---
title: Bauschke–Bendit–Moursi Modulus of Averagedness
url: https://www.emergentmind.com/topics/bauschke-bendit-moursi-modulus-of-averagedness
type: topic
---

# Bauschke–Bendit–Moursi Modulus of Averagedness

The Bauschke–Bendit–Moursi modulus of averagedness is a quantitative invariant of a nonexpansive operator on a real Hilbert space that measures the smallest averagedness parameter for which the operator can be represented as a convex combination of the identity and a nonexpansive mapping. In the notation of the cited works, this modulus appears as either $\alpha(T)$ or $k(T)$, and it has become a central device for exact computations for compositions of projections, for quantitative asymptotic regularity of composed averaged mappings, and for structural classifications of firmly nonexpansive, resolvent, and proximal operators [2303.13738], [2001.01513], [2507.19533].

## 1. Definition, notation, and equivalent characterizations

Let $H$ be a real Hilbert space with inner product $\langle \cdot,\cdot\rangle$ and norm $\|\cdot\|$. A mapping $T:H\to H$ is nonexpansive if
$$
\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).
$$
For $\alpha\in[0,1]$, $T$ is called $\alpha$-averaged if there exists a nonexpansive operator $R:H\to H$ such that
$$
T=(1-\alpha)\operatorname{Id}+\alpha R.
$$
The Bauschke–Bendit–Moursi modulus of averagedness is the minimal such parameter:
$$
\alpha(T):=\min\{\alpha\in[0,1]\mid T\text{ is }\alpha\text{-averaged}\},
$$
and, in the later notation,
$$
k(T):=\inf\{\alpha\in[0,1]\mid T\text{ is }\alpha\text{-averaged}\}.
$$
For the nonexpansive operators considered in these works, these formulas encode the same concept [2303.13738], [2507.19533].

For $\alpha\in(0,1]$, averagedness admits several equivalent characterizations. One widely used inequality is
$$
\|Tx-Ty\|^2 \le \|x-y\|^2-\frac{1-\alpha}{\alpha}\|( \operatorname{Id}-T)x-( \operatorname{Id}-T)y\|^2
\qquad (\forall x,y\in H).
$$
The case $\alpha=\tfrac12$ yields firm nonexpansiveness, so an operator is firmly nonexpansive if and only if it is $\tfrac12$-averaged. Equivalently,
$$
\|Tx-Ty\|^2\le \langle x-y,Tx-Ty\rangle \qquad (\forall x,y\in H).
$$
This establishes the threshold value $\tfrac12$ as the boundary between general averagedness and firm nonexpansiveness [2303.13738], [2507.19533].

In the linear setting, the modulus admits a closed-form variational characterization. If $T\neq \operatorname{Id}$ is linear and nonexpansive, then
$$
\alpha(T)=\sup_{z\neq Tz}\frac{\|z-Tz\|^2}{2\langle z,z-Tz\rangle}.
$$
For bounded linear nonexpansive $T$, the modulus is invariant under adjoints:
$$
\alpha(T)=\alpha(T^*).
$$
These formulas are particularly effective for exact computations [2303.13738].

## 2. Exact modulus for the composition of two orthogonal projections

A principal result of Bauschke, Bendit, and Moursi concerns the composition of orthogonal projections onto closed subspaces. Let $U,V\subset H$ be closed subspaces, let $P_U$ and $P_V$ be the corresponding orthogonal projections, and consider
$$
T:=P_VP_U.
$$
The exact modulus is expressed in terms of the cosine of the Friedrichs angle,
$$
c_F(U,V):=\sup\{\langle u,v\rangle \mid u\in U\cap (U\cap V)^\perp,\ v\in V\cap (U\cap V)^\perp,\ \|u\|\le 1,\ \|v\|\le 1\}\in[0,1].
$$
The exact formula is
$$
\alpha(P_VP_U)=\frac{2-c_F^2+c_F}{4-c_F^2}.
$$
This shows that while each projection is firmly nonexpansive, their composition is generally only averaged and no longer firmly nonexpansive [2303.13738].

