Basics of Functional Analysis with Bicomplex Scalars, and by Daniel Alpay

By Daniel Alpay

This booklet presents the principles for a rigorous idea of useful research with bicomplex scalars. It starts off with a close examine of bicomplex and hyperbolic numbers after which defines the concept of bicomplex modules. After introducing a few 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 lots of varied advancements, like the ordinary sensible research with complicated scalars and during this ebook it serves because the foundational fabric for the development and examine of a bicomplex model of the well-known Schur analysis.

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The space X = BC has both involutions, so that X 1,bar = C(j) and X 1,† = C(i). 4 Inner Products and Cartesian Decompositions s1 , t1 ⊗C(j) := s1 t1→ , 49 z 1 , w1 ⊗C(i) := z 1 w 1 , which are the usual inner products on C(j) and C(i). 1. Consider a bicomplex module X with an inner product ·, ·⊗. 14) = β1 (x, y) e + β2 (x, y) e† , where βν (x, y) = πν,i ( x, y⊗) ∗ C(i), βν (x, y) = πν,j ( x, y⊗) ∗ C(j), for ν = 1, 2. 15) Fν (x, y) := πν,j ( x, y⊗) = βν (x, y) ∗ C(j) . 20) are the usual inner products on the C(i)- and C(j)-linear spaces X e and X e† respectively.

2. Next, we deal with the linearity. Given μ = μ1 e + μ2 e† ∗ BC, then μ x, y⊗ X = (μ1 e + μ2 e† )(ex1 + e† x2 ), ey1 + e† y2 ⊗ X = e(μ1 x1 ) + e† (μ2 x2 ), ey1 + e† y2 ⊗ X = e μ1 x1 , y1 ⊗1 + e† μ2 x2 , y2 ⊗2 = eμ1 x1 , y1 ⊗1 + e† μ2 x2 , y2 ⊗2 = eμ1 + e† μ2 )(e x1 , y1 ⊗1 + e† x2 , y2 ⊗2 = μ x, y⊗ X . 3. The next step is to prove what the analog of being “Hermitian for an inner product” is in our situation. y, x⊗→X = ey1 + e† y2 , ex1 + e† x2 ⊗→X = e y1 , x1 ⊗1 + e† y2 , x2 ⊗2 → = e y1 , x1 ⊗1 + e† y2 , x2 ⊗2 = e x1 , y1 ⊗1 + e† x2 , y2 ⊗2 = x, y⊗ X .

Finally, we prove the non-degeneracy. x, x⊗ X = 0 ∀⊂ e∈x1 ∈21 + e† ∈x2 ∈22 = 0 ∀⊂ x1 = 0, x2 = 0 ∀⊂ x = 0. As already noted, the “inner product square” is a hyperbolic positive number, and this suggests the possibility of introducing a hyperbolic norm on an inner product BC-module consistent with the bicomplex inner product. Let us show that this is, indeed, possible. 8) where x, x⊗1/2 is the hyperbolic number given by x, x⊗1/2 = e x1 , x1 ⊗1 + e† x2 , x2 ⊗2 1/2 = e x1 , x1 ⊗1 1/2 1/2 + e† x2 , x2 ⊗2 = e · ∈x1 ∈1 + e† ∈x2 ∈2 ∗ D+ .

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