Properties of Multiplication Operators on K\"othe spaces

We make an exhaustive study of the properties of multiplication operator $M_u$ acting on K"othe spaces. We characterize the multiplication operators, acting on K"othe spaces, which are: bounded, injective, onto, bijective, Fredholm, compact, with closed range and with range finite. We also study the spectrum, the $n$-power and calculate the essential norm of $M_u$ and we study the algebra of multiplication operators.