Proving Closure of Set of Operators w/ Property P Under Addition

[Crot]

Could you please give me a hint on how to show that a set of operators with a property P is closed under addition? In other words, how one could prove that a sum of any two operators from the set still possesses this property P. The set is assumed to be infinite.

Any references, comments, suggestions, etc. will be kindly appreciated.

 

[mfb]

It depends on the set of operators and on P.

One general concept: Take an arbitrary sum of two operators, show that the sum is an operator which satisfies P.

 

[Crot]

Thank you for your answer. But, this concept is really too general.
Are there any other techniques?

 

[mfb]

Your question is extremely general. Just one step more general would be "how do you prove a mathematical statement".

All techniques are some sort of this proof, as it is the definition of "closed".