The paper also treats the relaxed composition
$$
T_\beta:=\bigl((1-\beta)\operatorname{Id}+\beta R_V\bigr)P_U,
\qquad R_V:=2P_V-\operatorname{Id},\quad \beta\in(0,1),
$$
for which
$$
\alpha(T_\beta)=
\frac{1+2\beta(1-c_F^2)+\sqrt{(1-2\beta)^2+4(1-\beta)\beta c_F^2}}
{2(2-\beta c_F^2)}.
$$
Setting $\beta=\tfrac12$ gives $T_{1/2}=P_VP_U$ and recovers the projection formula above [2303.13738].

The edge cases are explicit. If $U=V=H$, then $P_VP_U=\operatorname{Id}$ and the modulus is $0$. If $U\subseteq V$ properly or $V\subseteq U$ properly, then $c_F(U,V)=0$ and $\alpha(P_VP_U)=\tfrac12$. If $U\perp V$, then $c_F(U,V)=0$, $P_VP_U=0$, and the zero operator has modulus $\tfrac12$ [2303.13738].

For lines in $\mathbb{R}^2$ forming angle $\theta\in[0,\pi/2]$, $c_F=\cos\theta$, and
$$
\alpha(P_VP_U)=\frac{2-\cos^2\theta+\cos\theta}{4-\cos^2\theta}.
$$
As $\theta\to 0$, the modulus tends to $\tfrac23$; as $\theta\to \pi/2$, it tends to $\tfrac12$. In the worked example with
$$
U=\operatorname{span}\{(1,0)\},\qquad V=\operatorname{span}\{(1,1)\},
$$
one obtains
$$
P_UP_V=\frac12\begin{bmatrix}1&0\\1&0\end{bmatrix},
\qquad
\alpha\approx 0.6306,
$$
in agreement with the exact formula [2303.13738].

## 3. Composition laws and the sharpness of the Ogura–Yamada bound

For averaged mappings, the basic composition rule is encoded by the binary operation
$$
\alpha*\beta:=\frac{\alpha+\beta-2\alpha\beta}{1-\alpha\beta}.
$$
This operation is associative and commutative, and if $R_i$ is $\alpha_i$-averaged for $i=1,\dots,m$, then
$$
R_m\circ\cdots\circ R_1
$$
is $(\alpha_1*\cdots *\alpha_m)$-averaged [2001.01513].

For two operators this reproduces the Ogura–Yamada bound: if $T_1$ and $T_2$ are averaged with moduli $\alpha_1$ and $\alpha_2$, then
$$
\alpha(T_2T_1)\le
\frac{\alpha_1+\alpha_2-2\alpha_1\alpha_2}{1-\alpha_1\alpha_2}.
$$
In the later modulus notation,
$$
k(T_1T_2)\le
\frac{k(T_1)+k(T_2)-2k(T_1)k(T_2)}{1-k(T_1)k(T_2)}
$$
whenever $k(T_1)k(T_2)\neq 1$ [2303.13738], [2507.19533].

The Bauschke–Bendit–Moursi projection formula proves that this bound is sharp in general. In the relaxed projection setting,
$$
\alpha(T_\beta)\le \frac{1}{2-\beta},
$$
with equality if and only if $c_F(U,V)=1$. For $T=P_VP_U$, corresponding to $\beta=\tfrac12$, the bound gives
$$
\alpha(P_VP_U)\le \frac23,
$$
and equality occurs precisely when $c_F=1$. In finite-dimensional spaces one has $c_F<1$, hence $\alpha(P_VP_U)<\tfrac23$; in infinite-dimensional Hilbert spaces, $c_F=1$ can occur, and then the bound is attained [2303.13738].

A standard special case is the composition of $m$ firmly nonexpansive mappings. Since each has averagedness parameter $\tfrac12$,
$$
\underbrace{\tfrac12 * \cdots * \tfrac12}_{m\ \text{times}}=\frac{m}{m+1}.
$$
This shows that repeated composition drives the effective averagedness parameter toward $1$, even though each individual factor sits at the firm threshold $\tfrac12$ [2001.01513].

## 4. Quantitative asymptotic regularity and approximate fixed points

The proof-mining analysis in [2001.01513] places the modulus in a quantitative framework for inconsistent feasibility and asymptotic regularity. A mapping $R:X\to X$ has approximate fixed points, or arbitrarily small displacements, if
$$
\forall \varepsilon>0\ \exists p\in X \quad \|p-Rp\|\le \varepsilon.
$$
The quantitative assumption used there is stronger: for each component operator $R_i$ there exists a common function $K:(0,\infty)\to(0,\infty)$ such that
$$
\forall \varepsilon>0\ \exists p\in X \quad \|p\|\le K(\varepsilon)\ \text{and}\ \|p-R_ip\|\le \varepsilon.
$$
Under this hypothesis, if $R_1,\dots,R_m$ are averaged mappings on a Hilbert space, then their composition also has approximate fixed points and is asymptotically regular [2001.01513].

