Abstract
This talk investigates when a linear combination of powers of an $n$-potent operator $T$ (satisfying $T^n = T$) yields a projection. Starting from classical examples of projections as averages of isometric reflections, we develop the theory of "basic projections" — the fundamental building blocks — and enumerate all projections in $\operatorname{span}\{T, T^2, \ldots, T^{n-1}\}$ for small values of $n$. The talk concludes with an open question that was subsequently resolved in joint work with Easley and Monika (arXiv:2512.22497, 2025).
Motivation: Projections and Isometries
Projections and isometries are central objects in functional analysis for complementary reasons:
- Projections have a simple spectral structure (eigenvalues only $0$ and $1$) and arise as building blocks in the Spectral Theorem. Orthogonal projections appear in linear regression, QR decomposition, and multivariate analysis.
- Isometries are distance-preserving maps — the "isomorphisms" of metric geometry — and preserve many deep structural properties.
- The two are interconnected: averages of isometries often produce projections, and projections often arise from isometric structure.
Definitions
A linear map $T \colon X \to Y$ between normed spaces is an isometry if $\|Tx\| = \|x\|$ for every $x \in X$.
A bounded operator $P \colon X \to X$ is a projection if $P^2 = P$.
A bounded operator $R \colon X \to X$ is a reflection if $R^2 = I$.
Projections as Averages of Reflections
On $\mathbb{R}^3$ with $\|\cdot\|_\infty$, define $R(x_1,x_2,x_3) = (x_2,x_1,x_3)$. Then $R^2 = I$ and $R$ is an isometry. The average $$P = \tfrac{1}{2}(I+R) \colon (x_1,x_2,x_3) \mapsto \tfrac{1}{2}(x_1+x_2,\, x_1+x_2,\, 2x_3)$$ is a projection.
On $C[0,1]$, define $Rf(t) = f(1-t)$. Then $R^2 = I$ and $R$ is an isometry. The average $$P(f)(t) = \tfrac{1}{2}(f(t) + f(1-t))$$ is a projection.
General principle: If $R$ is an isometric reflection, then $P = \frac{1}{2}(I+R)$ is a projection.
Not every projection arises as $\frac{1}{2}(I+R)$. On $\mathbb{R}^3$ with $\|\cdot\|_\infty$, the projection $P(x_1,x_2,x_3) = (x_1+x_2, 0, x_3)$ would require $R(x_1,x_2,x_3) = (x_1+2x_2,-x_2,x_3)$, which is not an isometry.
Projections in the Convex Hull of Periodic Operators
For two isometries $T_0, T_1$ and positive $\lambda_0, \lambda_1$ with $\lambda_0 + \lambda_1 = 1$: if $Q = \lambda_0 T_0 + \lambda_1 T_1$ is a projection ($Q \neq I$), then $\lambda_0 = \lambda_1 = \frac{1}{2}$.
For $T^n = I$ and positive $\lambda_0, \ldots, \lambda_{n-1}$ summing to $1$: if $Q = \lambda_0 I + \lambda_1 T + \cdots + \lambda_{n-1} T^{n-1}$ is a projection, then $\lambda_0 = \cdots = \lambda_{n-1} = \dfrac{1}{n}$.
The $n$-Potent Problem: Basic Projections
An operator $T$ is $n$-potent if $T^n = T$.
Examples: projections ($n=2$), tripotents ($n=3$), periodic operators $T^m=I$ ($n=m+1$).
Let $T^{k+1} = T$ and $\lambda$ be a $k$-th root of unity. The projection $$Q_\lambda = \frac{1}{k}\sum_{i=1}^{k} \lambda^{-i} T^i$$ is called a basic projection.
Classification for small $n$
$n=2$ ($T^2 = T$): $Q = a_1 T$ is a projection $\Leftrightarrow$ $a_1 \in \{0,1\}$. Only projections: $0$ and $T$.
$n=3$ ($T^3 = T$): $Q = a_1 T + a_2 T^2$ is a projection $\Leftrightarrow$ $Q \in \left\{0,\; \dfrac{T+T^2}{2},\; \dfrac{-T+T^2}{2},\; T^2\right\}$.
| # | Expression | Type |
|---|---|---|
| 1 | $0$ | trivial |
| 2 | $\frac{1}{3}(T+T^2+T^3)$ | basic |
| 3 | $\frac{1}{3}(\omega T+\omega^2 T^2+T^3)$ | basic |
| 4 | $\frac{1}{3}(\omega^2 T+\omega T^2+T^3)$ | basic |
| 5 | $\alpha T+\beta T^2+\frac{2}{3}T^3$ | #2+#3 |
| 6 | $\beta T+\alpha T^2+\frac{2}{3}T^3$ | #2+#4 |
| 7 | $\frac{1}{3}(-T-T^2+2T^3)$ | #3+#4 |
| 8 | $T^3$ | sum of all basic |
where $\alpha = \frac{1+\sqrt{3}i}{6}$, $\beta = \frac{1-\sqrt{3}i}{6}$, $\omega = \frac{-1+\sqrt{3}i}{2}$. Exactly $2^3 = 8$ projections.
The Open Question and Its Resolution
For an $n$-potent operator $T$, is it possible to classify all projections in $\operatorname{span}\{T, T^2, \ldots, T^{n-1}\}$ via the basic projections?
Yes. Every projection in $\operatorname{comb}(T)$ is uniquely a sum $P_S = \sum_{\lambda \in S} P_{\{\lambda\}}$ of Riesz projections indexed by a subset $S \subseteq \sigma(T) \setminus \{0\}$. The full family forms a Boolean algebra with exactly $2^{|\sigma_0(T)|}$ elements — consistent with the count of $8 = 2^3$ for $n=4$ above.
References
- — Projections in the convex hull of isometries on absolutely continuous function spaces. J. Math. Anal. Appl., 2023. [Journal]
- — Projections in the algebra generated by an $n$-potent operator. Preprint, 2025. [arXiv:2512.22497]