S-functional Calculus Research Articles

The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define the functions of operators. The Fueter–Sce–Qian extension theorem is a two-step procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called S-functional calculus for noncommuting operators based on the S-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the monogenic spectrum, contains the Weyl functional calculus as a particular case. In this paper we show that the extension operator from slice hyperholomorphic functions to monogenic functions admits various possible factorizations that induce different function spaces. The integral representations in such spaces allow to define the associated functional calculi based on the S-spectrum. The function spaces and the associated functional calculi define the so-called fine structure of the spectral theories on the S-spectrum. Among the possible fine structures there are the harmonic and polyharmonic functions and the associated harmonic and polyharmonic functional calculi. The study of the fine structures depends on the dimension considered and in this paper we study in detail the case of dimension five, and we describe all of them. The five-dimensional case is of crucial importance because it allows to determine almost all the function spaces will also appear in dimension greater than five, but with different orders.

Universality property of the S-functional calculus, noncommuting matrix variables and Clifford operators

Spectral theory on the S-spectrum was born out of the need to give quaternionic quantum mechanics a precise mathematical foundation (Birkhoff and von Neumann [8] showed that a general set theoretic formulation of quantum mechanics can be realized on real, complex or quaternionic Hilbert spaces). Then it turned out that spectral theory on S-spectrum has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for n-tuples of noncommuting operators. With this paper we show that the spectral theory on the S-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. In fact, the S-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the S-functional calculus goes beyond quaternionic and Clifford analysis, indeed the S-functional calculus has a certain universality property. This fact makes the spectral theory on the S-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

Pseudo S-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε>0, the pseudo S-spectrum of T is defined asΛεS(T):=σS(T)⋃{q∈H∖σS(T):‖Qq(T)−1‖≥1ε}, where H denotes the division ring of quaternions, σS(T) is the S-spectrum of T and Qq(T)=T2−2Re(q)T+|q|2I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S-spectrum and explicitly compute the pseudo S-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε>0 the pseudo S-spectrum of the operator is the circularization of a closed disk in the complex plane centred at r with the radius ε. Further, we propose a G1-condition for quaternionic operators and prove some results in this setting.

Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators

Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defined a priori but must be chosen randomly. Spectral properties of an operator should hence be independent of this left multiplication. We show that results derived from functional calculi for intrinsic slice functions can be formulated without the assumption of a left multiplication. We develop the S-functional calculus in this setting and a new approach to spectral integration. This approach has a clear interpretation in terms of the right linear structure on the space and allows to formulate the spectral theorem without using any randomly chosen structure. Our techniques only apply to intrinsic slice functions, but only these functions are compatible with the basic intuition of a functional calculus that $f(T)$ should be defined by letting $f$ act on the spectral values of $T$. Using these tools, we develop a theory of quaternionic spectral operators. In particular, we show the existence of a canonical decomposition of such operator and discuss its behavior under the S-functional calculus. Finally, we show a relation with complex operator theory: if we embed the complex numbers into the quaternions, then complex and quaternionic operator theory are consistent. The symmetry of intrinsic slice functions guarantees that this compatibility is true for any imbedding of the complex numbers.

Quaternions and Functional Calculus

In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic C * -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions.

Fractional powers of vector operators and fractional Fourier’s law in a Hilbert space

In this paper we give a concrete application of the spectral theory based on the notion of S-spectrum to fractional diffusion process. Precisely, we consider the Fourier law for the propagation of the heat in non homogeneous materials, that is the heat flow is given by the vector operator:where , are orthogonal unit vectors in , a, b, c are given real valued functions that depend on the space variables , and possibly also on time. Using the -version of the S-functional calculus we have recently defined fractional powers of quaternionic operators, which contain, as a particular case, the vector operator T. Hence, we can define the non-local version , for , of the Fourier law defined by T. We will see in this paper how we have to interpret , when we introduce our new approach called: ‘The S-spectrum approach to fractional diffusion processes’. This new method allows us to enlarge the class of fractional diffusion and fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations and non commutative operator theory. Our theory applies not only to the heat diffusion process but also to Fick’s law and more in general it allows to compute the fractional powers of vector operators that arise in different fields of science and technology.

