---
title: "T^m"
source: https://www.jemoka.com/posts/kbhraising_operators_to_powers/
---

For an operator \(T \in \mathcal{L}(V)\), \(T^{n}\) would make sense. Instead of writing \(TTT\dots\), then, we just write \(T^{n}\).
constituents operator \(T \in \mathcal{L}(V)\) requirements \(T^{m} = T \dots T\) additional information \(T^{0}\) \begin{equation} T^{0} := I \in \mathcal{L}(V) \end{equation}
\(T^{-1}\) \begin{equation} T^{-m} = (T^{-1})^{m} \end{equation}
if \(T\) is invertable
usual rules of squaring \begin{equation} \begin{cases} T^{m}T^{n} = T^{m+n} \\ (T^{m})^{n} = T^{mn} \end{cases} \end{equation}
This can be shown by counting the number of times \(T\) is repeated by writing each \(T^{m}\) out.
