of spectrum.(II) In order to restore the analogy with the holomorphic functional calculus and the spectraltheorem in the quaternionic setting we have to replace the classical spectrum with the S -spectrum.In, fact the S -functional calculus and the quaternionic spectral theorem are both based on the S -spectrum. The Dirac operator factorizes the Laplace operator the monogenic functional calculus,based on the monogenic spectrum, has applications in harmonic analysis and in other related fields.Let us consider a Banach space V over R with norm k · k . It is possible to endow V with anoperation of multiplication by elements of R n which gives a two-sided module over R n and by V n we indicate the two-sided Banach module over R n given by V ⊗ R n .We start with the definition of a functional calculus for ( n +1)-tuples of not necessarily commutingoperators using slice hyperholomorphic functions. So we consider the paravector operator T = T + n X j =1 e j T j , where T µ ∈ B ( V ) for µ = 0 , , ..., n , and where B ( V ) is the space of all bounded R -linear operatorsacting on V . The notion of S -spectrum follows from the Cauchy formula of slice hyperholomorphicfunctions and from some not trivial considerations on the fact that we can replace in the Cauchykernel S − L ( s, x ) the paravector x by the paravector operator T also in the case the components( T , T , ..., T n ) of T do not commute among themselves. Remark . We make a crucial observation which justifies the definition of S -spectrum and of S -resolvent operator. With the procedure of Remark 3.8 it is natural to replace the paravector x by the paravector operator T = T + T e + ... + T n e n with bounded not necessarily commutingcomponents T ℓ , ℓ = 0 , ..., n in the Cauchy kernel series. We obtain ∞ X m =0 T m s − − m = − ( T − s ) T + | s | I ) − ( T − s I ) , k T k < | s | . even though the components of T do not commute. From this relation we justify the definition ofthe S -resolvent operator and of the S -spectrum. The quaternionic setting is just a particular case.We have the following definition. efinition 3.11 ( S -spectrum) . Let T ∈ B ( V n ) be a paravector operator. We define the S -spectrum σ S ( T ) of T as: σ S ( T ) = { s ∈ R n +1 : T − s ) T + | s | I is not invertible in B ( V n ) } where I denotes the identity operator. The S -resolvent set of T is defined as ρ S ( T ) = H \ σ S ( T ) . Definition 3.12.
Let T ∈ B ( V n ) be a paravector operator and s ∈ ρ S ( T ). We define the left S -resolvent operator as S − L ( s, T ) := − ( T − s ) T + | s | I ) − ( T − s I ) . (7)A similar definition can be given for the right resolvent operator. Definition 3.13.
We denote by SH Lσ S ( T ) the set of slice hyperholomorphic functions defined onthe axially symmetric set U that contains the S -spectrum of T .A crucial result for the definition of the S -functional calculus is that integral12 π Z ∂ ( U ∩ C I ) S − L ( s, T ) ds I f ( s ) , for f ∈ SH Lσ S ( T ) (8)depends neither on U nor on the imaginary unit I ∈ S , so the S -functional calculus turns out tobe well defined. Definition 3.14 ( S -functional calculus) . Let T ∈ B ( V n ) and let U ⊂ H be as above. We set ds I = − Ids and we define the S -functional calculus as f ( T ) := 12 π Z ∂ ( U ∩ C I ) S − L ( s, T ) ds I f ( s ) , for f ∈ SH Lσ S ( T ) . (9)Observe that the definition of the S -functional calculus is very natural for non commuting op-erators in noncommutative spectral theory. The heart of the general version of the S -functionalcalculus can be found in the original papers [1, 33, 36, 37] and its commutative version [35]. Warning . In the monogenic setting the natural functional calculus is for vector operators that iswhen we set T = 0 in the paravector operator T = T + P nj =1 e j T j . The reason will be clear in thesequel, but to point out this fact we use the symbol A = ( A , . . . , A n ) or A = P nj =1 e j A j insteadof ( T , . . . , T n ) or T = P nj =1 e j T j .Using the Cauchy integral formula for monogenic functions, we establish the monogenic functionalcalculus for the n -tuple A = ( A , . . . , A n ) of bounded linear operators on a