The BBM viewpoint enters through the composite averagedness parameter
$$
\alpha_*:=\alpha_1*\cdots *\alpha_m.
$$
If $R$ is $\alpha$-averaged, then it is strongly nonexpansive with explicit modulus
$$
w_\alpha(b,\varepsilon)=\frac{1-\alpha}{4b\,\alpha}\varepsilon^2.
$$
Hence the composition $R:=R_m\circ\cdots\circ R_1$ is strongly nonexpansive with modulus $w_{\alpha_*}$. This modulus, together with the recursive approximate-fixed-point bound $V(m,\{\alpha_i\},K,\varepsilon)$ constructed in Theorem 2.3, feeds into the general rate constructor $Q$ of Theorem 2.5 and yields the explicit uniform rate
$$
E_{m,\{\alpha_i\},K,b,d}(\varepsilon)
=
Q\bigl(V(m,\{\alpha_i\},K,\varepsilon),\,b,\,d,\,\varepsilon,\,w_{\alpha_*}\bigr)
$$
for asymptotic regularity of the Picard iteration [2001.01513].

For cyclic projections onto closed convex sets, each projection is firmly nonexpansive, so $\alpha_i=\tfrac12$ and $\alpha_*=\frac{m}{m+1}$. The corresponding strong nonexpansivity modulus becomes
$$
w_{\alpha_*}(b,\varepsilon)=\frac{1}{4bm}\varepsilon^2.
$$
A plausible implication is that the modulus of averagedness does not merely certify nonexpansiveness of a composition; it also determines quantitative constants in asymptotic regularity estimates [2001.01513].

## 5. Threshold phenomena and operator classifications

The 2025 classification results organize nonexpansive and firmly nonexpansive operators around the threshold value $\tfrac12$. The paper introduces the terms “normally nonexpansive” for $k(T)<\tfrac12$ and “specially nonexpansive” for $k(T)=\tfrac12$. Projections onto a proper nonempty closed convex set $C\subsetneq X$ satisfy $k(P_C)=\tfrac12$, constant mappings have $k(T)=\tfrac12$, and orthogonal isometries $A\neq \operatorname{Id}$ satisfy $k(A)=1$ [2507.19533].

The central structural theorem is that if $T$ is normally nonexpansive, then $T$ is a bi-Lipschitz homeomorphism of $X$. Writing
$$
k(T)=\frac12-\alpha \qquad \text{with } \alpha\in(0,\tfrac12),
$$
one has the explicit two-sided estimate
$$
2\alpha \|x-y\|\le \|Tx-Ty\|\le \|x-y\|.
$$
Conversely, if a nonexpansive $T$ is not bijective, then $k(T)\ge \tfrac12$ [2507.19533].

Several basic properties of the modulus support this classification. It is translation invariant:
$$
k(T+v)=k(T), \qquad k(T(\cdot+v))=k(T).
$$
It behaves exactly under relaxation:
$$
k\bigl((1-\lambda)\operatorname{Id}+\lambda T\bigr)=\lambda k(T).
$$
It is convex with respect to convex combinations of nonexpansive operators. Moreover, $k(T)=0$ if and only if $T=\operatorname{Id}+v$ for some $v\in X$; if $\operatorname{Fix}(T)\neq \varnothing$, then $k(T)=0$ if and only if $T=\operatorname{Id}$ [2507.19533].

The same work studies the limiting operator
$$
T_\infty x:=w\text{-}\lim_{n\to\infty}T^n x
$$
for an $\alpha$-averaged mapping $T$ with $\operatorname{Fix}(T)\neq\varnothing$. One has
$$
\operatorname{Fix}(T)=\operatorname{Fix}(T_\infty)=\operatorname{ran}(T_\infty),
\qquad
(T_\infty)^2=T_\infty,
$$
and $k(T_\infty)\in[\tfrac12,1]$. Furthermore,
$$
T_\infty=P_{\operatorname{Fix} T}
\iff k(T_\infty)\le \frac12
\iff k(T_\infty)=\frac12.
$$
This locates projection-valued asymptotic limits exactly at the threshold [2507.19533].

