Quaternionic Wiener algebras, factorization and applications. (English) Zbl 1464.47051
The author presents an extension of certain theorems from complex analysis and operator theory to the skew field of quaternions. This consists in an almost periodic extension of the quaternionic Wiener algebra, the proof of a Wiener-Lévy-type theorem for it, factorization, and the Riemann-Hilbert problem for matrix-valued discrete and continuous algebras. The applications include solvability of two classes of quaternionic functional equations as well as rational matrix functions and canonical factorization.
Reviewer: Georgi Hristov Georgiev (Shumen)
MSC:
47S05 | Quaternionic operator theory |
47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
15A66 | Clifford algebras, spinors |
13J05 | Power series rings |
15A09 | Theory of matrix inversion and generalized inverses |
30E25 | Boundary value problems in the complex plane |
30G35 | Functions of hypercomplex variables and generalized variables |
47B39 | Linear difference operators |
Keywords:
quaternions; Wiener algebras; Wiener-Hopf factorization; Riemann-Hilbert problem; difference equations; convolution equations; rational matrix functionsReferences:
[1] | Alpay, D., Colombo, F., Kimsey, D.P., Sabadini, I.: Wiener algebra for the quaternions. Mediter. J. Math. 13, 2463-2482. Birkhäuser/Springer Basel AG, Basel (2016) · Zbl 1377.46034 |
[2] | Alpay, D., Colombo, F., Sabadini, I.: Slice Hyperholomorphic Schur Analysis. Operator Theory: Advances and Applications, vol. 256. Birkhäuser/Springer Basel AG, Basel (2017) · Zbl 1366.30001 |
[3] | Bart, H., Gohberg, I., Kaashoek, M.A., Ran, A.C.M.: A State Space Approach to Canonical Factorization with Applications. Operator Theory: Advances and Applications, Vol. 200, p. 59-60. Birkhäuser/Springer Basel AG, Basel (2010) · Zbl 1203.47001 |
[4] | Clancy, K., Gohberg, I.: Factorization of Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol. 3, p. 64-71. Birkhäuser/Springer Basel AG, Basel (1981) · Zbl 0474.47023 |
[5] | Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative Functional Calculus. Theory and Applications of Slice Hyperholomorphic Functions. Progress in Mathematics, vol. 289. Birkhäuser/Springer Basel AG, Basel (2011) · Zbl 1228.47001 |
[6] | Gohberg, I.C., Fel’dman, I.A.: Convolution equations and projection methods for their solution. Translation of Mathematical Monographs, vol. 41, p. 157-167. American Mathematical Society (1974) · Zbl 0278.45008 |
[7] | Gohberg, I., Manojlovic, N., dos Santos, A.F.: Factorization and Integrable Systems. Operator Theory: Advances and Applications, Vol. 141, pp. 62-69. Birkhäuser/Springer Basel AG, Basel (2003) · Zbl 1012.00042 |
[8] | Zhang, F.: Quaternions and Matrices of Quaternions. Linear Algebra and its Applications, Vol. 251, pp. 21-57. Elsevier Science Inc., Amsterdam (1997) · Zbl 0873.15008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.