Banach space X bysubstituting the n -tuple A for the vector x ∈ R n .In the following for the monogenic functional calculus we limit ourselves to the most simple casewhen n is odd and the n -tuple A = ( A , . . . , A n ) of bounded linear operators commute amongthemselves. Such restrictions can be removed but one needs to do further considerations. Remark . If n is odd, A is a commutative n -tuple, that is, A j A k = A k A j for j, k = 1 , · · · , n ,and each operator A j has real spectrum σ ( A j ) ⊂ R for j = 1 , · · · , n , then for suitable ω ∈ R n +1 ,the expression G ω ( A ) := 1 σ n ω I − A | ω I − A | n +1 (10)makes sense as an element of B ( V n ) and it is called the monogenic resolvent. emark . For an even integer m we have | ω I − A | − m = (cid:16) (cid:16) ω I + n X j =1 ( ω j I − A j ) (cid:17) − (cid:17) m/ and ω I − A = ω I − n X j =1 ( ω j I − A j ) e j for ω = ω + P nj =1 ω j . Observe that the operator ω I + n X j =1 ( ω j I − A j ) is invertible in B ( V ) for each ω = 0. Definition 3.17 (Monogenic spectrum) . The function ω G ω ( A ) , is defined on the set R n +1 \ ( { } × γ ( A )) where γ ( A ) = { ( ω , . . . , ω n ) | n X j =1 ( ω j I − A j ) is not invertible in B ( V ) } is called the monogenic spectrum. Definition 3.18 (The monogenic functional calculus) . Let n be an odd number and let us assumethat A = ( A , . . . , A n ) is a commutative n -tuple of bounded linear operators (that is A j A k = A k A j for j, k = 1 , · · · , n ), and each operator A j has real spectrum σ ( A j ) ⊂ R for j = 1 , · · · , n . If f is amonogenic function on an open set that contains U ⊂ R n +1 with γ ( A ) ⊂ U . Then we define themonogenic functional calculus as f ( A ) = Z ∂U G ω ( A ) η ( ω ) f ( ω ) dS ( ω )where G ω ( A ) is the monogenic resolvent operator (10), η ( ω ) is the outer unit normal to ∂U and dS ( ω ) is the scalar element of surface area on ∂U .In the case m = 2 , , , ... the operator | ω I − A | − m needs to be defined in a suitable way. Thedirect formulation employs Taylor’s functional calculus, but by using the plane wave decompositionof the Cauchy kernel, the case of even n and noncommuting operators can be treated simultaneously.For an n -tuple ( A , ...., A n ) of commuting bounded linear operators on a Banach space V with realspectra, the nonempty compact subset γ ( A ) of R n coincides with Taylor’s joint spectrum definedin terms of the Koszul complex.The Cauchy formula of slice hyperholomorphic functions allows to define the notion of S -spectrum, while the Cauchy formula for monogenic functions induces the notion of monogenic pectrum, as illustrated by the diagram: SH ( U ) T F S −−−−→ M ( U ) y y Slice Cauchy F ormula M onogenic Cauchy F ormula y y S − Spectrum M onogenic Spectrum y y S − F unctional calculus M onogenic F unctional Calculus
In the above diagram we have replaced the set of intrinsic functions N by the larger set of slicehyperholomorphic functions SH . This is clearly possible because the map T F S is the Laplaceoperator or its powers.We finally recall that the quaternionic spectral theorem is based on the S -spectrum and noton the monogenic spectrum. In 2015 (and published in 2016) the quaternionic spectral theoremfor quaternionic normal operators was finally proved, see [2] (see also [3]). Later on perturbationresults of quaternionic normal operators were proved in [16]. Beyond the spectral theorem thereare more recent developments in the direction of the characteristic operator functions, see [5] andthe theory of quaternionic spectral operators was developed in [56].Finally, we wish to give an idea of the structure of the quaternionic spectral theorem. For acomplete treatment see [24]. If T ∈ B ( H ) is a bounded normal quaternionic linear operator, ona quaternionic Hilbert space H , then there exist three quaternionic linear operators A , J , B suchthat T = A + J B , where A is self-adjoint and B is positive, J is an anti self-adjoint partial isometry(called imaginary operator). Moreover, A , B and J mutually commute.There exists a unique spectral measure E I on σ S ( T ) ∩ C + I so that for any slice continuous intrinsicfunction f = f + f I we have: h f ( T ) x, y i = Z σ S ( T ) ∩ C + I f ( q ) d h E I ( q ) x, y i + Z σ S ( T ) ∩ C + I f ( q ) d h J E I ( q ) x, y i , x, y ∈ H . (11)This theorem extends to the case of unbounded operators as well and holds true for a larger classof functions that are not necessarily continuous.4. Interaction of the hyperholomorphic spectral theories
