There is a surprisingly frequent need to compute the quantity $O = 1 + G$ where both $O$ and $G$ are operators. These operators can be algebraic, often in the form of a matrix, or a derivative operator, and can be either finite- or infinite-dimensional. One typical example consists of constructing the local representation of a Lie group operator ‘near’ the identity – a construction that shows up most everywhere in quantum mechanics and in general relativity.
The interesting next step is to determine what is the inverse operator $O^{-1}$ in terms of $G$ such that
\[ O^{-1} O = 1 \; , \]which translates to
\[ (1+G)^{-1} (1+G) = 1 \; .\]To proceed, we can `decorated’ the right hand side with a infinite string of suggestive zeros
\[ (1+G)^{-1} (1 + G) = 1 + (G – G) + (G^2 – G^2) + \cdots \; .\]Next we indulge in some inspired rearrangement of terms to get
\[ (1+G)^{-1} (1 + G) = \left[ (1 + G) – (G + G^2) + (G^2 + G^3) – (G^3 + G^4) \cdots \right] \; . \]We can now factor $(1+G)$ out to the right to get
\[ (1+G)^{-1} (1+G) = \left[ 1 – G + G^2 – G^3 + \cdots \right] (1 + G) \; .\]Comparing the two sides we conclude that the quantity in the brackets must then be a series representation of the inverse
\[ (1+G)^{-1} = 1 – G + G^2 – G^3 + \cdots \; .\]If the construction is correct, then the expansion must also serve as a right inverse. That is to say,
\[ (1 + G) (1 + G)^{-1} = 1 \; .\]Substituting the form determined above yields
\[ (1 + G) (1 – G + G^2 – G^3 + \cdots) = 1 + G – G – G^2 + G^2 + G^3 – G^3 – G^4 + \cdots \; . \]It is interesting to note that this operator series is formally similar the Taylor series expansion for the geometric series:
\[ \frac{1}{1+x} = 1 – x + x^2 – x^3 + \cdots \; .\]Both series have limited domains of applicability. For the scalar function just expanded, it is well-known that the condition on $x$ for the expansion to be within the radius of convergence is $|x| < 1$, this latter the condition being the domain of applicability.
For the operator version, the corresponding condition isn’t nearly as clear and that will be the subject of next month’s blog.