---
title: Resolvent Compositions in Monotone Operator Theory
url: https://www.emergentmind.com/topics/resolvent-compositions
type: topic
---

# Resolvent Compositions in Monotone Operator Theory

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 [2406.01041][2209.06200]. 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 \(u=J_{\lambda A}(x)\) if and only if \(x-u\in \lambda A(u)\). By Minty’s theorem, \(J_{\lambda A}\) is firmly nonexpansive, hence nonexpansive, with
\[
\|J_{\lambda A}x - J_{\lambda A}y\|^2
+\|(x - J_{\lambda A}x) - (y - J_{\lambda A}y)\|^2
\le \|x-y\|^2
\]
for all \(x,y\in\mathcal H\) [2406.01041].

For two operators, one standard construction is
\[
T=J_{A/\beta_1}\circ J_{B/\beta_2},
\]
with \(\beta_1,\beta_2>0\). In the formulation of Wang and Bauschke, the associated fixed-point set is
\[
E=\{x\in H\mid 0\in A(x)/\beta_2 + B(x)/\beta_1\}=(A/\beta_2+B/\beta_1)^{-1}(0),
\]
and
\[
\Fix(J_{A/\beta_1}J_{B/\beta_2})=E.
\]
Moreover, if \(E\neq\varnothing\), the iterates \(x_{n+1}=J_{A/\beta_1}(J_{B/\beta_2}(x_n))\) converge weakly to a point of \(E\) [1003.4793].

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 \(A_1,\dots,A_n\) with resolvents \(J_i=J_{\lambda_iA_i}\), define
\[
T=J_n\circ J_{n-1}\circ\cdots\circ J_1.
\]
An \(n\)-tuple \((x_1,\dots,x_n)\in\mathcal H^n\) is a cycle if
\[
x_{i+1}=J_i(x_i),\quad i=1,\dots,n,\qquad x_{n+1}:=x_1.
\]
Then \(x_1\) satisfies the fixed-point equation
\[
x_1=(J_n\circ\cdots\circ J_1)(x_1),
\]
so \(x_1\in \Fix\,T\). Conversely, once \(x_1\in\Fix\,T\) is known, the cycle is recovered by
\[
x_2=J_1(x_1),\;x_3=J_2(x_2),\;\dots,\;x_n=J_{n-1}(x_{n-1}),\;x_1=J_n(x_n).
\]
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 [2406.01041].

The full cycle set is
\[
C=\bigl\{(x_1,\dots,x_n)\in\mathcal H^n\mid x_{i+1}=J_i(x_i),\;x_{n+1}=x_1\bigr\}.
\]
In the product space \(\mathcal X=\mathcal H^n\), with
\[
J_A(x_1,\dots,x_n)=(J_1x_1,\dots,J_nx_n),\qquad
R(x_1,\dots,x_n)=(x_n,x_1,\dots,x_{n-1}),
\]
one has
\[
C=\Fix(J_A\circ R).
\]
If \(C\neq\varnothing\), the projection onto the first coordinate,
\[
Q(x_1,\dots,x_n)=x_1,
\]
satisfies
\[
Q(C)=\Fix\,T,
\]
and \(Q:C\to\Fix\,T\) is bijective with inverse
\[
Q^{-1}(x)=\bigl(x,\;J_1x,\;J_2J_1x,\;\dots,\;J_{n-1}\cdots J_1x\bigr).
\]
Accordingly,
\[
\Fix\,T\neq\varnothing \Longleftrightarrow C\neq\varnothing
\]
[2406.01041].

For two resolvents, Wang and Bauschke also establish a finer correspondence between compositions and averages. With
\[
E=\Fix(J_{A/\beta_1}J_{B/\beta_2}),\quad
F=\Fix(J_{B/\beta_2}J_{A/\beta_1}),\quad
S=\{(x,y)\mid x=J_{A/\beta_1}(y),\;y=J_{B/\beta_2}(x)\},
\]
and the averaged mapping
\[
R=\lambda_1J_{A/\beta_1}+\lambda_2J_{B/\beta_2},
\]
the sets \(E\), \(F\), \(S\), and \(\Fix R\) 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 [1003.4793].

