Revisitation of the Inheritance of Linear Combinations Of Tripotent，Idempotent and Involutory Operators

The inheritance of linear combinations of idempotent、involutory and tripotent operators are studied. Using block operator matrix methods, a universal proof of this kind of combinations is given. The sufficient and necessary conditions such that a linear combination of mutually commutative tripotent operators is tripotent are obtained. At last, this problem is revisited. It shows that this method is very powerful in dealing with linear combinations of multi-commutative potents. Some known results are generalized with a short proof.