Consider a signal 3^n. Take its Z transform, which is z/(z-3). Now i know that in real sense, Z is a delay operator. We can model a system such that Z/(Z-3) is an operator and 3^n is its output, when given a particular input x(n). You mention the ROC of the system to be |Z|>3, which is understandable in mathematical sense, because we form a binomial expression in Z , and for that expression to be valid, it must converge thus subsequently yielding |Z|>3 as the condition for that expression to make sense.

BUT! Here comes the exciting part of my doubt

does |Z|>3 makes physical sense???????

i know Z is an operator. How can an operator be a number as dictatated by ROC???? From what i know operators act on numbers. Operators are not numbers themselves. Operators are independent of numbers