Reading the following statement :
Theorem :
An operator $A$ has the property $P$ if $\underset{\lambda \rightarrow 0}{% \lim }\left( A-\lambda \right) ^{-1}$ exists.
Does the reader understand that $\left( A-\lambda \right) ^{-1}$ is defined on a neighborhood of $0$ ?
Or the correct way to announce the theorem must be like :
Theorem :
An operator $A$ has the property $P$ if there is some neighborood $V\subset% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion $ of $0$ such that $\left( A-\lambda \right) ^{-1}$ and $\underset{\lambda \rightarrow 0}{\lim }\left( A-\lambda \right) ^{-1}$ exist.
Short announcements are more beautiful. If there are any rules to follow, can somebody tell me about ?
I think a good habit when writing down a theorem is to add the essential details about the objects involved in the statement. Other properties may be added elsewhere or not added at all if not strictly important.
In both cases, it is not clear where the operator $A$ lives. That is an essential thing to add so as to make the reader able to understand why property $P$ is meaningful for $A$ or why $(A-\lambda)^{-1}$ makes sense. I think it is clear that $(A-\lambda)^{-1}$ must be defined around $\lambda = 0$ in order to be able to take the limit.