Now we formulate the Fueter-Sce-Qian theorem in integral form and we use it to define the F -functional calculus. This gives a version of the monogenic functional calculus for n -tuples ofcommuting operators but it is based on the S -spectrum instead of the monogenic spectrum. Thiscalculus was introduced in [38] and further investigated in [21, 34].It is important to recall that the monogenic functional calculus is defined for n -tuples of oper-ators A j , j = 1 , ..., n that have real spectrum considered as operators A j : V → V on the realBanach space V . The F -functional calculus has advantages and disadvantages with respect to themonogenic functional calculus. Precisely, the F -functional calculus allows to consider a much largerclass of operators because it does not require that the spectrum of the operators A j , j = 1 , ..., n has to be real. Moreover, this calculus allows to consider paravector operators and not only vectoroperators as the monogenic functional calculus imposes. n the other hand, from the hyperholomorphic functions point of view the F -functional calculusis less general with respect to the monogenic functional calculus because it works for the subset ofmonogenic function given by˘ M = { ˘ f | ˘ f ( x ) = ∆ n − f ( x ) for f ∈ SH ( U ) } . We now show how the Fueter-Sce mapping theorem provides an alternative way to define thefunctional calculus for monogenic functions. The main idea is to apply the Fueter-Sce operator T F S to the slice hyperholomorphic Cauchy kernel as illustrated by the diagram: SH ( U ) AM ( U ) y Slice Cauchy F ormula T F S −−−−→ F ueter − Sce theorem in integral f rom y y S − F unctional calculus F − f unctional calculus This method generates an integral transform, called the Fueter-Sce mapping theorem in integralform, that allows to define the so called F -functional calculus. This calculus uses slice hyperholo-morphic functions and the commutative version of the S -spectrum and now we show how it works.We point out that the operator T F S has a kernel and one has to pay attention to this fact withthe definition of the F -functional calculus, more details are given in [24].Now observe that one can apply the powers of the Laplace operators to both sides of (4) so thatwe have ∆ h f ( x ) = 12 π Z ∂ ( U ∩ C I ) ∆ h S − L ( s, x ) ds I f ( s ) . In general, it is not easy to compute ∆ h f and when we apply ∆ h to the Cauchy kernel written inthe form (5), we do not get a simple formula. However, S − L ( s, x ) can be written in two equivalentways as follows. Proposition 4.1.
Let x , s ∈ R n +1 (or in H in the quaternionic case) be such that x − x Re( s ) + | s | = 0 . Then the following identity holds: S − L ( s, x ) = − ( x − x Re( s ) + | s | ) − ( x − s ) = ( s − ¯ x )( s − x ) s + | x | ) − . (12)If we use the second expression for the Cauchy kernel we find a very simple expression for∆ h S − L ( s, x ). In fact, we have: Theorem 4.2.
Let x , s ∈ R n +1 be such that x − x Re( s ) + | s | = 0 . Let S − L ( s, x ) = ( s − ¯ x )( s − x ) s + | x | ) − be the slice monogenic Cauchy kernel and let ∆ = P ni =0 ∂ ∂x i be the Laplace operator in the variables ( x , x , ..., x n ) . Then, for h ≥ , we have: ∆ h S − L ( s, x ) = C n,h ( s − ¯ x )( s − x ) s + | x | ) − ( h +1) , (13) where C n,h := ( − h h Y ℓ =1 (2 ℓ ) h Y ℓ =1 ( n − (2 ℓ − . he function ∆ h S − ( s, x ) is slice hyperholomorphic in s for any h ∈ N but is monogenic in x ifand only if h = ( n + 1) /
2, namely if and only if h equals the Sce’s exponent. We define the kernel F L ( s, x ) := ∆ n − S − L ( s, x ) = γ n ( s − ¯ x )( s − x ) s + | x | ) − n +12 , where γ n := ( − ( n − / ( n − / ( n − (cid:16) n − (cid:17) ! (14)which can be used to obtain the Fueter-Sce mapping theorem in integral form. Theorem 4.3.
