# Resolvents and an anchor point

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

We keep the standing data of [Minimal invariant sets and diametral sequences](minimal-invariant-sets-and-diametral-sequences.md): a minimal weakly compact convex set \(K\) of diameter \(1\) in a reflexive space \(X\), with a fixed-point-free nonexpansive \(T:K\to K\), the dense sequence \((z_i)\), \(Y=\overline{\operatorname{span}}(K-K)\), and the free ultrafilter \(\mathcal U\) on \(\mathbb N\). This lesson produces a point \(x\in K\) with a strong return property (Lemma 6.1) [OpenAI-K, Section 3]: we may choose nonexpansive maps \(G_1,G_2,\dots\) one at a time, each required, at a precision chosen after seeing all previous choices, to make \(G_i(x)\) an approximate fixed point; and after finitely many steps a convex combination of the points \(G_i(x)\) is as close to \(x\) as we wish. The maps come from a family generated by the resolvents of \(T\) (Section 1), and \(x\) is a fixed point of an idempotent in a compact semigroup of maps (Section 5), found by a lemma of Ellis.

We use: Lemmas 1.1, 1.2 and 2.1 of the first lesson; Tychonoff's theorem, [Theorem 2.3 of the weak topologies lesson](course:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#OA-FND-WT-02), and Mazur's theorem there (Theorem 4.1); the strict separation theorem, [Theorem 6.3 of the Banach space lesson](course:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#OA-FND-HB-06); and Zorn's lemma.

## 1. Resolvents

**Lemma 1.1.** For every \(n\ge1\) and \(a\in K\) there is exactly one \(R_n(a)\in K\) with
\[
\begin{aligned}
&R_n(a)=\frac an\\
&\quad +\Bigl(1-\frac1n\Bigr)TR_n(a).
\end{aligned}\tag{1.1}
\]
The map \(R_n:K\to K\) is nonexpansive, and for \(a\in K\)
\[
\begin{aligned}
&\|TR_n(a)-R_n(a)\|\le\frac1n,\\
&\|R_n(a)-a\|\\
&\quad \le(n-1)\|Ta-a\|.
\end{aligned}\tag{1.2}
\]

**Proof.** \(R_1(a)=a\). For \(n>1\), the map \(u\mapsto\frac an+(1-\frac1n)Tu\) sends \(K\) into \(K\) by convexity and has Lipschitz constant \(1-\frac1n<1\); \(K\) is closed in a Banach space, hence complete, and Banach's contraction principle gives exactly one solution. For \(u=R_n(a)\) and \(u'=R_n(a')\), subtracting the two equations shows that \(\|u-u'\|\) is at most \(\frac1n\|a-a'\|+(1-\frac1n)\|u-u'\|\), so \(\|u-u'\|\le\|a-a'\|\). From (1.1), \(Tu-u=\frac1n(Tu-a)\), and \(\|Tu-a\|\le\operatorname{diam}K=1\). Finally \(u-a=(1-\frac1n)(Tu-a)\), so \(\|u-a\|\) is at most \((1-\frac1n)(\|Tu-Ta\|+\|Ta-a\|)\), which is at most \((1-\frac1n)(\|u-a\|+\|Ta-a\|)\), and rearranging gives the second estimate. \(\square\)

## 2. A family of maps with controlled displacement

Let \(\mathcal G\) be the smallest set of maps \(K\to K\) containing the identity and all \(R_n\) and closed under composition and pointwise finite convex combinations. By convexity of \(K\) these are self-maps of \(K\).

**Lemma 2.1.** Every \(G\in\mathcal G\) is nonexpansive, and there is a constant \(c_G\ge0\) with
\[
\begin{gathered}
\|G(a)-a\|\le c_G\|Ta-a\|\\
(a\in K).
\end{gathered}\tag{2.1}
\]

**Proof.** Both properties hold for the identity (\(c=0\)) and for \(R_n\) (\(c=n-1\), by (1.2)), and they are preserved by the two operations. A convex combination \(\sum_r\lambda_rG_r\) is nonexpansive and satisfies (2.1) with \(\sum_r\lambda_rc_{G_r}\). For a composition,
\[
\begin{aligned}
&\|G_1(G_2(a))-a\|\\
&\quad \le\|G_1(G_2(a))-G_1(a)\|\\
&\quad +\|G_1(a)-a\|\le\|G_2(a)-a\|\\
&\quad +\|G_1(a)-a\|,
\end{aligned}
\]
so \(c_{G_1\circ G_2}=c_{G_1}+c_{G_2}\) works. \(\square\)

For \(m\ge1\) let \(\mathcal H_m=\{R_n\circ G:n\ge m,\ G\in\mathcal G\}\). These sets are nonempty, contained in \(\mathcal G\), decreasing in \(m\), and by (1.2)
\[
\begin{gathered}
\|Th(a)-h(a)\|\le\frac1m\\
(h\in\mathcal H_m,\ a\in K).
\end{gathered}\tag{2.2}
\]
Every value of a map in \(\mathcal H_m\) is a \(\frac1m\)-approximate fixed point.

