Bauschke–Bendit–Moursi Modulus of Averagedness

• The Bauschke–Bendit–Moursi modulus is defined as the minimal averagedness parameter for nonexpansive operators in Hilbert spaces, offering a precise measure for operator decomposition.
• It provides exact formulas for compositions of projections by linking the modulus to the cosine of the Friedrichs angle, thereby refining asymptotic regularity and convergence analysis.
• The framework underpins composition laws and operator classifications, revealing practical implications for the design and analysis of averaged mappings and proximal operators.

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 (Bauschke et al., 2023, Sipos, 2020, Song et al., 22 Jul 2025).

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 $k(T)$0 such that

$k(T)$1

The Bauschke–Bendit–Moursi modulus of averagedness is the minimal such parameter:

$k(T)$2

and, in the later notation,

$k(T)$3

For the nonexpansive operators considered in these works, these formulas encode the same concept (Bauschke et al., 2023, Song et al., 22 Jul 2025).

For $k(T)$4, averagedness admits several equivalent characterizations. One widely used inequality is

$k(T)$5

The case $k(T)$6 yields firm nonexpansiveness, so an operator is firmly nonexpansive if and only if it is $k(T)$7-averaged. Equivalently,

$k(T)$8

This establishes the threshold value $k(T)$9 as the boundary between general averagedness and firm nonexpansiveness (Bauschke et al., 2023, Song et al., 22 Jul 2025).

In the linear setting, the modulus admits a closed-form variational characterization. If $H$0 is linear and nonexpansive, then

$H$1

For bounded linear nonexpansive $H$2, the modulus is invariant under adjoints:

$H$3

These formulas are particularly effective for exact computations (Bauschke et al., 2023).

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 $H$4 be closed subspaces, let $H$5 and $H$6 be the corresponding orthogonal projections, and consider

$H$7

The exact modulus is expressed in terms of the cosine of the Friedrichs angle,

$H$8

The exact formula is

$H$9

This shows that while each projection is firmly nonexpansive, their composition is generally only averaged and no longer firmly nonexpansive (Bauschke et al., 2023).

The paper also treats the relaxed composition

$\langle \cdot,\cdot\rangle$0

for which

$\langle \cdot,\cdot\rangle$1

Setting $\langle \cdot,\cdot\rangle$2 gives $\langle \cdot,\cdot\rangle$3 and recovers the projection formula above (Bauschke et al., 2023).

The edge cases are explicit. If $\langle \cdot,\cdot\rangle$4, then $\langle \cdot,\cdot\rangle$5 and the modulus is $\langle \cdot,\cdot\rangle$6. If $\langle \cdot,\cdot\rangle$7 properly or $\langle \cdot,\cdot\rangle$8 properly, then $\langle \cdot,\cdot\rangle$9 and $\|\cdot\|$0. If $\|\cdot\|$1, then $\|\cdot\|$2, $\|\cdot\|$3, and the zero operator has modulus $\|\cdot\|$4 (Bauschke et al., 2023).

For lines in $\|\cdot\|$5 forming angle $\|\cdot\|$6, $\|\cdot\|$7, and

$\|\cdot\|$8

As $\|\cdot\|$9, the modulus tends to $T:H\to H$0; as $T:H\to H$1, it tends to $T:H\to H$2. In the worked example with

$T:H\to H$3

one obtains

$T:H\to H$4

in agreement with the exact formula (Bauschke et al., 2023).

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

For averaged mappings, the basic composition rule is encoded by the binary operation

$T:H\to H$5

This operation is associative and commutative, and if $T:H\to H$6 is $T:H\to H$7-averaged for $T:H\to H$8, then

$T:H\to H$9

is $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$0-averaged (Sipos, 2020).

For two operators this reproduces the Ogura–Yamada bound: if $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$1 and $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$2 are averaged with moduli $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$3 and $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$4, then

$\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$5

In the later modulus notation,

$\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$6

whenever $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$7 (Bauschke et al., 2023, Song et al., 22 Jul 2025).

The Bauschke–Bendit–Moursi projection formula proves that this bound is sharp in general. In the relaxed projection setting,

$\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$8

with equality if and only if $\|Tx-Ty\|\le \|x-y\| \qquad (\forall x,y\in H).$9. For $\alpha\in[0,1]$0, corresponding to $\alpha\in[0,1]$1, the bound gives

$\alpha\in[0,1]$2

and equality occurs precisely when $\alpha\in[0,1]$3. In finite-dimensional spaces one has $\alpha\in[0,1]$4, hence $\alpha\in[0,1]$5; in infinite-dimensional Hilbert spaces, $\alpha\in[0,1]$6 can occur, and then the bound is attained (Bauschke et al., 2023).

A standard special case is the composition of $\alpha\in[0,1]$7 firmly nonexpansive mappings. Since each has averagedness parameter $\alpha\in[0,1]$8,

$\alpha\in[0,1]$9

This shows that repeated composition drives the effective averagedness parameter toward $T$0, even though each individual factor sits at the firm threshold $T$1 (Sipos, 2020).