Let n be an odd number. Let f be a slice hyperholomorphic function defined inan open set that contains U , where U is a bounded axially symmetric open set. Suppose that theboundary of U ∩ C I consists of a finite number of rectifiable Jordan curves for any I ∈ S . Then, if x ∈ U , the function ˘ f ( x ) , given by ˘ f ( x ) = ∆ n − f ( x ) is monogenic and it admits the integral representation ˘ f ( x ) = 12 π Z ∂ ( U ∩ C I ) F L ( s, x ) ds I f ( s ) , ds I = ds/I, (15) where the integral depends neither on U nor on the imaginary unit I ∈ S . In the sequel, we will consider bounded paravector operators T , with commuting components T ℓ ∈ B ( V ) for ℓ = 0 , , . . . , n . Such subset of B ( V n ) will be denoted by BC , ( V n ). The F -functionalcalculus is based on the commutative version of the S -spectrum given by σ S ( T ) = { s ∈ R n +1 : s I − ( T + T ) s + T T is not invertible in B ( V n ) } where the operator T is defined by T = T − T e − · · · − T n e n . We observe that for historical reasons the commutative version of the S -spectrum is sometimescalled F -spectrum because it is used for the F -functional calculus. So we define the F -resolventoperators. Definition 4.4 ( F -resolvent operators) . Let n be an odd number and let T ∈ BC , ( V n ). For s ∈ ρ S ( T ) we define the left F -resolvent operator by F L ( s, T ) := γ n ( sI − T )( s I − ( T + T ) s + T T ) − n +12 , (16)and the constants γ n are given in (14). Definition 4.5 (The F -functional calculus for bounded operators) . Let n be an odd number, let T = T + T e + · · · + T n e n ∈ BC , ( V n ) and set ds I = ds/I , for I ∈ S . Let SH Lσ S ( T ) and U be asin Definition 3.13. We define ˘ f ( T ) := 12 π Z ∂ ( U ∩ C I ) F L ( s, T ) ds I f ( s ) . (17)The definition of the F -functional calculus is well posed since the integrals in (17) depends neitheron U and nor on the imaginary unit I ∈ S .We conclude this section with some considerations on the hyperholomorphic functional calculito show the difference with respect the the complex case.(I) The product rule holds for the S -functional calculus but just in the case one of the twofunctions is intrinsic function. For the monogenic functional calculus the product rule does nothold. This is due to the fact the that product of two monogenic functions is not monogenic. Forthe F - functional calculus the product rule does not hold. II) Regarding the compatibility with polynomials we have: that the S -functional calculus andthe monogenic functional calculus are compatible with slice hyperholomorphic polynomials andwith monogenic polynomials, respectively. For the F -functional calculus the compatibility withpolynomials holds if we consider ˘ P ( q ) = ∆ P ( q )where ˘ P ( q ) is a monogenic (or Fueter) and P is a slice monogenic polynomials q → T ⇒ ˘ P ( q ) ⇒ ˘ P ( T )(III) The spectral properties of the operator T can be deduced by the S -functional calculus andthe quaternionic spectral theorem for which T x = λx = ⇒ f ( T ) x = f ( λ ) x (18)when we use intrinsic functions.5. The S -spectrum approach to fractional diffusion problems An important extension of the S -functional calculus to unbounded sectorial operators is the H ∞ -functional calculus which is one of the ways to define functions of unbounded operators. The H ∞ -functional calculus has been used to define fractional powers of paravector operators andof quaternionic linear operators that define fractional Fourier laws for nonhomogeneous materialin the theory of heat propagation. For the original contributions on fractional powers of vectoroperators and of quaternionic operators and of the H ∞ -functional calculus based on the S -spectrumsee [7, 20, 22]. For a systematic and recent treatment of quaternionic spectral theory on the S -spectrum and the fractional diffusion problems based on techniques on the S -spectrum see the books[23, 24] published in 2019. Moreover, in the monograph [46], published 2011, one can find also thefoundations of the spectral theory on the S -spectrum for n -tuples of noncommuting operators.The theory on the fractional powers of quaternionic operators has been recently applied tophysical problems and in particular to generate the fractional Fourier law for the heat equationthat is collected in the papers [19, 28, 29, 17, 18], here we give an overview of some of our results.We denote by x := ( x , x , x ) a generic point in R (we warn the reader that the symbol x is alsoused to denote the imaginary part of a quaternion. It will be clear from the context which is themeaning of the symbol x otherwise it will be specified). Let Ω ⊂ R bounded or unbounded domain(with C boundary), the heat equation for nonhomogeneous materials with the associated initial-boundary conditions descibes the evolution of the heat. Precisely, we determine v : Ω × (0 , τ ] → R (for τ >
0) such that ∂ t v ( x, t ) + div T ( x ) v ( x, t ) = 0 , ( x, t ) ∈ Ω × (0 , τ ] v ( x,
0) = f ( x ) , x ∈ Ω v ( x, t ) = 0 , x ∈ ∂ Ω t ∈ [0 , τ ] , (19)where f is a given datum and T ( x ) = a ( x ) ∂ x a ( x ) ∂ x a ( x ) ∂ x (20)where we suppose that the coefficients a , a , a : Ω ⊂ R → R of T belong to C (Ω) and they arenot necessarily constant. We also consider the heat equation for nonhomogeneous materials with obin boundary conditions, that consists in finding v : Ω × (0 , τ ] → R (for τ >
0) such that ∂ t v ( x, t ) + div T ( x ) v ( x, t ) = 0 , ( x, t ) ∈ Ω × (0 , τ ] v ( x,
0) = f ( x ) , x ∈ Ω b ( x ) v ( x, t ) + P ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ v ( x, t ) = 0 , ( x, t ) ∈ ∂ Ω × (0 , τ ] , (21)where n = ( n , n , n ) is the outward unit normal vector to ∂ Ω, and b : ∂ Ω → R is a givencontinuous function. From the physical point of view, if we call q ( x, t ) the flux of the quantitydescribed by v ( x, t ) at the instant t , the Fourier’s law states that q ( x, t ) = T ( x )( v ( x, t )).The simpler case is when we consider Ω = R and the homogeneous diffusion problem is theconsequence of Fourier’s law q ( x, t ) = −∇ v ( x, t )and of the conservation of the energy ∂ t v ( x, t ) + div( q ( x, t )) = 0 . In this case T is reduced to the negative gradient operator T = −∇ and observing that div ◦∇ = ∆the fractional diffusion model is obtained by replacing in the heat equation the Laplace operatorby its fractional powers( − ∆) α u ( x ) := Z R u ( x ) − u ( y ) | x − y | α dV ( y ) , for α ∈ (0 , . The fractional versions of the evolution equation in R is given by ∂ t v ( x, t ) + ( − ∆) α v ( x, t ) = 0 . (22)We observe that the fractional diffusion problem modifies both the Fourier’s law and the conser-vation of the energy. Using the quaternionic functional calculus we are able to define the fractionalpowers of vector operators, such as ∇ or T , in a bounded or unbounded domain Ω of R . Denoted,just for the moment, by T α or ∇ α these fractional operators, we can define the fractional diffusionproblem (22) in the divergence form ∂ t v ( x, t ) + div T α ( x ) v ( x, t ) = 0 , ( x, t ) ∈ Ω × (0 , τ ] . (23)The boundary conditions one has to associate with the fractional evolution problem are a very deli-cate issue and will not discussed here. We just mention that the most natural boundary conditionsare v = 0 at infinity in the case Ω = R . Remark . The boundary condition that we have to assume to generate the fractional powers of T are given by a ( x ) v ( x, t ) + X ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ v ( x, t ) = 0 , ( x, t ) ∈ ∂ Ω × (0 , τ ] , (24)where a : ∂ Ω → R is a given continuous function. These Robin-like boundary condition differs fromthe boundary condition in (21) by a power two on the coefficients a ℓ ’s. This is due to the fact thatin (21) the boundary condition rise from a physical condition on the flux through the boundary,instead the boundary condition in (24) naturally comes from the definition of T α . In any case, thetwo boundary conditions are related when T has coefficients that become constant on the boundary ∂ Ω. Indeed, suppose that there exists a constant µ such that the functions a , a , a satisfy theconditions a ( x ) = a ( x ) = a ( x ) = µ for all x ∈ ∂ Ω (25)and the coefficients a and b are such that a ( x ) = µb ( x ) for all x ∈ ∂ Ω . (26) hen the relation X ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ v ( x ) + b ( x ) v ( x ) = 0is equivalent to X ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ v ( x ) + a ( x ) v ( x ) = 0when x ∈ ∂ Ω . For, using (25) and (26), we have X ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ + a ( x ) I = µ X ℓ =1 n ℓ ( x ) ∂ x ℓ + µb ( x ) I = µ (cid:16) X ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ + b ( x ) I (cid:17) . This kind of approach has several advantages. • It generates the fractional Fourier law from the Fourier law q ( x, t ) = T α ( x ) v ( x, t ) (27)using the boundary conditions of the problem and without modifying the conservation ofenergy law. • We can define the fractional heat equation for nonhomogeneous materials. • The fractional differential equation remains in the divergence form, so the definition of aweak solution in obtained in a simple way. • It turns out that the approach through the quaternionic functional calculus for defining thefractional heat equations is consistent with the classical one. Indeed, we have that for any α ∈ (0 ,
1) 2 div( ∇ α ) = ( − ∆) + α . Now we present how the quaternions can be used to describe the vector operators. Let e ℓ , for ℓ = 1 , ,
3, be an orthogonal basis for the quaternions H . We identify the vector operator T ,described in (20), with the quaternionic gradient operator with non constant coefficients a ( x ) ∂ x a ( x ) ∂ x a ( x ) ∂ x ≡ X ℓ =1 e ℓ T ℓ , (28)where the components T ℓ , ℓ = 1 , ,
3, are defined by T ℓ := a ℓ ( x ) ∂ x ℓ , x ∈ Ω. From the physicalpoint of view the operator T , defined in (28), can represent the Fourier law for nonhomogeneousmaterials, but it can represent also different physical laws. Our goal is to generate the fractionalpowers of T , that we denote with P α ( T ) for α ∈ (0 , T ℓ , for ℓ = 1 , , Remark . The notation P α ( T ), for the fractional powers of T ,is more precise with respect tothe formal notation T α is a sense that will be clear just in the following with the precise definition.The formal notation T α is used in (23) and (24) (see (27)). Definition 5.3.
