Characterisation of Multiplication Operator on Bicomplex Lorentz Spaces With Hyperbolic Norm

Abstract

The multiplication operator M (u) f = u. f within the bicomplex Lorentz space L-p,L-q (BC) (Omega, R, theta), is investigated. It is initially established that M-u is D-bounded if and only if the function u is essentially D-bounded. Subsequently, it is proved that the collection of all Dbounded multiplication operators on BC-Lorentz spaces forms a maximal abelian sub-algebra within the Banach algebra of all bounded linear operators on L-p,L-q (BC) (Omega, R, theta). Additionally, a necessary and sufficient condition for the compactness of M-u is provided. Finally, by introducing a condition for a multiplication operator to exhibit a closed range, the author identifies some conditions equivalent to M-u being a Fredholm operator.