Abstract
Let $T \in B(X)$ be an $n$-potent operator, i.e., $T^n = T$ for some integer $n \geq 2$. We give a complete classification of all projections lying in the algebra $\operatorname{comb}(T) = \operatorname{span}\{T, T^2, \ldots, T^{n-1}\}$. Using an explicit resolvent expansion, we derive closed-form formulas for all Riesz projections of $T$ — expressed as discrete Fourier transforms of $\{T, T^2, \ldots, T^{n-1}\}$ — without contour integration. We show that every projection in $\operatorname{comb}(T)$ is a sum of Riesz projections indexed by a subset $S \subseteq \sigma(T) \setminus \{0\}$, and that the full family of such projections forms a Boolean algebra isomorphic to $\mathcal{P}(\sigma_0(T))$, with exactly $2^{|\sigma_0(T)|}$ elements. This unifies and generalizes results of Bikchentaev–Yakushev (2011) and Botelho–Dey–Easley (2023).
Motivation
Thread 1: Projections in the convex hull of isometries
- (JMAA 2008): $\lambda T_1 + (1-\lambda)T_2$ a projection on $C(\Omega)$ forces $\lambda = \tfrac{1}{2}$; always a generalized bi-circular projection.
- (Bull. Malays. Math. Sci. Soc., 2021): the unique projection in $\operatorname{co}\{U_1, U_2, U_3\}$ is $\tfrac{1}{3}(U_1 + U_2 + U_3)$.
- (JMAA 2023): for $U^m = I$ periodic, the unique projection in $\operatorname{co}\{U^k\}_{k=0}^{m-1}$ is the Cesàro mean $\tfrac{1}{m}\sum_{k=0}^{m-1} U^k$.
Pattern: the convex hull yields at most one projection.
Thread 2: Projections from a single operator
- (LAA 2011): for $T^3 = T$, characterized projections in $\operatorname{span}\{T, T^2\}$ — the $n=3$ case of our theorem.
- (JMAA 2022): when is $T = \lambda_1 P + \lambda_2(I - P)$ an isometry?
For general $n$-potent $T$ ($T^n = T$, $n \geq 2$), the projections lying in the algebra $\operatorname{comb}(T) = \operatorname{span}\{T, T^2, \ldots, T^{n-1}\}$ have not been systematically characterized.
Setup and Spectral Preliminaries
$T \in B(X)$ is $n$-potent if $T^n = T$ for some integer $n \geq 2$.
Examples: projections ($n=2$), tripotents ($n=3$), periodic operators $T^m = I$ ($n = m+1$).
If $T$ is $n$-potent then $$\sigma(T) \subseteq \{0, 1, \omega, \ldots, \omega^{n-2}\}, \qquad \omega = e^{2\pi i/(n-1)},$$ and every eigenvalue is semisimple.
For $\lambda \in \sigma(T)$ isolated: $$P_{\{\lambda\}} := -\frac{1}{2\pi i} \oint_{\Gamma_\lambda} (T - zI)^{-1}\,dz.$$
Key properties: $P_{\{\lambda\}}^2 = P_{\{\lambda\}}$, $P_{\{\lambda\}} P_{\{\mu\}} = 0$ for $\lambda \neq \mu$, $\sum_\lambda P_{\{\lambda\}} = I$, $\operatorname{Ran}(P_{\{\lambda\}}) = \ker(T - \lambda I)$.
$T$ is $n$-potent if and only if $T = \sum_{\lambda \in \sigma(T)} \lambda\, P_{\{\lambda\}}$, where $\{P_{\{\lambda\}}\}$ is a unique finite resolution of identity.
Main Results
For $z \in \rho(T)$: $$(zI - T)^{-1} = \frac{I}{z} + \frac{T^{n-1}}{z(z^{n-1}-1)} + \sum_{k=1}^{n-2} \frac{z^{n-k-2}}{z^{n-1}-1} T^k.$$
With $\omega = e^{2\pi i/(n-1)}$, for each $j = 0, \ldots, n-2$: $$\omega^j \in \sigma(T) \;\Longleftrightarrow\; \boxed{P_{\{\omega^j\}} = \frac{1}{n-1} \sum_{k=1}^{n-1} \omega^{-jk} T^k}$$ is a nonzero projection satisfying $T P_{\{\omega^j\}} = \omega^j P_{\{\omega^j\}}$.