The vector part of the fractional powers P α ( T ) is called the fractional Fourier lawassociated with T .Now we present the general theory of the S -spectrum to construct the fractional power of aquaternionic right linear operator. Let V be a two-sided quaternionic Banach space and K ( V ) theset of closed quaternionic right linear operators on V . The Banach space of all bounded right linear perators on V is indicated by the symbol B ( V ) and is endowed with the natural operator norm.For T ∈ K ( V ), we define the operator associated with the S -spectrum as: Q s ( T ) := T − s ) T + | s | I , for s ∈ H (29)where Q s ( T ) : dom ( T ) → V , where dom ( T ) is the domain of T . We define the S -resolvent setof T as ρ S ( T ) := { s ∈ H : Q s ( T ) is invertible and Q s ( T ) − ∈ B ( V ) } and the S -spectrum of T as σ S ( T ) := H \ ρ S ( T ) . The operator Q s ( T ) − is called the pseudo S -resolvent operator. For s ∈ ρ S ( T ), the left S -resolventoperator is defined as S − L ( s, T ) := Q s ( T ) − s − T Q s ( T ) − (30)and the right S -resolvent operator is given by S − R ( s, T ) := − ( T − I s ) Q s ( T ) − . (31)The fractional powers of T , denoted by P α ( T ), are defined as follows: for any I ∈ S , for α ∈ (0 , v ∈ dom( T ) we set P α ( T ) v := 12 π Z − I R S − L ( s, T ) ds I s α − T v, (32)or P α ( T ) v := 12 π Z − I R s α − ds I S − R ( s, T ) T v, (33)where ds j = ds/I . These formulas are a consequence of the quaternionic version of the H ∞ -functional calculus based on the S -spectrum, see the book [23] for more details. For the generation ofthe fractional powers P α ( T ) a crucial assumption on the S -resolvent operators is that, for s ∈ H \{ } with Re( s ) = 0, the estimates (cid:13)(cid:13) S − L ( s, T ) (cid:13)(cid:13) B ( V ) ≤ Θ | s | and (cid:13)(cid:13) S − R ( s, T ) (cid:13)(cid:13) B ( V ) ≤ Θ | s | , (34)hold with a constant Θ > s . It is important to observethat the conditions (34) assure that the integrals (32) and (33) are convergent and so the fractionalpowers are well defined.For the definition of the fractional powers of the operator T we can use equivalently the integralrepresentation in (32) or the one in (33). Moreover, they correspond to a modified version ofBalakrishnan’s formula that takes only spectral points with positive real part into account.We want to apply the previous theory to the case V := L (Ω , H ) and T ∈ K ( L (Ω , H )) defined asin (28) (dom( T ) ⊂ L (Ω , H ) is a densely subset). A crucial problem is to determine the conditionson the coefficients a , a , a : Ω ⊂ R → R such that (32) and (33) are convergent. This problemis splitted into two problems • the first is to find appropriate conditions for the coefficients a i ’s such that the purely imag-inary quaternions are in the S -resolvent set ρ S ( T ) (i.e. Q s ( T ) : dom( T ) → L (Ω , H ) isinvertible and bounded). This is a necessary condition, see formulas (32) and (33). Then,since in the quaternionic case the map s s α , for α ∈ (0 ,