4. Quantitative asymptotic regularity and approximate fixed points

The proof-mining analysis in (Sipos, 2020) places the modulus in a quantitative framework for inconsistent feasibility and asymptotic regularity. A mapping $T$2 has approximate fixed points, or arbitrarily small displacements, if

$T$3

The quantitative assumption used there is stronger: for each component operator $T$4 there exists a common function $T$5 such that

$T$6

Under this hypothesis, if $T$7 are averaged mappings on a Hilbert space, then their composition also has approximate fixed points and is asymptotically regular (Sipos, 2020).

The BBM viewpoint enters through the composite averagedness parameter

$T$8

If $T$9 is $\alpha$0-averaged, then it is strongly nonexpansive with explicit modulus

$\alpha$1

Hence the composition $\alpha$2 is strongly nonexpansive with modulus $\alpha$3. This modulus, together with the recursive approximate-fixed-point bound $\alpha$4 constructed in Theorem 2.3, feeds into the general rate constructor $\alpha$5 of Theorem 2.5 and yields the explicit uniform rate

$\alpha$6

for asymptotic regularity of the Picard iteration (Sipos, 2020).

For cyclic projections onto closed convex sets, each projection is firmly nonexpansive, so $\alpha$7 and $\alpha$8. The corresponding strong nonexpansivity modulus becomes

$\alpha$9

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 (Sipos, 2020).

5. Threshold phenomena and operator classifications

The 2025 classification results organize nonexpansive and firmly nonexpansive operators around the threshold value $k(T)$00. The paper introduces the terms “normally nonexpansive” for $k(T)$01 and “specially nonexpansive” for $k(T)$02. Projections onto a proper nonempty closed convex set $k(T)$03 satisfy $k(T)$04, constant mappings have $k(T)$05, and orthogonal isometries $k(T)$06 satisfy $k(T)$07 (Song et al., 22 Jul 2025).

The central structural theorem is that if $k(T)$08 is normally nonexpansive, then $k(T)$09 is a bi-Lipschitz homeomorphism of $k(T)$10. Writing

$k(T)$11

one has the explicit two-sided estimate

$k(T)$12

Conversely, if a nonexpansive $k(T)$13 is not bijective, then $k(T)$14 (Song et al., 22 Jul 2025).

Several basic properties of the modulus support this classification. It is translation invariant:

$k(T)$15

It behaves exactly under relaxation:

$k(T)$16

It is convex with respect to convex combinations of nonexpansive operators. Moreover, $k(T)$17 if and only if $k(T)$18 for some $k(T)$19; if $k(T)$20, then $k(T)$21 if and only if $k(T)$22 (Song et al., 22 Jul 2025).

The same work studies the limiting operator

$k(T)$23

for an $k(T)$24-averaged mapping $k(T)$25 with $k(T)$26. One has

$k(T)$27

and $k(T)$28. Furthermore,

$k(T)$29

This locates projection-valued asymptotic limits exactly at the threshold (Song et al., 22 Jul 2025).

6. Resolvents, proximal operators, and related developments

For a maximally monotone operator $k(T)$30, the resolvent

$k(T)$31

is firmly nonexpansive, and the reflected resolvent is

$k(T)$32

The BBM modulus satisfies

$k(T)$33

Using Yosida regularization, the paper proves the identities

$k(T)$34

and in particular

$k(T)$35

These lead to the exact formula

$k(T)$36

where $k(T)$37 is the cocoercive value and $k(T)$38 is the monotone value of the inverse. Consequently, $k(T)$39 is normally nonexpansive if and only if $k(T)$40 is single-valued with full domain and cocoercive (Song et al., 22 Jul 2025).

For proximal mappings,

$k(T)$41

the same paper gives a sharp characterization. For $k(T)$42, the following are equivalent: $k(T)$43 is normally nonexpansive; $k(T)$44 is $k(T)$45-smooth on $k(T)$46 for some $k(T)$47; $k(T)$48 is $k(T)$49-strongly convex for some $k(T)$50; $k(T)$51 is a Banach contraction; and $k(T)$52 is a bi-Lipschitz homeomorphism of $k(T)$53. The modulus is

$k(T)$54

and if $k(T)$55 is $k(T)$56-smooth, then

$k(T)$57

More generally,

$k(T)$58

If $k(T)$59, then $k(T)$60 (Song et al., 22 Jul 2025).

These formulas include familiar special cases: for a nonempty closed convex set $k(T)$61,

$k(T)$62

For distinct closed subspaces $k(T)$63, the Douglas–Rachford operator

$k(T)$64

satisfies $k(T)$65. For the product-space subspace configuration

$k(T)$66

one obtains

$k(T)$67

which is consistent with the two-projection exact formulas in the earlier work (Song et al., 22 Jul 2025, Bauschke et al., 2023).

An open direction remains from the projection-composition analysis: the paper conjectures that the Ogura–Yamada bound remains sharp for the doubly underrelaxed composition

$k(T)$68

but an exact modulus formula was left open (Bauschke et al., 2023). 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.