On Some Properties of the Quaternionic Functional Calculus

By Fabrizio Colombo, Irene Sabadini. 2009

94 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/dfd6bf32cbf31d138cd19386dd4ef6bd35cd212f

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A Cauchy kernel for slice regular functions (2009, background): “In the quaternionic case, the functional calculus is associated to quaternionic operators (see for example [4,7,2]), while in the Clifford algebras case it deals with n-tuples of non-commuting operators (see [1,5]).”
Extension results for slice regular functions of a quaternionic variable (2009, background): “…of a regular function at a point q belonging to an axially symmetric s-domain is related to the values that the function assumes at two conjugate points on a same plane LJ and belonging to the 2-sphere associated to q (see Theorem 2.26 in [1], as well as [5]): Theorem 3.1 (Representation Formula).”
Entire slice regular functions (2015, background): “The S-functional calculus, see [9, 45, 48, 49, 59, 60, 61, 71], is the analogue of the Riesz-Dunford functional calculus in the quaternionic setting.”
Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting (2011, background): “For some results on the quaternionic functional calculus we refer the reader to [34, 37, 39, 41].”
A nonconstant coefficients differential operator associated to slice monogenic functions (2012, background): “Slice monogenic functions are defined on paravectors in the Euclidean space and have values in a Clifford algebra, but there exists a parallel theory for functions of a quaternionic variable (see [14]) associated to a quaternionic functional calculus (see [3]).”
The Fueter mapping theorem in integral form and the ℱ‐functional calculus (2010, methodology): “In our treatment we will consider not only the case of functions of several operators but also the quaternionic version of our results, see [11], based on the so-called slice regular functions.”
The $H^\infty$ functional calculus based on the $S$-spectrum for quaternionic operators and for $n$-tuples of noncommuting operators (2015, unknown): “which is the quaternionic version of the Riesz-Dunford functional calculus, see [6, 12, 13, 20].”
The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum (2014, unknown): “To show the deep difference between complex spectral theory and the quaternionic spectral theory, we recall the quaternionic version of the Riesz-Dunford functional calculus, which suggests the notion of S-spectrum, see [15, 16].”
The inverse Fueter mapping theorem (2011, background): “Its main importance is that it allows a natural definition of a functional calculus for n-tuples of non necessarily commuting operators, see [6] and [12] and the book [14] in which it is also developed the theory of the quaternionic functional calculus see also [7].”
Regular Functions of a Quaternionic Variable (2013, background): “Moreover, we want to mention that the theory or regular quaternionic functions presented in this thesis and its generalizations to other hypercomplex algebras have been applied to construct new theories of functional calculus in non-commutative settings in [6, 7, 9, 8, 10, 13].”
A New Resolvent Equation for the $$S$$S-Functional Calculus (2013, unknown): “The calculus admits a quaternionic version, which works for quaternionic linear operators and is based on slice hyperholomorphic (or slice regular) functions defined on subsets of the real algebra of quaternions H with values in the quaternions, see [11, 13].”
Power series and analyticity over the quaternions (2009, background): “Hence the theory of regular quaternionic functions, which is quickly developing, reveals a new feature that reminds the nice properties of holomorphic complex functions.”
Pontryagin-de Branges-Rovnyak spaces of slice hyperholomorphic functions (2012, unknown): “[16, 18, 21]; while slice monogenic functions admit a functional calculus for n-tuples of operators, see [24, 20, 22].”