is it possible to define an extension method that at the same time is an operator? I want for a fixed class add the possibility to use a known operator that actually can\'t
No, you can't have an extension method which is also an operator. Extension methods can only be declared in static classes, which can't have instances and according to the C# spec,
User-defined operator declarations always require at least one of the parameters to be of the class or struct type that contains the operator declaration. [7.3.2]
Therefore, it is impossible for an extension method to also be an overloaded operator.
Additionally, you can't override System.String since it is a sealed class.