What is the proof that linear operators can be treated as variables?

Question

I understand what a linear operator is, but I don't understand why you can just treat it as a variable.

Answer 1

If the question is why can operators $f(A)$ for various functions $f$ be introduced for a linear operator $A$, this is called functional calculus for linear operators and can be done in some generality, but not to the extent of taking an arbitrary function of complex numbers and extending it to linear operators.

If the question is why is composition of linear operators analogous to multiplication of numbers, this is true only for collections of operators that all commute with each other.

Answer 2

That is just the postfix operator $$ .' $$ replaced by a prefix operator $$ D . $$ So $$ x'' + 3 x' = (x')' + 3 x' = D (D x) + 3 D x = (D + 3 ) (D x) = ((D+3) D) x = (D(D+3)) x $$ where certain algebraic properties were used, like associativity or commutativity.