By Daniel Alpay
This booklet presents the principles for a rigorous idea of sensible research with bicomplex scalars. It starts off with an in depth examine of bicomplex and hyperbolic numbers after which defines the thought of bicomplex modules. After introducing a couple of norms and internal items on such modules (some of which seem during this quantity for the 1st time), the authors strengthen the speculation of linear functionals and linear operators on bicomplex modules. All of this can serve for lots of various advancements, like the traditional sensible research with advanced scalars and during this e-book it serves because the foundational fabric for the development and research of a bicomplex model of the well-known Schur analysis.
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Additional resources for Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis
2 Constructing a BC-Module from Two Complex Linear Spaces Let X 1 , X 2 be C(i)-linear spaces. We want to construct a BC-module X such that, with suitable meaning of the symbols, X = eX 1 + e† X 2 . 7) This means that we have to give a precise meaning to the symbols eX 1 and e† X 2 . For this purpose, consider the C(i)-linear space e C(i) which is a C(i)-linear subspace of BC, and define the tensor products eX 1 := e C(i) ⊗C(i) X 1 and e† X 2 := e† C(i) ⊗C(i) X 2 . It is clear that both eX 1 and e† X 2 are C(i)-linear spaces.
Struppa, A. Vajiac, Bicomplex numbers and their holomorphic functions. In preparation 4. G. Krantz, Function Theory of Several Complex Variables (AMS Chelsea Publishing, Providence, Rhode Island, 1992) 5. R. Gervais Lavoie, L. Marchildon, D. Rochon, Finite-dimensional bicomplex Hilbert spaces. Adv. Appl. Clifford Algebras 21(3), 561–581 (2011) Chapter 3 BC-Modules Abstract The chapter deals with the definition of a bicomplex module and some structures induced on it by the properties of bicomplex scalars.
11. The last inequality in the statement of this corollary is equivalent to the system ⎞ ⎟ π1 − i π2 > 0, ⎠ π1 + i π2 > 0 . 2 Bicomplex Matrices 29 It is known that in the case of complex matrices every eigenvector corresponds to only one eigenvalue. This is not the case for bicomplex matrices. Specifically, a bicomplex eigenvector can correspond to an infinite family of bicomplex eigenvalues. We restrict our considerations to hyperbolic positive matrices. Let A be such a matrix, and let π = γ1 e + γ2 e† be one of its eigenvalues (in particular π is a non zero divisor in D+ ).
Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis by Daniel Alpay