Note: This is a discrete Fourier transform of $\{T, T^2, \ldots, T^{n-1}\}$ in $\mathbb{Z}_{n-1}$. No contour integration needed in practice.
Let $T$ be $n$-potent and let $P = \sum_{i=1}^{n-1} a_i T^i \in \operatorname{comb}(T)$ be a projection. Then $P$ has a unique decomposition $$P = \sum_{\lambda \in \sigma(T)} \beta_\lambda P_{\{\lambda\}}, \qquad \beta_\lambda = \sum_{i=1}^{n-1} a_i \lambda^i.$$ Moreover, $\beta_\lambda \in \{0,1\}$ for every $\lambda$, and $\operatorname{Ran}(P) = \bigoplus_{\beta_\lambda = 1} P_{\{\lambda\}}(X)$.
Converse: For every $S \subseteq \sigma_0(T)$, the operator $P_S := \sum_{\lambda \in S} P_{\{\lambda\}}$ lies in $\operatorname{comb}(T)$ and is a projection.
$\{\text{projections in } \operatorname{comb}(T)\} \;\longleftrightarrow\; \mathcal{P}(\sigma_0(T))$
The map $S \mapsto P_S$ is a bijection $\mathcal{P}(\sigma_0(T)) \longleftrightarrow \{\text{projections in } \operatorname{comb}(T)\}$, and $\{P_S\}$ is a Boolean algebra: $$P_S \vee P_R = P_{S \cup R}, \qquad P_S \wedge P_R = P_{S \cap R}, \qquad P_S^c = P_{\sigma_0(T) \setminus S}.$$ There are exactly $2^{|\sigma_0(T)|}$ projections in $\operatorname{comb}(T)$. For $n=5$ with full spectrum: $2^4 = \mathbf{16}$ projections.
Illustrative Example: 5-Potent Operator
Let $T$ be 5-potent: $\sigma(T) \subseteq \{0, 1, i, -1, -i\}$, $\omega = i$.
$$P_{\{i\}} = \tfrac{1}{4}(-iT - T^2 + iT^3 + T^4), \qquad P_{\{-1\}} = \tfrac{1}{4}(-T + T^2 - T^3 + T^4).$$
$$P_{\{i,-1\}} = P_{\{i\}} + P_{\{-1\}} = \Bigl(-\tfrac{1+i}{4}\Bigr)T + \Bigl(\tfrac{-1+i}{4}\Bigr)T^3 + \tfrac{1}{2}T^4.$$
Recovery: Given $P = \bigl(-\tfrac{1+i}{4}\bigr)T + \bigl(\tfrac{-1+i}{4}\bigr)T^3 + \tfrac{1}{2}T^4$, compute $\beta_\lambda = \sum_j a_j \lambda^j$:
$$\beta_0 = 0,\;\beta_1 = 0,\;\beta_i = 1,\;\beta_{-1} = 1,\;\beta_{-i} = 0 \;\Longrightarrow\; P = P_{\{i\}} + P_{\{-1\}}. \checkmark$$
For $n=3$: $\sigma_0(T) \subseteq \{1,-1\}$ and $T = P_{\{1\}} - P_{\{-1\}}$, recovering Bikchentaev–Yakushev (2011) exactly. Our formula generalizes this to all $n$.
Open Questions
- Hermitian Riesz projections: When are the $P_{\{\lambda\}}$ bicircular (Hermitian)? On Hilbert spaces, when are they orthogonal?
- Convex hull vs. algebra: Which $P_S \in \operatorname{comb}(T)$ are also in $\operatorname{co}\{T^k\}$? Characterize the intersection.
- $C^*$-algebra extension: Classify projections in $\operatorname{Alg}(T, T^*)$ on Hilbert spaces.
Related Work
- — Projections as convex combinations of isometries on $C(\Omega)$. J. Math. Anal. Appl., 2008.
- — Projections in linear spans of tripotents. Linear Algebra Appl., 2011.
- — Generalized circular projections. J. Math. Anal. Appl., 2022.
- — 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]