On power series expansions of the S-resolvent operator and the Taylor formula

The $S$-functional calculus is based on the theory of slice hyperholomorphic functions and it defines functions of $n$-tuples of not necessarily commuting operators or of quaternionic operators. This calculus relays on the notion of $S$-spectrum and of $S$-resolvent operator. Since most of the properties that hold for the Riesz-Dunford functional calculus extend to the S-functional calculus it can be considered its non commutative version. In this paper we show that the Taylor formula of the Riesz-Dunford functional calculus can be generalized to the S-functional calculus, the proof is not a trivial extension of the classical case because there are several obstructions due to the non commutativity of the setting in which we work that have to be overcome. To prove the Taylor formula we need to introduce a new series expansion of the $S$-resolvent operators associated to the sum of two $n$-tuples of operators. This result is a crucial step in the proof of our main results,but it is also of independent interest because it gives a new series expansion for the $S$-resolvent operators. This paper is devoted to researchers working in operators theory and hypercomplex analysis.

The H∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators

In this paper we extend the H∞ functional calculus to quaternionic operators and to n-tuples of noncommuting operators using the theory of slice hyperholomorphic functions and the associated functional calculus, called S-functional calculus. The S-functional calculus has two versions: one for quaternionic-valued functions and one for Clifford algebra-valued functions and can be considered the Riesz–Dunford functional calculus based on slice hyperholomorphicity, because it shares with it the most important properties. The S-functional calculus is based on the notion of S-spectrum which, in the case of quaternionic normal operators on a Hilbert space, is also the notion of spectrum that appears in the quaternionic spectral theorem. The main purpose of this paper is to construct the H∞ functional calculus based on the notion of S-spectrum for both quaternionic operators and for n-tuples of noncommuting operators. We remark that the H∞ functional calculus for (n+1)-tuples of operators applies, in particular, to the Dirac operator.

TheF-functional calculus for unbounded operators

In the recent years the theory of slice hyperholomorphic functions has become an important tool to study two functional calculi for n -tuples of operators and also for its applications to Schur analysis. In particular, using the Cauchy formula for slice hyperholomorphic functions, it is possible to give the Fueter–Sce mapping theorem an integral representation. With this integral representation it has been defined a monogenic functional calculus for n -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 containing unbounded operators and we obtain an integral representation formula analogous to the one of the Riesz–Dunford functional calculus for unbounded operators acting on a complex Banach space. As we will see, it is not an easy task to provide the correct definition of the F -functional calculus in the unbounded case. This paper is addressed to a double audience, precisely to people with interests in hypercomplex analysis and also to people working in operator theory.

On some notions of spectra for quaternionic operators and for n-tuples of operators

The S-spectrum has been introduced for the definition of the S-functional calculus that includes both the quaternionic functional calculus and a calculus for n-tuples of nonnecessarily commuting operators. The notion of right spectrum for right linear quaternionic operators has been widely used in the literature, especially in the context of quaternionic quantum mechanics. Moreover, several results in linear algebra, like the spectral theorem for quaternionic matrices, involve the right spectrum. In this Note we prove that the two notions of S-spectrum and of right spectrum coincide.

Condition numbers of large matrices, and analytic capacities

Given an operator T : X −→ X on a Banach space X, we compare the condition number of T ,C N(T )= � T �·� T −1 � , and the spectral condition number defined as SCN(T )= � T �· r(T −1 ), where r(·) stands for the spectral radius. For a set Y of operators, we put Φ(∆) = sup{CN(T ): T ∈ Y, SCN(T ) ≤ ∆} ,∆ ∈ (1, ∞), and say that Y is spectrally Φ-conditioned. As Y we consider certain sets of (n × n)- matrices or, more generally, algebraic operators with deg(T ) ≤ n that admit a specific functional calculus. In particular, the following sets are included: Hilbert (Banach) space power bounded matrices (operators), polynomially bounded matrices, Kreiss type matrices, Tadmor-Ritt type matrices, and matrices (operators) admitting a Besov class B s-functional calculus. The above function Φ is estimated in terms of the analytic capacity capA(·) related to the corresponding function class A.I n particular, for A = B s p,q, the quantity Φ(∆) is equivalent to ∆ n n s as ∆ −→ ∞ (or as n −→ ∞ )f or s> 0, and is bounded by ∆ n (log(n)) 1/q for s =0 .