## 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 \(H\) and \(G\), \(L\in\mathcal B(H,G)\), and \(B:G\to 2^G\), the resolvent composition is
\[
L\rhd B
=
L^*\Bigl(L\circ J_B\circ L^*\Bigr)^{-1}-\Id_H,
\]
equivalently
\[
J_{L\rhd B}=L\circ J_B\circ L^*.
\]
The associated cocomposition is
\[
L\lhd B=(L\rhd B)^{-1},
\]
with
\[
J_{L\lhd B}=\Id_H-L\circ L^*+L\circ J_B\circ L^*.
\]
If \(B\) is monotone and \(\|L\|\le 1\), then \(L\rhd B\) is monotone; if \(B\) is maximally monotone and \(\|L\|\le 1\), then \(L\rhd B\) is maximally monotone. The same framework provides identities for domain, range, and zeros, including
\[
\zer(L\rhd B)=\{x\in H\mid L^*x\in \zer B\},
\]
and the associativity-type law
\[
Q\rhd(L\rhd M)=(L\circ Q)\rhd M
\]
[2209.06200].

The same paper links operator composition to convex analysis through the proximal composition. For a proper function \(g:G\to(-\infty,+\infty]\), one defines a proximal composition \(L\rhd_{\mathrm{prox}} g\) so that, under \(\|L\|\le 1\),
\[
\partial(L\rhd_{\mathrm{prox}} g)=L\rhd(\partial g),
\qquad
\prox_{L\rhd_{\mathrm{prox}} g}=L\circ \prox_g\circ L^*.
\]
Under injectivity of \(L\),
\[
\partial(f\circ L)=L^*\rhd(\partial f).
\]
The framework recovers the resolvent average and more general resolvent mixtures as concrete instances [2209.06200].

A parametrized variant is developed through
\[
R^\lambda_{L\odot A}
=
L^*\triangleright\bigl(A+\lambda^{-1}\Id_G\bigr)-\lambda^{-1}\Id_H,
\]
where \(L^*\triangleright C=(L^*\circ C^{-1}\circ L)^{-1}\) denotes parallel composition. Its classical resolvent can be written in closed form:
\[
J^{\,R^\lambda}_{\gamma}
=
(\Id_H+\gamma R^\lambda)^{-1}
=
L^*\circ J^A_\gamma\circ L.
\]
With the perturbation operator \(M=\Id_G-\lambda LL^*\), one has
\[
R^\lambda_{L\odot A}=L^*\triangleright(A+\lambda^{-1}M).
\]
If \(\|L\|\le 1\) and \(A\) is maximally monotone, then both \(R^\lambda_{L\odot A}\) and the corresponding cocomposition are maximally monotone. The same analysis provides graph-convergence and \(\rho\)-Hausdorff convergence as \(L\), \(A\), or \(\lambda\) vary, together with the limits \(R^\lambda_{L\odot A}\to L^*\circ A\circ L\) as \(\lambda\downarrow 0\) and \(R^\lambda_{L\odot A}\to 0\) as \(\lambda\uparrow+\infty\) whenever \(0\in\zer A\) [2410.01090].

A related line studies the resolvent of the parallel composition \(L\triangleright A=(L\circ A^{-1}\circ L^*)^{-1}\). Under a qualification condition ensuring maximal monotonicity of \(LA^{-1}L^*\), the resolvent admits
\[
J_{U(L\triangleright A)}=L\,(A+L^*U^{-1}L)^{-1}L^*U^{-1},
\]
where \(U\) is self-adjoint and strongly monotone. In convex optimization, this yields
\[
\prox_{\gamma(f\triangleright L)}(u)
=
L(\Id+\gamma L^*L+\gamma\partial f)^{-1}L^*u
=
u-L^*\prox_{(1/\gamma)f^*}(Lu/\gamma),
\]
together with a generalized Moreau decomposition [2109.06771].

## 4. Positive-operator resolvent compositions and metric geometry

For positive linear operators, resolvent composition acquires an order-theoretic and metric structure. Let \(A\in\mathcal S(G)\) be strictly positive and \(L\in(H,G)\) satisfy \(0<\|L\|\le 1\). The resolvent cocomposition is
\[
R^{\rm co}_{A,L}(\gamma)
=
L^*\bigl(A^{-1}+\gamma(\Id_G-LL^*)\bigr)^{-1}L
\in\mathscr P(H),
\]
and the resolvent composition is
\[
R_{A,L}(\gamma)
=
\Bigl(L\proxcc{1/\gamma}(A^{-1})\Bigr)^{-1}
=
L^*\pushfwd\bigl(A+\tfrac1\gamma\Id_G\bigr)-\tfrac1\gamma\Id_H.
\]
These operations generalize the resolvent average and are monotonicity-preserving [2509.07251].

Their behavior is controlled by the Löwner partial order. If \(A\preceq B\), then
\[
R^{\rm co}_{A,L}(\gamma)\preceq R^{\rm co}_{B,L}(\gamma),
\]
and similarly for \(R_{A,L}(\gamma)\). They also satisfy the bounds
\[
L^*\circ A\circ L
\succeq
R^{\rm co}_{A,L}(\gamma)
\succeq
\frac1{1+\gamma\|A\|}(L^*\circ A\circ L),
\]
and, when \(L\) is bounded below,
\[
L^*\pushfwd A
\preceq
R_{A,L}(\gamma)
\preceq
\Bigl(1+\frac{\|A^{-1}\|}{\gamma}\Bigr)L^*\pushfwd A.
\]
For \(0<\rho<\gamma\),
\[
R^{\rm co}_{A,L}(\gamma)\preceq R^{\rm co}_{A,L}(\rho),
\qquad
R_{A,L}(\gamma)\preceq R_{A,L}(\rho).
\]
Asymptotically,
\[
R^{\rm co}_{A,L}(\gamma)\to L^*\circ A\circ L
\quad\text{as }\gamma\to 0^+,
\]
and if \(L\) is bounded below,
\[
R_{A,L}(\gamma)\to L^*\pushfwd A
\quad\text{as }\gamma\to+\infty
\]
in operator norm [2509.07251].

The same paper proves nonexpansiveness in the Thompson metric \(d_T\): if \(L\) is bounded below and \(0<\|L\|\le 1\), then
\[
d_T(R^{\rm co}_{A,L}(\gamma),R^{\rm co}_{B,L}(\gamma))
\le d_T(A,B),
\qquad
d_T(R_{A,L}(\gamma),R_{B,L}(\gamma))
\le d_T(A,B).
\]
It also introduces a geometric interpolation family \(\mathcal L_\gamma(L,A)\) that interpolates between \(L^*\pushfwd A\) and \(L^*\circ A\circ L\), and studies nonlinear equations such as
\[
X=R^{\rm co}_{X\#_t B,L}(\gamma),
\qquad
X=R^{\rm co}_{B^*X^tB,L}(\gamma),
\]
for which unique fixed points follow from contraction properties in the Thompson metric [2509.07251].

## 5. Algorithmic realizations and convergence guarantees

Resolvent compositions also serve as algorithmic primitives for structured monotone inclusions. For
\[
0\in \sum_{i=1}^n A_i x + K^*B(Kx),
\]
with \(A_i:\mathcal H\rightrightarrows\mathcal H\), \(B:\mathcal G\rightrightarrows\mathcal G\) maximally monotone, and \(K:\mathcal H\to\mathcal G\) bounded linear, a minimal-lifting scheme operates in \(\mathcal H^n\times\mathcal G\). With variables \(u=(u_1,\dots,u_n)\) and \(w\), one first forms
\[
x_1^k=u_1^k-\gamma_1K^*w^k,\qquad
x_i^k=u_i^k+\gamma_i(x_{i-1}^k-u_{i-1}^k)\;\;(i=2,\dots,n),
\]
then evaluates
\[
p_i^k=J_{\gamma_iA_i}(2x_i^k-u_i^k),
\]
updates \(u_i^{k+1}=u_i^k+(\lambda/\gamma_i)(p_i^k-x_i^k)\), and finally computes
\[
q^k=J_{\delta B}(Kx_n^k+w^k),\qquad
w^{k+1}=w^k+\mu(Kx_n^k-q^k).
\]
Each iteration uses each resolvent \(J_{\gamma_iA_i}\) exactly once, \(J_{\delta B}\) exactly once, and one application each of \(K\) and \(K^*\). When \(K=I\), the scheme reduces exactly to the minimal-lifting splitting of Malitsky–Tam. For \(n=2\), there is a 1-fold lifting method using only one primal variable \(u\) and one dual variable \(w\), while still calling \(J_{A_1}\), \(J_{A_2}\), and \(J_B\) exactly once per iteration [2111.09757].

Under the step-size conditions
\[
0<\gamma_i\|K\|^2<1,\qquad 0<\delta\|K\|^2<1,\qquad 0<\lambda,\mu<2,
\]
the sequence \((u^k,w^k)\) converges weakly to a fixed point of the induced operator, and the shadow \(x_n^k\) converges to a solution \(x^*\) of
\[
0\in \sum_{i=1}^n A_i(x^*)+K^*B(Kx^*).
\]
The ergodic averages
\[
\bar x^k=\frac1k\sum_{j=1}^k x_n^j
\]
satisfy an \(O(1/k)\) decay of the natural gap-function. The same type of convergence holds for the 1-fold lifting scheme when \(n=2\) [2111.09757].

A complementary algorithmic direction computes the resolvent of a composite operator
\[
T=C^T\mathcal M C.
\]
To find \(x=J_{\lambda C^T\mathcal M C}(y)\), Adly and Le introduce the fixed-point map
\[
\mathcal Q(v)
=
\mathcal M_{1/\mu}\!\Bigl(Cy+(I-\lambda\mu CC^T)v\Bigr)
=
\frac1\mu\bigl(I-J_{(1/\mu)\mathcal M}\bigr)\Bigl(Cy+(I-\lambda\mu CC^T)v\Bigr),
\]
with recovery formula
\[
J_{\lambda C^T\mathcal M C}(y)=y-\lambda\mu C^Tv.
\]
If \(0<\lambda\mu\le 2/\|C\|^2\), then \(\mathcal Q\) is nonexpansive; if \(\|I-\lambda\mu CC^T\|<1\), it is a strict contraction. The Krasnosel'skii–Mann iteration
\[
v_{k+1}=(1-\alpha_k)v_k+\alpha_k\mathcal Q(v_k),\qquad
x_k=y-\lambda\mu C^Tv_k,
\]
converges weakly under \(\sum_k\alpha_k(1-\alpha_k)=+\infty\), strongly if \(\inf_k\alpha_k>0\), and linearly when \(\|I-\lambda\mu CC^T\|<1\). Each iteration requires one evaluation of \(J_{(1/\mu)\mathcal M}\), one \(C\)-multiply, and one \(C^T\)-multiply [2502.01664].

## 6. Applications, special cases, and unresolved directions

The two-operator case is the main testing ground. If \(A=\partial\iota_C\) and \(B=\partial\iota_D\) are normal-cone operators of closed convex sets \(C,D\subset\mathcal H\), then
\[
J_A=P_C,\qquad J_B=P_D,\qquad T=P_D\circ P_C.
\]
The fixed points of \(T\) are exactly those \(x\) with \(x=P_D(P_Cx)\), and the corresponding cycle is \((x,P_Cx)\) [2406.01041]. In the earlier two-resolvent theory, the set of least-squares minimizers of \(\lambda_1d_C^2+\lambda_2d_D^2\), equivalently \(\Fix(\lambda_1P_C+\lambda_2P_D)\), is in natural bijection with the alternating-projection cycles
\[
S=\{(x,y):x=P_C(y),\;y=P_D(x)\},
\]
which gives a partial answer to Byrne’s question in the two-set case [1003.4793].

Linear-operator-based compositions broaden the application range. The resolvent average is recovered by taking a diagonal product operator and an embedding \(L:x\mapsto(x,\dots,x)\), leading to
\[
L\rhd B=\Bigl(\sum_{k=1}^p \omega_k J_{B_k}\Bigr)-\Id_H.
\]
More generally, direct sums of maps \(L_k\) yield the resolvent mixture
\[
x\longmapsto \Bigl(\sum_{k=1}^p \omega_k L_k^*J_{B_k}L_k\Bigr)-\Id_H.
\]
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 \(g(L^*x)\) [2209.06200].

Algorithmically, the same perspective connects classical splitting methods. In the parametrized family \(R^\lambda_{L\odot A}\), the choice \(G=H\oplus H\), \(Lx=(x,x)\), and \(A(u,v)=A_1u\oplus A_2v\) gives
\[
R^\lambda_{L\odot A}(x)=J^{A_1}_\lambda(x)+J^{A_2}_\lambda(x)-x,
\]
which is exactly the classical Douglas–Rachford splitting operator [2410.01090]. In optimization, the composite Lasso example
\[
\min_{x\in\mathbb R^d} f(x)+g(Ax)
\]
has optimality condition \(0\in \partial f(x)+A^T\partial g(Ax)\); in the notation of the minimal-lifting scheme this is the case \(n=1\), and the method reduces to the primal–dual algorithm of Chambolle–Pock [2111.09757].

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 [1003.4793]. 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.

Source: https://www.emergentmind.com/topics/resolvent-compositions