Does there exists in physics an operator satisfies: $A^{-1}(t)=A(-t+ i \beta)$ , $\beta$ is a real number non-null?

Let $$t$$ be a real number such that present the time. Really am interesting to know if there exists an operator satisfies the below property:

$$A^{-1}(t)=A(-t+ i \beta)$$ $$\beta$$ is a real number non-null and $$A^{-1}$$ is the compositional inverse of $$A$$ , For instance I have got only the Unitary Operator as exponential form which it's used widely in quantum mechanics as shown here but it satisfies only :$$U^{-1}(t)=U(-t)$$

Note 01: In Have edited my question without changing the meaning of it and for the given answer , I didn't meant by U in my titled equality the unitary operator but it is an operator which i search on it

Note 02: The motivation of this question is to know more about chaotic operators

• Can you provide a bit more physical context? What other properties do you want $U(t)$ to satisfy? – By Symmetry Aug 10 '19 at 15:17
• Ok , for Other property i want U = U* to get a hermitian operator , I will add that in my question i missed that – zeraoulia rafik Aug 10 '19 at 15:18
• @BySymmetry, Another reformulation of the titled question is : Does the Hermitian operator satisfy the titled equality ? – zeraoulia rafik Aug 10 '19 at 15:25
• can some connection be made to KMS boundary condition? – Sunyam Aug 10 '19 at 15:39
• Could you elaborate in what context does this requirement appears? how did you come up with this rule? – lurscher Aug 10 '19 at 18:48

I am a bit confused by your desideratum, but if you had a unitary operator $$W(t)=\exp (itH)$$, so $$W(t) W(-t)=1\!\! 1,$$ You could define $$U(t)\equiv W(t) ~~e^{-\beta H/2},$$ so that $$U(t) U(-t+i\beta) = W(t) ~~e^{-\beta H/2} W(-t+i\beta) ~~e^{-\beta H/2}= e^{ i H (t- i \beta/2 -t +i\beta -i\beta/2 )}=1\!\! 1 ,$$ no?
• @zeraouliarafik That doesnt make any sense. If $U\colon \mathbb R\to\mathcal H$, then the compositional inverse is $U^{-1}\colon\mathcal H\to\mathbb R$, and so you cannot possibly have $U^{-1}(t)=U(-t+i\beta)$. – AccidentalFourierTransform Aug 10 '19 at 17:40