1) is not defined for s ∈ ( −∞ , s α inorder to avoid this problem. For this reason it is of great importance to assume the condi-tion Re( s ) ≥ −∞ , • The second crucial fact is to determine the conditions on the coefficients a i ’s such that theestimate (34) for the S -resolvent operator of T holds true. oth these problems are solved by considering the following approach. According to the initialcondition of the boundary-value problems we invert the operator Q s ( T ) on the space H (Ω , H ),when we consider the Dirichlet boundary condition, and on the space H := { u ∈ H (Ω) | Z Ω u ( x ) dV ( x ) = 0 } when we consider the Robin boundary condition. The invertibility of Q s ( T ) is thus reduced to solvein a weak sense the following two partial differential equations: given F ∈ L (Ω , H ) and s ∈ H \ { } such that Re( s ) = 0 ( Q s ( T )( u ) = (cid:0) T − s T + | s | I (cid:1) u ( x ) = F ( x ) , x ∈ Ω ,u ∈ H (Ω , H ) , (35)and Q s ( T )( u ) = (cid:0) T − s T + | s | I (cid:1) u ( x ) = F ( x ) , x ∈ Ω ,u ∈ H (Ω , H ) ,b ( x ) v ( x ) + P ℓ =1 a ℓ ( x ) n ℓ ( x ) ∂ x ℓ v ( x ) = 0 , x ∈ ∂ Ω . (36)To solve in the weak sense (35) (resp. (36)) means that for any F ∈ L (Ω , H ) we have to find u F ∈ H (Ω , H ) (resp. u F ∈ H (Ω , H )) such that: for any v ∈ H (Ω , H ) (resp. v ∈ H (Ω , H )) we have h Q s ( T )( u F ) , v i = ( F, v ) L = Z Ω F v dV ( x ) , (37)where the angle-brackets means that Q s ( T ) is applied to u F in the sense of distribution. If we solve(37), we can define Q s ( T ) − ( F ) := u F In order to solve (37) we apply the Lax-Milgram Lemma to the sesquilinear form: h Q s ( T )( u F ) , v i .Thus it is crucial to prove an explicit formula for the left hand side of (37). This formula can bededuced from an arguments of integration by parts and using the Dirichlet boundary condition for(35): h Q s ( T )( u ) , v i = X ℓ =1 Z Ω a ℓ ( x ) ∂ x ℓ ( u ( x )) a ℓ ( x ) ∂ x ℓ ( v ( x )) dV ( x )+12 X ℓ =1 Z Ω ∂ x ℓ ( u ( x )) ∂ x ℓ (cid:0) a ℓ ( x ) (cid:1) v ( x ) dV ( x ) + (Vect( Q s ( T )) u, v ) L + | s | ( u, v ) L (:= b s, ( u, v )) (38)or the Robin boundary condition for (36) h Q s ( T )( u ) , v i = X ℓ =1 Z Ω a ℓ ( x ) ∂ x ℓ ( u ( x )) a ℓ ( x ) ∂ x ℓ ( v ( x )) dV ( x )+ 12 X ℓ =1 Z Ω ∂ x ℓ ( u ( x )) ∂ x ℓ (cid:0) a ℓ ( x ) (cid:1) v ( x ) dV ( x ) + (Vect( Q s ( T )) u, v ) L + Z ∂ Ω a ( x ) u ( x ) v ( x ) dS ( x ) + | s | ( u, v ) L (=: b s, ( u, v )) . (39)In [19], [28], [29], [17] and [18] we found suitable conditions for the coefficients a i ’s such that thetwo sesquilinear forms b s, and b s, are coercive and continuous when Ω is bounded or unbounded.In conclusion by the Lax-Milgram lemma we obtain the solvability of the equation (37) (i.e. the nvertibility of Q s ( t )) and the estimate (34) for the S -resolvent operator. Thus the problem of theconvergence of (32) and (33) is solved. In the next three paragraphs we summarize the conditionswe found on the coefficients of T to obtain the convergence of (32) and (33) . The fractional Fourier’s law in the problem (23) with Ω bounded. In the paper [19] it wasconsidered the commutative Fourier’s law T com , that is an operator of the form T com = a ( x ) ∂ x e + a ( x ) ∂ x e + a ( x ) ∂ x e (40)where the real operators a ( x ) ∂ x , a ( x ) ∂ x and a ( x ) ∂ x commute among themselves. It hasbeen shown that if the coefficients a ℓ : Ω → R , for ℓ = 1 , , C (Ω , R ) and if a ℓ , for ℓ = 1 , , T com by the more general Fourier’s law T ( x ) = a ( x ) ∂ x e + a ( x ) ∂ x e + a ( x ) ∂ x e (41)where now the real operators a ( x ) ∂ x , a ( x ) ∂ x e and a ( x ) ∂ x do not commute among themselves.In this case the conditions for the existence of the fractional powers are more complicated.The main result is summarized in the following theorem (see for more details Theorems 4 .
1, 4 . . Theorem 5.4.
