Continuous Slice Functional Calculus in Quaternionic Hilbert Spaces

@article{Ghiloni2013ContinuousSF,
  title={Continuous Slice Functional Calculus in Quaternionic Hilbert Spaces},
  author={R. Ghiloni and Valter Moretti and A. Perotti},
  journal={Reviews in Mathematical Physics},
  year={2013},
  volume={25},
  pages={1350006}
}
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… Expand
Slice Functional Calculus in Quaternionic Hilbert Spaces
We propose a continuous functional calculus in quaternionic Hilbert spaces. The class of continuous functions considered is the one of slice quaternionic functions. Slice functions generalize theExpand
Functions of the infinitesimal generator of a strongly continuous quaternionic group
The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper,Expand
Singularities of slice regular functions over real alternative ⁎-algebras
Abstract The main goal of this work is classifying the singularities of slice regular functions over a real alternative ⁎-algebra A. This function theory has been introduced in 2011 as aExpand
Fractional powers of quaternionic operators and Kato's formula using slice hyperholomorphicity
In this paper we introduce fractional powers of quaternionic operators. Their definition is based on the theory of slice-hyperholomorphic functions and on the $S$-resolvent operators of theExpand
The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum
In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of S-spectrum. The proof technique consists of first establishing a spectral theorem forExpand
Spectral characterization of quaternionic positive definite functions on the real line
This paper is concerned with the spectral characteristics of quaternionic positive definite functions on the real line. We generalize the Stone's theorem to the case of a right quaternionic linearExpand
1 2 O ct 2 01 7 SPECTRAL REPRESENTATIONS OF NORMAL OPERATORS IN QUATERNIONIC HILBERT SPACES VIA
The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann’s foundational works in the thirties. The absence of a suitable quaternionicExpand
Perturbation of normal quaternionic operators
The theory of quaternionic operators has applications in several different fields such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. TheExpand
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. ThisExpand
A Representation of Weyl-Heisenberg Lie Algebra in the Quaternionic Setting
Using a left multiplication defined on a right quaternionic Hilbert space, linear self-adjoint momentum operators on a right quaternionic Hilbert space are defined in complete analogy with theirExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 42 REFERENCES
Quaternion Quantum Mechanics: Second Quantization and Gauge Fields
Abstract Recent work on algebraic chromodynamics has indicated the importance of a systematic study of quaternion structures in quantum mechanics. A quaternionic Hilbert module, a closed linearExpand
On quaternionic functional analysis
  • C. Ng
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
  • 2007
Abstract In this paper, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion B*-algebrasExpand
Quantum mechanics of the quaternionic Hilbert spaces based upon the imprimitivity theorem
Abstract We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem.Expand
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 includedExpand
Representations of a class of real *-algebras as algebras of quaternion-valued functions
For a compact Hausdorff space X , let C{X, H) denote the set of all quaternion-valued functions on X . It is proved that if a real B* -algebra A satisfies the following conditions: (i) the spectrumExpand
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 regularExpand
Foundations of Quaternion Quantum Mechanics
A new kind of quantum mechanics using inner products, matrix elements, and coefficients assuming values that are quaternionic (and thus noncommutative) instead of complex is developed. This is theExpand
Spectral Theory and Quantum Mechanics: With an Introduction to the Algebraic Formulation
Introduction and mathematical backgrounds.- Normed and Banach spaces, examples and applications.- Hilbert spaces and bounded operators.- Families of compact operators on Hilbert spaces andExpand
A New Theory of Regular Functions of a Quaternionic Variable
Abstract In this paper we develop the fundamental elements and results of a new theory of regular functions of one quaternionic variable. The theory we describe follows a classical idea of Cullen,Expand
Slice monogenic functions
In this paper we offer a new definition of monogenicity for functions defined on ℝn+1 with values in the Clifford algebra ℝn following an idea inspired by the recent papers [6], [7]. This new classExpand
...
1
2
3
4
5
...