Fixed product preserving mappings on Banach algebras

In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the elements are zero or identity element. First show if such map preserves a finite-rank operator, then it must also product. several instances, is enough product preserving be scalar multiple of an algebra homomorphism. Second, explore more general problem concerning existence and relationship Lastly, motivated by Kaplansky's on invertibility preservers, invertible either homomorphisms antihomomorphisms multiplied left