( 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.
[ w = z_1 + z_2 \mathbfj = \alpha \cdot \mathbfe_1 + \beta \cdot \mathbfe_2 ] where [ \mathbfe_1 = \frac1 + \mathbfk2, \quad \mathbfe_2 = \frac1 - \mathbfk2 ] satisfy ( \mathbfe_1^2 = \mathbfe_1, \ \mathbfe_2^2 = \mathbfe_2, \ \mathbfe_1 \mathbfe_2 = 0, \ \mathbfe_1 + \mathbfe_2 = 1 ), and ( \alpha = z_1 - i z_2, \ \beta = z_1 + i z_2 ) are complex numbers.
The bicomplex spectrum of ( T ) is: [ \sigma_\mathbbBC(T) = \lambda \in \mathbbBC : \lambda I - T \text is not invertible . ] In idempotent form: [ \sigma_\mathbbBC(T) = \sigma_\mathbbC(T_1) \mathbfe 1 + \sigma \mathbbC(T_2) \mathbfe_2 ] where the sum is in the sense of idempotent decomposition: ( \alpha \mathbfe_1 + \beta \mathbfe_2 : \alpha \in \sigma(T_1), \beta \in \sigma(T_2) ). Basics of Functional Analysis with Bicomplex Sc...
This decomposition is the key to bicomplex analysis: it reduces bicomplex problems to two independent complex problems . In classical functional analysis, we work with vector spaces over ( \mathbbR ) or ( \mathbbC ). Over ( \mathbbBC ), a bicomplex module replaces the vector space, but caution: ( \mathbbBC ) is not a division algebra (it has zero divisors, e.g., ( \mathbfe_1 \cdot \mathbfe_2 = 0 ) but neither factor is zero). Hence, we cannot define a bicomplex-valued norm in the usual sense—the triangle inequality fails due to zero divisors.
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. ] ( T ) is bounded if there exists
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 ).
Every bicomplex number has a unique :
[ \mathbbBC = z_1 + z_2 \mathbfj \mid z_1, z_2 \in \mathbbC ]
It sounds like you’re looking for a feature article or an in-depth explanatory piece on (likely short for Bicomplex Scalars or Bicomplex Numbers ). The bicomplex spectrum of ( T ) is: