Resolvent Compositions in Monotone Operator Theory

• Resolvent compositions are algebraic constructions that combine resolvents of maximally monotone operators to yield fixed-point maps and effective splitting methods.
• They enable cyclic operator compositions whose fixed points encode periodic cycles and can be analyzed using Attouch–Thera duality in Hilbert spaces.
• By integrating bounded linear maps with set-valued operators, resolvent compositions produce explicit resolvent formulas and strong convergence guarantees in algorithmic applications.

Resolvent compositions are constructions in monotone operator theory that derive new operators, fixed-point maps, and splitting schemes from resolvents of maximally monotone operators. In a real Hilbert space $\mathcal H$, the basic object is the resolvent

$J_{\lambda A}=(\Id+\lambda A)^{-1},$

which is single-valued for a maximally monotone $A$ and is firmly nonexpansive. In the cited literature, the term encompasses both ordinary cyclic compositions such as $J_n\circ\cdots\circ J_1$, whose fixed points encode periodic cycles, and operator constructions such as $L\rhd B$ or $R^\lambda_{L\odot A}$, which combine a bounded linear map with a set-valued operator while preserving monotonicity and yielding explicit resolvent formulas (Alwadani, 2024, Combettes, 2022). This suggests that the unifying theme is not a single definition but the preservation of resolvent structure under algebraically useful compositions.

1. Classical resolvents and cyclic composition

Let $A:\mathcal H\rightrightarrows \mathcal H$ be monotone, meaning

$$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$

and maximally monotone when its graph is not properly contained in the graph of a strictly larger monotone operator. For $\lambda>0$, the resolvent is

$J_{\lambda A}=(\Id+\lambda A)^{-1},$

and $J_{\lambda A}=(\Id+\lambda A)^{-1},$0 if and only if $J_{\lambda A}=(\Id+\lambda A)^{-1},$1. By Minty’s theorem, $J_{\lambda A}=(\Id+\lambda A)^{-1},$2 is firmly nonexpansive, hence nonexpansive, with

$J_{\lambda A}=(\Id+\lambda A)^{-1},$3

for all $J_{\lambda A}=(\Id+\lambda A)^{-1},$4 (Alwadani, 2024).

For two operators, one standard construction is

$J_{\lambda A}=(\Id+\lambda A)^{-1},$5

with $J_{\lambda A}=(\Id+\lambda A)^{-1},$6. In the formulation of Wang and Bauschke, the associated fixed-point set is

$J_{\lambda A}=(\Id+\lambda A)^{-1},$7

and

$J_{\lambda A}=(\Id+\lambda A)^{-1},$8

Moreover, if $J_{\lambda A}=(\Id+\lambda A)^{-1},$9, the iterates $A$0 converge weakly to a point of $A$1 (Wang et al., 2010).

These constructions place resolvent compositions at the intersection of fixed-point theory and monotone inclusions. Firm nonexpansiveness supplies the geometric control, while the composition encodes coupled operator equations without leaving the Hilbertian setting.

2. Cycles, fixed-point sets, and product-space geometry

For maximally monotone operators $A$2 with resolvents $A$3, define

$A$4

An $A$5-tuple $A$6 is a cycle if

$A$7

Then $A$8 satisfies the fixed-point equation

$A$9

so $J_n\circ\cdots\circ J_1$0. Conversely, once $J_n\circ\cdots\circ J_1$1 is known, the cycle is recovered by

$J_n\circ\cdots\circ J_1$2

Using Attouch Thera duality, the 2024 analysis shows that cycles defined by resolvent operators can be formulated in Hilbert space as the solution to a fixed-point equation, and relates them explicitly to fixed-point sets of the composition of resolvents (Alwadani, 2024).

The full cycle set is

$J_n\circ\cdots\circ J_1$3

In the product space $J_n\circ\cdots\circ J_1$4, with

$J_n\circ\cdots\circ J_1$5

one has

$J_n\circ\cdots\circ J_1$6

If $J_n\circ\cdots\circ J_1$7, the projection onto the first coordinate,

$J_n\circ\cdots\circ J_1$8

satisfies

$J_n\circ\cdots\circ J_1$9

and $L\rhd B$0 is bijective with inverse

$L\rhd B$1

Accordingly,

$L\rhd B$2

(Alwadani, 2024).

For two resolvents, Wang and Bauschke also establish a finer correspondence between compositions and averages. With

$L\rhd B$3

and the averaged mapping

$L\rhd B$4

the sets $L\rhd B$5, $L\rhd B$6, $L\rhd B$7, and $L\rhd B$8 are in one-to-one, continuous-inverse correspondence. In the convex subdifferential case, this yields the statement that minimizers of proximal-average functions can be recovered from fixed points of compositions of two proximal mappings, and for indicator-function specializations it recovers least-squares solutions from cycles of alternating projections (Wang et al., 2010).

3. Resolvent composition as an operator-valued construction

A second major meaning of resolvent composition combines a bounded linear operator with a set-valued operator while retaining an explicit resolvent. For real Hilbert spaces $L\rhd B$9 and $R^\lambda_{L\odot A}$0, $R^\lambda_{L\odot A}$1, and $R^\lambda_{L\odot A}$2, the resolvent composition is

$R^\lambda_{L\odot A}$3

equivalently

$R^\lambda_{L\odot A}$4

The associated cocomposition is

$R^\lambda_{L\odot A}$5

with

$R^\lambda_{L\odot A}$6

If $R^\lambda_{L\odot A}$7 is monotone and $R^\lambda_{L\odot A}$8, then $R^\lambda_{L\odot A}$9 is monotone; if $A:\mathcal H\rightrightarrows \mathcal H$0 is maximally monotone and $A:\mathcal H\rightrightarrows \mathcal H$1, then $A:\mathcal H\rightrightarrows \mathcal H$2 is maximally monotone. The same framework provides identities for domain, range, and zeros, including

$A:\mathcal H\rightrightarrows \mathcal H$3

and the associativity-type law

$A:\mathcal H\rightrightarrows \mathcal H$4

(Combettes, 2022).

The same paper links operator composition to convex analysis through the proximal composition. For a proper function $A:\mathcal H\rightrightarrows \mathcal H$5, one defines a proximal composition $A:\mathcal H\rightrightarrows \mathcal H$6 so that, under $A:\mathcal H\rightrightarrows \mathcal H$7,

$A:\mathcal H\rightrightarrows \mathcal H$8

Under injectivity of $A:\mathcal H\rightrightarrows \mathcal H$9,

$$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$0

The framework recovers the resolvent average and more general resolvent mixtures as concrete instances (Combettes, 2022).

A parametrized variant is developed through

$$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$1

where $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$2 denotes parallel composition. Its classical resolvent can be written in closed form: $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$3 With the perturbation operator $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$4, one has

$$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$5

If $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$6 and $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$7 is maximally monotone, then both $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$8 and the corresponding cocomposition are maximally monotone. The same analysis provides graph-convergence and $$\langle x-y,\;u-v\rangle\ge 0 \quad\text{whenever }u\in Ax,\;v\in Ay,$$9-Hausdorff convergence as $\lambda>0$0, $\lambda>0$1, or $\lambda>0$2 vary, together with the limits $\lambda>0$3 as $\lambda>0$4 and $\lambda>0$5 as $\lambda>0$6 whenever $\lambda>0$7 (Cornejo, 2024).

A related line studies the resolvent of the parallel composition $\lambda>0$8. Under a qualification condition ensuring maximal monotonicity of $\lambda>0$9, the resolvent admits

$J_{\lambda A}=(\Id+\lambda A)^{-1},$0

where $J_{\lambda A}=(\Id+\lambda A)^{-1},$1 is self-adjoint and strongly monotone. In convex optimization, this yields

$J_{\lambda A}=(\Id+\lambda A)^{-1},$2

together with a generalized Moreau decomposition (Briceño-Arias et al., 2021).

4. Positive-operator resolvent compositions and metric geometry

For positive linear operators, resolvent composition acquires an order-theoretic and metric structure. Let $J_{\lambda A}=(\Id+\lambda A)^{-1},$3 be strictly positive and $J_{\lambda A}=(\Id+\lambda A)^{-1},$4 satisfy $J_{\lambda A}=(\Id+\lambda A)^{-1},$5. The resolvent cocomposition is

$J_{\lambda A}=(\Id+\lambda A)^{-1},$6

and the resolvent composition is

$J_{\lambda A}=(\Id+\lambda A)^{-1},$7

These operations generalize the resolvent average and are monotonicity-preserving (Cornejo, 8 Sep 2025).

Their behavior is controlled by the Löwner partial order. If $J_{\lambda A}=(\Id+\lambda A)^{-1},$8, then

$J_{\lambda A}=(\Id+\lambda A)^{-1},$9

and similarly for $J_{\lambda A}=(\Id+\lambda A)^{-1},$00. They also satisfy the bounds

$J_{\lambda A}=(\Id+\lambda A)^{-1},$01

and, when $J_{\lambda A}=(\Id+\lambda A)^{-1},$02 is bounded below,

$J_{\lambda A}=(\Id+\lambda A)^{-1},$03

For $J_{\lambda A}=(\Id+\lambda A)^{-1},$04,

$J_{\lambda A}=(\Id+\lambda A)^{-1},$05

Asymptotically,

$J_{\lambda A}=(\Id+\lambda A)^{-1},$06

and if $J_{\lambda A}=(\Id+\lambda A)^{-1},$07 is bounded below,

$J_{\lambda A}=(\Id+\lambda A)^{-1},$08

in operator norm (Cornejo, 8 Sep 2025).

The same paper proves nonexpansiveness in the Thompson metric $J_{\lambda A}=(\Id+\lambda A)^{-1},$09: if $J_{\lambda A}=(\Id+\lambda A)^{-1},$10 is bounded below and $J_{\lambda A}=(\Id+\lambda A)^{-1},$11, then

$J_{\lambda A}=(\Id+\lambda A)^{-1},$12

It also introduces a geometric interpolation family $J_{\lambda A}=(\Id+\lambda A)^{-1},$13 that interpolates between $J_{\lambda A}=(\Id+\lambda A)^{-1},$14 and $J_{\lambda A}=(\Id+\lambda A)^{-1},$15, and studies nonlinear equations such as

$J_{\lambda A}=(\Id+\lambda A)^{-1},$16

for which unique fixed points follow from contraction properties in the Thompson metric (Cornejo, 8 Sep 2025).

5. Algorithmic realizations and convergence guarantees

Resolvent compositions also serve as algorithmic primitives for structured monotone inclusions. For

$J_{\lambda A}=(\Id+\lambda A)^{-1},$17

with $J_{\lambda A}=(\Id+\lambda A)^{-1},$18, $J_{\lambda A}=(\Id+\lambda A)^{-1},$19 maximally monotone, and $J_{\lambda A}=(\Id+\lambda A)^{-1},$20 bounded linear, a minimal-lifting scheme operates in $J_{\lambda A}=(\Id+\lambda A)^{-1},$21. With variables $J_{\lambda A}=(\Id+\lambda A)^{-1},$22 and $J_{\lambda A}=(\Id+\lambda A)^{-1},$23, one first forms

$J_{\lambda A}=(\Id+\lambda A)^{-1},$24

then evaluates

$J_{\lambda A}=(\Id+\lambda A)^{-1},$25

updates $J_{\lambda A}=(\Id+\lambda A)^{-1},$26, and finally computes

$J_{\lambda A}=(\Id+\lambda A)^{-1},$27

Each iteration uses each resolvent $J_{\lambda A}=(\Id+\lambda A)^{-1},$28 exactly once, $J_{\lambda A}=(\Id+\lambda A)^{-1},$29 exactly once, and one application each of $J_{\lambda A}=(\Id+\lambda A)^{-1},$30 and $J_{\lambda A}=(\Id+\lambda A)^{-1},$31. When $J_{\lambda A}=(\Id+\lambda A)^{-1},$32, the scheme reduces exactly to the minimal-lifting splitting of Malitsky–Tam. For $J_{\lambda A}=(\Id+\lambda A)^{-1},$33, there is a 1-fold lifting method using only one primal variable $J_{\lambda A}=(\Id+\lambda A)^{-1},$34 and one dual variable $J_{\lambda A}=(\Id+\lambda A)^{-1},$35, while still calling $J_{\lambda A}=(\Id+\lambda A)^{-1},$36, $J_{\lambda A}=(\Id+\lambda A)^{-1},$37, and $J_{\lambda A}=(\Id+\lambda A)^{-1},$38 exactly once per iteration (Briceño-Arias, 2021).

Under the step-size conditions

$J_{\lambda A}=(\Id+\lambda A)^{-1},$39

the sequence $J_{\lambda A}=(\Id+\lambda A)^{-1},$40 converges weakly to a fixed point of the induced operator, and the shadow $J_{\lambda A}=(\Id+\lambda A)^{-1},$41 converges to a solution $J_{\lambda A}=(\Id+\lambda A)^{-1},$42 of

$J_{\lambda A}=(\Id+\lambda A)^{-1},$43

The ergodic averages

$J_{\lambda A}=(\Id+\lambda A)^{-1},$44

satisfy an $J_{\lambda A}=(\Id+\lambda A)^{-1},$45 decay of the natural gap-function. The same type of convergence holds for the 1-fold lifting scheme when $J_{\lambda A}=(\Id+\lambda A)^{-1},$46 (Briceño-Arias, 2021).

A complementary algorithmic direction computes the resolvent of a composite operator

$J_{\lambda A}=(\Id+\lambda A)^{-1},$47

To find $J_{\lambda A}=(\Id+\lambda A)^{-1},$48, Adly and Le introduce the fixed-point map

$J_{\lambda A}=(\Id+\lambda A)^{-1},$49

with recovery formula

$J_{\lambda A}=(\Id+\lambda A)^{-1},$50

If $J_{\lambda A}=(\Id+\lambda A)^{-1},$51, then $J_{\lambda A}=(\Id+\lambda A)^{-1},$52 is nonexpansive; if $J_{\lambda A}=(\Id+\lambda A)^{-1},$53, it is a strict contraction. The Krasnosel'skii–Mann iteration

$J_{\lambda A}=(\Id+\lambda A)^{-1},$54

converges weakly under $J_{\lambda A}=(\Id+\lambda A)^{-1},$55, strongly if $J_{\lambda A}=(\Id+\lambda A)^{-1},$56, and linearly when $J_{\lambda A}=(\Id+\lambda A)^{-1},$57. Each iteration requires one evaluation of $J_{\lambda A}=(\Id+\lambda A)^{-1},$58, one $J_{\lambda A}=(\Id+\lambda A)^{-1},$59-multiply, and one $J_{\lambda A}=(\Id+\lambda A)^{-1},$60-multiply (Adly et al., 1 Feb 2025).

6. Applications, special cases, and unresolved directions

The two-operator case is the main testing ground. If $J_{\lambda A}=(\Id+\lambda A)^{-1},$61 and $J_{\lambda A}=(\Id+\lambda A)^{-1},$62 are normal-cone operators of closed convex sets $J_{\lambda A}=(\Id+\lambda A)^{-1},$63, then

$J_{\lambda A}=(\Id+\lambda A)^{-1},$64

The fixed points of $J_{\lambda A}=(\Id+\lambda A)^{-1},$65 are exactly those $J_{\lambda A}=(\Id+\lambda A)^{-1},$66 with $J_{\lambda A}=(\Id+\lambda A)^{-1},$67, and the corresponding cycle is $J_{\lambda A}=(\Id+\lambda A)^{-1},$68 (Alwadani, 2024). In the earlier two-resolvent theory, the set of least-squares minimizers of $J_{\lambda A}=(\Id+\lambda A)^{-1},$69, equivalently $J_{\lambda A}=(\Id+\lambda A)^{-1},$70, is in natural bijection with the alternating-projection cycles

$J_{\lambda A}=(\Id+\lambda A)^{-1},$71

which gives a partial answer to Byrne’s question in the two-set case (Wang et al., 2010).

Linear-operator-based compositions broaden the application range. The resolvent average is recovered by taking a diagonal product operator and an embedding $J_{\lambda A}=(\Id+\lambda A)^{-1},$72, leading to

$J_{\lambda A}=(\Id+\lambda A)^{-1},$73

More generally, direct sums of maps $J_{\lambda A}=(\Id+\lambda A)^{-1},$74 yield the resolvent mixture

$J_{\lambda A}=(\Id+\lambda A)^{-1},$75

Examples given in the literature include projectors onto subspaces, frame-analysis constructions with diagonal proximal maps, relaxations of inconsistent split problems, and convex optimization models based on $J_{\lambda A}=(\Id+\lambda A)^{-1},$76 (Combettes, 2022).

Algorithmically, the same perspective connects classical splitting methods. In the parametrized family $J_{\lambda A}=(\Id+\lambda A)^{-1},$77, the choice $J_{\lambda A}=(\Id+\lambda A)^{-1},$78, $J_{\lambda A}=(\Id+\lambda A)^{-1},$79, and $J_{\lambda A}=(\Id+\lambda A)^{-1},$80 gives

$J_{\lambda A}=(\Id+\lambda A)^{-1},$81

which is exactly the classical Douglas–Rachford splitting operator (Cornejo, 2024). In optimization, the composite Lasso example

$J_{\lambda A}=(\Id+\lambda A)^{-1},$82

has optimality condition $J_{\lambda A}=(\Id+\lambda A)^{-1},$83; in the notation of the minimal-lifting scheme this is the case $J_{\lambda A}=(\Id+\lambda A)^{-1},$84, and the method reduces to the primal–dual algorithm of Chambolle–Pock (Briceño-Arias, 2021).

A recurring structural issue is order dependence. For more than two resolvents, extension is nontrivial; cycles may depend on the order of composition and no simple average-to-composition recovery is known. The multi-set theory remains open (Wang et al., 2010). This suggests that the two principal branches of the subject—cyclic compositions of resolvents and linear-operator-induced resolvent compositions—share a common fixed-point and monotonicity calculus, but diverge sharply in their higher-order geometry and in the types of invariance that remain available.