Riccardo Ghiloni

Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well–known Fueter’s construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C 1 .

CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

We study in detail the zero set of a slice regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.

A New Approach to Slice Regularity on Real Algebras

We expose the main results of a theory of slice regular functions on a real alternative algebra A, based on a well-known Fueter construction. Our general theory includes the theory of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits us to extend the range of these function theories and to obtain new results. In particular, we show that a fundamental theorem of algebra with multiplicities holds for an ample class of polynomials with coefficients in A. We give several examples to illustrate some interesting aspects of the theory.

Spectral Properties of Compact Normal Quaternionic Operators

General, especially spectral, features of compact normal operators in quaternionic Hilbert spaces are studied and some results are established which generalize well-known properties of compact normal operators in complex Hilbert spaces. More precisely, it is proved that the norm of such an operator always coincides with the maximum of the set of absolute values of the eigenvalues (exploiting the notion of spherical eigenvalue). Moreover the structure of the spectral decomposition of a generic compact normal operator T is discussed also proving a spectral characterization theorem for compact normal operators.

Power and spherical series over real alternative *-algebras

We study two types of series over a real alternative

Semigroups over real alternative *-algebras: Generation theorems and spherical sectorial operators

^*

Construction of a Finite Element Basis of the First de Rham Cohomology Group and Numerical Solution of 3D Magnetostatic Problems

-algebra

The algebra of slice functions

A

Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMs

. The first type are series of the form