# Generalization of $e^{t A}$ to $e^{t^{\alpha}A}$

I have already ask this question in here without any response. How to express $(S_{\alpha}(t))_{t \geq 0}$ where $S_{\alpha}(t)=e^{t^{\alpha} A}$ as a "one parameter semigroup" as it was done for the operator $S(t)=e^{t A}$ for $\alpha=1$?

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It isn't a one-parameter semigroup for $\alpha \neq 1$. The product law fails. –  Nik Weaver May 2 at 3:18
@Nik Weaver. But I want to know which nice properties can be satisfied by this operator. –  Jonas May 2 at 6:44
It is not clear for me what your question is. You consider the same family of operators, but change the time flow. Hence the product law fails, but most properties (unitary, norm continuous, stability, etc) are inherited. –  András Bátkai May 2 at 7:22
I know that the question is ambiguous. I want to know if it is possible to express that operator with an appropriate algebraic structure. As we has done in the case \alpha=1 with one-parameter semigroup. –  Jonas May 2 at 12:48