Let Ω be a bounded C -domain in R , let T = P i =1 a i ( x ) ∂ x i e i with a i ∈ C (Ω) forany i = 1 , , and set F ( a ,a ,a ) := X i =1 e i ∂ x i ( a i ) . Let a , a , a ≥ m > , and assume that min { inf x ∈ Ω a , inf x ∈ Ω a , inf x ∈ Ω a } − (2 max { sup x ∈ Ω a , sup x ∈ Ω a , sup x ∈ Ω a } ) / C Ω k F ( a ,a ,a ) k L ∞ > and − k F ( a ,a ,a ) k L ∞ (cid:16) C max { sup x ∈ Ω (1 /a ) , sup x ∈ Ω (1 /a ) , sup x ∈ Ω (1 /a ) } (cid:17) > , (43) where C Ω in the Poincar´e constant of Ω and k F ( a ,a ,a ) k L ∞ := sup x ∈ Ω ( | ∂ x ( a ) | + | ∂ x ( a ) | + | ∂ x ( a ) | ) . Then for any α ∈ (0 , and for any v ∈ dom( T ) , the integrals (32) and (33) converge absolutely. The sesquilinear form b s, ( u, v ), defined in (38), associated with the invertibility of the operator Q s ( T ) := T − s T + | s | I , with homogeneous Dirichlet boundary conditions, has to be consideredwith care. We summarized in Remark 5.5 some considerations associated with b s, ( u, v ). Remark . We point out some fact that appear in the application of the Lax–Milgram lemmaaccording to the dimension n of Ω.(I) The quadratic form b s, ( u, v ) associated to the operator Q s ( T ) is in general degenerate on H (Ω , H ).(II) In dimension n = 3 , when Ω is a C bounded set in R and a = 0, a = 0, a = 0, it turnsout that b s, ( u, v ) is continuous and coercive under suitable conditions on the coefficients of Y × Y ,where Y := { v ∈ H (Ω , H ) : v = v = v = v } is a closed subspace of H (Ω , H ) and the S -resolvent operators satisfy suitable growth conditionswhich ensure the existence of the fractional powers. III) In dimension n = 2, when Ω is a C bounded set in R and a = 0, a = 0, it turns out that b s, ( u, v ) is continuous and coercive under suitable conditions on the coefficients of X × X , where X := { v ∈ H (Ω , H ) : v = v and v = v } is a closed subspace of H (Ω , H ) and the S -resolvent operators satisfy suitable growth conditionswhich ensure the existence of the fractional powers.(IV) If we consider the quadratic form in dimension n = 3, that is when Ω is a C bounded set in R and a = 0, a = 0, a = 0, then the quadratic form is not coercive because of a = 0. It seemsthat this case cannot be treated using Lax–Milgram Lemma, but a suitable method for degenerateequations has to be used.(V) In [28] we proved, under more restrictive hypothesis on the coefficients a i ’s, that the sesquilin-ear form b s, ( u, v ) is continuous and coercive in H (Ω , H ).(VI) From the physical point of view the case for a = 0, a = 0, a = 0, in dimension n = 3 isthe case in which the conductivity is the direction z goes to zero.(VII) The proofs for the continuity and coercivity are similar in any dimension, the estimate forthe S -resolvent operators have some differences according to the fact that we work in Y or in X . The fractional Fourier’s law in the problem (24) with Ω bounded. Regarding the initial boundaryvalue problem of Robin-type, the conditions on the coefficients a i ’s depend on two positive constantswhich appear in the following two inequalities: • for all u ∈ H (Ω , R ) the following inequality holds: k u k H / ( ∂ Ω , R ) ≤ C ∂ Ω k u k H (Ω , R ) . (44)where C ∂ Ω does not depend on u • for all u ∈ H (Ω , R ) the following inequality holds: (cid:13)(cid:13)(cid:13)(cid:13) u − | Ω | − Z Ω u ( x ) dx (cid:13)(cid:13)(cid:13)(cid:13) L (Ω , R ) ≤ C P k∇ u k L (Ω , R ) , (45)where C P does not depend on u .We obtained in [18] the following result (see Theorems 4 .
4, 5 . . Theorem 5.6.
Let Ω be a bounded domain in R with boundary ∂ Ω of class C . Assume that a ∈ C ( ∂ Ω , R ) and let T be the operator defined in (20) with coefficients a , a , a ∈ C (Ω , R ) .Define the following constants: C T := min ℓ =1 , , inf x ∈ Ω ( a ℓ ( x )) , C ′ T := X i,ℓ =1 k a ℓ ∂ x ℓ a i k ∞ , K a, Ω := C ∂ Ω k a k ∞ , where k · k ∞ denotes the sup norm and C ∂ Ω is the constant in (44) . Moreover, assume that C T − C ′ T C P − K a, Ω (cid:16) C P (cid:17) > and C T > , (46) where C P is the constant in (45). Then for any α ∈ (0 , and for any v ∈ dom( T ) , the integrals (32) and (33) converge absolutely.Remark . In [17] we treated the case of the operator T with commutative coefficients and of Ωbounded with Robin-type boundary condition. The fractional Fourier’s law in the problem (23) with Ω unbounded. Regarding the initial bound-ary value problem of Dirichlet-type for the unbounded domains, we obtained in [18] the followingresult (see Theorems 4 .
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Email address : [email protected] (JG) (was PhD student at) Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9,20133 Milano, Italy Email address : [email protected] (SP) Politecnico di Milano, Dipartimento di Matematica, Via E. Bonardi, 9, 20133 Milano, Italy Email address : [email protected]@polimi.it