Properties
Suppose that T is a bounded operator on the normed vector space X. If the Neumann series converges in the operator norm, then Id – T is invertible and its inverse is the series:
- ,
where is the identity operator in X. To see why, consider the partial sums
- .
Then we have
One case in which convergence is guaranteed is when X is a Banach space and |T| < 1 in the operator norm. However, there are also results which give weaker conditions under which the series converges.
Read more about this topic: Neumann Series
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