Definition:Neumann Series

Definition

Let $T$ be an operator.

Let $\circ$ denote the composition.

Let $I$ be the identity mapping.

For any $k \in \N$ let $T^k = \underbrace{T \circ T \circ \ldots \circ T \circ T}_{k \text{ times} }$ and $T^0 = I$

Then the series $\ds \sum_{k \mathop = 0}^\infty T^k$ are known as Neumann series.

Also see

• Results about Neumann series can be found here.

Source of Name

This entry was named for Carl Gottfried Neumann.

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations