By Daniel Alpay
This publication presents the rules for a rigorous concept of practical research with bicomplex scalars. It starts off with an in depth learn of bicomplex and hyperbolic numbers after which defines the suggestion 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 strengthen the speculation of linear functionals and linear operators on bicomplex modules. All of this can serve for lots of diverse advancements, like the traditional sensible research with complicated scalars and during this booklet 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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Additional info for Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis
It is easy to verify that the hyperbolic norm satisfies the properties one would expect. (I) ∈x∈D = ∈ex1 + e† x2 ∈D = 0 if and only if x1 = 0 and x2 = 0. (II) Given μ = μ1 e + μ2 e† ∗ BC, one has: ∈μx∈D = μx, μx⊗1/2 = (μμ→ )1/2 · x, x⊗1/2 = |μ1 |e + |μ2 |e† · ∈x∈D = |μ|k · ∈x∈D . 3 Bicomplex Modules with Inner Product 45 ∈x + y∈D = x + y, x + y⊗1/2 = x1 + y1 , x1 + y1 ⊗1 · e + x2 + y2 , x2 + y2 ⊗2 · e† = ∈x1 + y1 ∈21 · e + ∈x2 + y2 ∈22 · e† 1/2 1/2 (∈x1 ∈1 + ∈y1 ∈1 )2 · e + (∈x2 ∈2 + ∈y2 ∈2 )2 · e† 1/2 = ∈x∈D + ∈y∈D .
1) one can define two mappings P : X ∗ X, Q:X∗X by D. 1007/978-3-319-05110-9_3, © The Author(s) 2014 31 3 BC-Modules 32 P(x) := e x Q(x) := e† x . Since P + Q = I dX , P ≥ Q = Q ≥ P = 0, P 2 = P and Q 2 = Q, then the operators P and Q are mutually complementary projectors. 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† .
Then U is unitary if and only if its idempotent components are complex unitary matrices or, equivalently, its cartesian components satisfy t t t t U1 U 1 + U2 U 2 = U 1 U1 + U 2 U2 = I n and t t t U2 U 1 = U1 U 2 , t U 1 U2 = U 2 U1 . Proof: t t t t Since U ∗ t = U 1 − j U 2 = U 1 e + U 2 e† , then for the idempotent representation we have: t t U1 U 1 e + U2 U 2 e † = e I n + e † I n and t t U 1 U1 e + U 2 U2 e † = e I n + e † I n . This means that U1 and U2 are complex unitary matrices. Similarly for the cartesian representation we have: ⎛ t t t t U1 U 1 + U2 U 2 + j U2 U 1 − U1 U 2 = I n , ⎛ t t t t U 1 U1 + U 2 U2 + j U 1 U2 − U 2 U1 = I n , 30 2 Bicomplex Functions and Matrices which means that t t t t U1 U 1 + U2 U 2 = U 1 U1 + U 2 U2 = I n and t t U2 U 1 = U1 U 2 , The result follows.
Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis by Daniel Alpay