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Equivalence of slice semi-regular functions via Sylvester operators.
- Source :
-
Linear Algebra & its Applications . Dec2020, Vol. 607, p151-189. 39p. - Publication Year :
- 2020
-
Abstract
- The aim of this paper is to study some features of slice semi-regular functions SEM (Ω) on a circular domain Ω contained in the skew-symmetric algebra of quaternions H via the analysis of a family of linear operators built from left and right ⁎-multiplication on SEM (Ω) ; this class of operators includes the family of Sylvester-type operators S f , g. Our goal is achieved by a strategy based on a matrix interpretation of these operators as we show that SEM (Ω) can be seen as a 4-dimensional vector space on the field SEM R (Ω). We then study the rank of S f , g and describe its kernel and image when it is not invertible, finding meaningful differences in the cases when the rank is either 2 or 3. By using these results, we are able to characterize when the functions f and g are either equivalent under ⁎-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. In this way, information about the operator obtained by linear algebra techniques give as a significant application the solution of a problem in an area of function theory which had a remarkable development in the last decade (see [16]). We also provide a complete classification of idempotents and zero divisors on product domains of H. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 607
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 146118913
- Full Text :
- https://doi.org/10.1016/j.laa.2020.08.009