Basics Of Functional Analysis With Bicomplex Sc... Online

A is defined as: [ |w|_\mathbfk = \sqrtw \cdot \barw = \sqrt(z_1 + z_2 \mathbfj)(\barz_1 - z_2 \mathbfj) = \sqrt z_1 \barz_1 + z_2 \barz_2 + \mathbfk (z_2 \barz_1 - z_1 \barz_2) ] which takes values in ( \mathbbR \oplus \mathbbR \mathbfk ) (the hyperbolic numbers). But careful: this is not real-valued. To get a real norm, one composes with a “hyperbolic absolute value.”

In idempotent form: ( T = T_1 \mathbfe_1 + T_2 \mathbfe_2 ), where ( T_1, T_2 ) are complex linear operators between ( X_1, Y_1 ) and ( X_2, Y_2 ). Basics of Functional Analysis with Bicomplex Sc...

with componentwise addition and multiplication. Equivalently, introduce an independent imaginary unit ( \mathbfj ) (where ( \mathbfj^2 = -1 ), commuting with ( i )), and write: A is defined as: [ |w|_\mathbfk = \sqrtw

( T ) is bounded if there exists ( M > 0 ) such that ( | T x | \leq M | x | ) for all ( x ). This is equivalent to ( T_1 ) and ( T_2 ) being bounded complex operators. with componentwise addition and multiplication

This decomposition is the of the theory: every bicomplex functional analytic result follows from applying complex functional analysis to each idempotent component. 4. Bicomplex Linear Operators Let ( X, Y ) be bicomplex Banach spaces. A map ( T: X \to Y ) is bicomplex linear if: [ T(\lambda x + \mu y) = \lambda T(x) + \mu T(y), \quad \forall \lambda, \mu \in \mathbbBC, \ x,y \in X. ]