# 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 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 at 15:18
• @BySymmetry, Another reformulation of the titled question is : Does the Hermitian operator satisfy the titled equality ? – zeraoulia rafik Aug 10 at 15:25
• can some connection be made to KMS boundary condition? – Sunyam Aug 10 at 15:39
• Could you elaborate in what context does this requirement appears? how did you come up with this rule? – lurscher Aug 10 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?

• But in my question U^(-1) is compositional inverse of U not a reciprocal (multiplicative) inverse . – zeraoulia rafik Aug 10 at 17:01
• @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 at 17:40
• @zeraouliarafik your baseline QM example is definitely multiplicative, no? – Cosmas Zachos Aug 10 at 17:43
• No, i asked for which operator such that the titled equality Holds , Probably it will hold for some partition of Rational numbers like (-1,1) – zeraoulia rafik Aug 10 at 17:46
• @zeraouliarafik how is that relevant? unitary or not, the equation only makes sense if the inverse is in the multiplicative sense. (Also, in QM operators are always linear, so composition is the same as multiplication...) – AccidentalFourierTransform Aug 10 at 19:49