T ^ ^ ( x ) | r ⟩ ⟩ = | r + x ⟩ ⟩ Its adjoint expression is: ⟨ ⟨ r | ( T ^ ^ ( x ) ) † † = ⟨ ⟨ r + x | Using the results above, ( T ^ ^ ( x ) ) † † = ( T ^ ^ ( x ) ) − − 1 = T ^ ^ ( − − x ) : ⟨ ⟨ r | T ^ ^ ( − − x ) = ⟨ ⟨ r + x | Replacing x by − − x , ⟨ ⟨ r | T ^ ^ ( x ) = ⟨ ⟨ r − − x |