## 6. Resolvents, proximal operators, and related developments

For a maximally monotone operator $A:X\rightrightarrows X$, the resolvent
$$
J_A:=(\operatorname{Id}+A)^{-1}
$$
is firmly nonexpansive, and the reflected resolvent is
$$
R_A:=2J_A-\operatorname{Id}.
$$
The BBM modulus satisfies
$$
k(R_A)=2k(J_A).
$$
Using Yosida regularization, the paper proves the identities
$$
J_{\alpha Y_\mu(A)}=\frac{\mu}{\mu+\alpha}\operatorname{Id}
+\frac{\alpha}{\mu+\alpha}J_{(\mu+\alpha)A},
$$
and in particular
$$
J_{Y_\mu(A)}=\frac{\mu}{\mu+1}\operatorname{Id}
+\frac{1}{\mu+1}J_{(\mu+1)A}.
$$
These lead to the exact formula
$$
k(J_{\alpha A})
=
\frac12\,\frac{\alpha}{\alpha+c(A)}
=
\frac12\,\frac{\alpha}{\alpha+m(A^{-1})},
$$
where $c(A)$ is the cocoercive value and $m(A^{-1})$ is the monotone value of the inverse. Consequently, $J_A$ is normally nonexpansive if and only if $A$ is single-valued with full domain and cocoercive [2507.19533].

For proximal mappings,
$$
\operatorname{prox}_f:=(\operatorname{Id}+\partial f)^{-1},
$$
the same paper gives a sharp characterization. For $f\in \Gamma_0(X)$, the following are equivalent: $\operatorname{prox}_f$ is normally nonexpansive; $f$ is $L$-smooth on $X$ for some $L>0$; $f^*$ is $(1/L)$-strongly convex for some $L>0$; $\operatorname{prox}_{f^*}$ is a Banach contraction; and $\operatorname{prox}_f$ is a bi-Lipschitz homeomorphism of $X$. The modulus is
$$
k(\operatorname{prox}_f)=
\frac12\,\frac{1}{1+[\ell(\partial f)]^{-1}},
$$
and if $f$ is $L$-smooth, then
$$
k(\operatorname{prox}_f)=\frac12\,\frac{L}{1+L}.
$$
More generally,
$$
k(\operatorname{prox}_{\alpha f})=\frac12\,\frac{\alpha \ell(\nabla f)}{1+\alpha \ell(\nabla f)}.
$$
If $\operatorname{dom} f\neq X$, then $k(\operatorname{prox}_f)=\tfrac12$ [2507.19533].

These formulas include familiar special cases: for a nonempty closed convex set $C\subsetneq X$,
$$
k(P_C)=\frac12,\qquad k(R_C)=1,\qquad
k((1-\lambda)\operatorname{Id}+\lambda P_C)=\frac{\lambda}{2},\qquad
k((1-\lambda)\operatorname{Id}+\lambda R_C)=\lambda.
$$
For distinct closed subspaces $U,V$, the Douglas–Rachford operator
$$
T_{U,V}:=\frac{\operatorname{Id}+R_UR_V}{2}
$$
satisfies $k(T_{U,V})=\tfrac12$. For the product-space subspace configuration
$$
U=X\times\{0\},\qquad
V=\{(y,(\tan\theta)y)\mid y\in X\},\quad \theta\in(0,\pi/2),
$$
one obtains
$$
k(P_VP_U)=\frac{1+\cos\theta}{2+\cos\theta}\in \Bigl[\frac12,\frac23\Bigr],
$$
which is consistent with the two-projection exact formulas in the earlier work [2507.19533], [2303.13738].

An open direction remains from the projection-composition analysis: the paper conjectures that the Ogura–Yamada bound remains sharp for the doubly underrelaxed composition
$$
\bigl((1-\beta)\operatorname{Id}+\beta R_V\bigr)\bigl((1-\alpha)\operatorname{Id}+\alpha R_U\bigr),
$$
but an exact modulus formula was left open [2303.13738]. This suggests that the BBM modulus has already reached a mature exact theory for several fundamental operator classes, while still leaving nontrivial composition problems unresolved.

Source: https://www.emergentmind.com/topics/bauschke-bendit-moursi-modulus-of-averagedness