By Daniel Alpay

This publication presents the rules for a rigorous conception of sensible research with bicomplex scalars. It starts with an in depth research of bicomplex and hyperbolic numbers after which defines the proposal of bicomplex modules. After introducing a couple of norms and internal items on such modules (some of which look during this quantity for the 1st time), the authors improve the idea of linear functionals and linear operators on bicomplex modules. All of this can serve for plenty of various advancements, like the ordinary practical research with complicated scalars and during this booklet it serves because the foundational fabric for the development and learn of a bicomplex model of the well-known Schur analysis.

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**Extra info for Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis**

**Example text**

Consider now the cartesian product eX 1 × e† X 2 , where as usual, eX 1 is seen as eX 1 × {0} and e† X 2 is seen as {0} × e† X 2 . Since eX 1 × e† X 2 is an additive abelian group we have endowed the sum eX 1 + e† X 2 with a meaning: for any x1 √ X 1 , x2 √ X 2 ex1 + e† x2 := (ex1 , e† x2 ) √ eX 1 × e† X 2 . 7) is defined already but for the moment as a C(i)linear space only. Let us endow it with the structure of BC-module. Given λ = β1 e + β2 e† √ BC and ex1 + e† x2 √ eX 1 + e† X 2 , we set: λ(ex1 + e† x2 ) = (β1 e + β2 e† ) · (ex1 + e† x2 ) := e(β1 x1 ) + e† (β2 x2 ).

1) is called the idempotent decomposition of X , and it plays an extremely important role in what follows. In particular, it allows to realize component-wise the operations on X : if x = ex + e† x, y = ey + e† y and if λ = λ1 e + λ2 e† then x + y = (ex + ey) + (e† x + e† y), λx = λ1 xe + λ2 xe† . In what follows we will write X C(i) or X C(j) whenever X is considered as a C(i) or C(j) linear space respectively. Since X e and X e† are R-, C(i)– and C(j)–linear spaces as well as BC-modules, we have that X = X e ≤ X e† where the direct sum ≤ can be understood in the sense of R-, C(i)– or C(j)–linear spaces, as well as BC-modules.

1 (Inner product). Let X be a BC-module. A mapping · , ·⊗ : X × X → BC is said to be a BC-inner, or BC-scalar, product on X if it satisfies the following properties: 1. x, y + z⊗ = x, y⊗ + x, z⊗ for all x, y, z ∗ X ; 2. μ x, y⊗ = μ x, y⊗ for all μ ∗ BC, for all x, y ∗ X ; 3. x, y⊗ = y, x⊗→ for all x, y ∗ X ; 4. x, x⊗ ∗ D+ , and x, x⊗ = 0 if and only if x = 0. Note that property (3) implies the following: a + b j = x, x⊗ = x, x⊗→ = a − b j with a, b ∗ C(i), hence a ∗ R, b = i b1 , with b1 ∗ R, that is, x, x⊗ ∗ D.