## 3. The pointwise weak topology

Let \(K^K\) be the set of all maps \(K\to K\), with the product topology of the weak topology on each factor (*pointwise weak topology*). By Lemma 1.1 of the first lesson and Tychonoff's theorem it is compact, and it is Hausdorff. Closures in this space are written \(\overline{\mathcal B}^{\rm pw}\).

**Lemma 3.1** (finite approximation). Let \(\mathcal B\subseteq K^K\) be a nonempty set of nonexpansive maps closed under pointwise convex combinations. Every \(h\in\overline{\mathcal B}^{\rm pw}\) is nonexpansive. For \(a_1,\dots,a_q\in K\) and \(\delta>0\) there is \(g\in\mathcal B\) with \(\|g(a_i)-h(a_i)\|<\delta\) for all \(i\).

**Proof.** For fixed \(a,b\in K\), the set of \(h\) with \(\|h(a)-h(b)\|\le\|a-b\|\) is closed in \(K^K\): evaluation at \(a\) and \(b\) is continuous, subtraction is weakly continuous, and closed balls are weakly closed. Intersecting over all pairs gives the first statement. For the second, the set \(V=\{(g(a_1),\dots,g(a_q)):g\in\mathcal B\}\subseteq X^q\) is convex, and \((h(a_1),\dots,h(a_q))\) lies in its closure for the product of the weak topologies, by continuity of evaluation. On \(X^q\) with the maximum norm, every bounded linear functional is a sum of functionals of the coordinates, so this product topology is the weak topology of \(X^q\). By Mazur's theorem the weak closure of the convex set \(V\) equals its norm closure, and a point of \(V\) within \(\delta\) of the tuple in the maximum norm gives \(g\). \(\square\)

## 4. An idempotent

A *semigroup* is a set with an associative multiplication.

**Lemma 4.1** (Ellis). Let \(S\) be a nonempty compact Hausdorff semigroup in which every right translation \(h\mapsto hp\) is continuous. Then \(S\) contains an element \(p\) with \(p^2=p\).

**Proof.** Order the nonempty closed subsemigroups of \(S\) by reverse inclusion. The intersection of a chain is nonempty by compactness and is again a closed subsemigroup, so Zorn's lemma gives a minimal one, \(S'\). Fix \(p\in S'\). The set \(S'p\) is compact, being the image of \(S'\) under right translation, hence closed; it lies in \(S'\) and is a subsemigroup, since \((ap)(bp)=(apb)p\). By minimality \(S'p=S'\), so \(E=\{h\in S':hp=p\}\) is nonempty. It is closed, as the preimage of \(\{p\}\) under right translation, and a subsemigroup, since \(hp=kp=p\) gives \((hk)p=h(kp)=p\). By minimality \(E=S'\); in particular \(pp=p\). \(\square\)

## 5. The anchor point

Put \(A_m=\overline{\operatorname{conv}\mathcal H_m}^{\rm pw}\), where the convex hull is taken pointwise, and \(A=\bigcap_{m\ge1}A_m\).

**Proposition 5.1.** \(A\) is a nonempty compact semigroup of nonexpansive maps under composition and contains an idempotent \(P\). There is \(x\in K\) with \(P(x)=x\) such that the nonempty weakly compact set
\[
\begin{gathered}
D_x=\bigcap_{m\ge1}B_m,\\
B_m=\overline{\{h(x):h\in\mathcal H_m\}}^{\rm weak},
\end{gathered}
\]
satisfies \(x\in\overline{\operatorname{conv}}\,D_x\) (norm closure).

**Proof.** The sets \(A_m\) are nonempty, closed and decreasing in the compact space \(K^K\), so \(A\) is nonempty and compact; its members are nonexpansive by Lemma 3.1, since \(\operatorname{conv}\mathcal H_m\subseteq\mathcal G\) consists of nonexpansive maps.

*\(\operatorname{conv}\mathcal H_m\) is closed under composition.* If \(g=\sum_r\lambda_r\,R_{n_r}\circ G_r\) with \(n_r\ge m\) and \(k\in\operatorname{conv}\mathcal H_m\subseteq\mathcal G\), then \(g\circ k=\sum_r\lambda_r\,R_{n_r}\circ(G_r\circ k)\) pointwise, and \(G_r\circ k\in\mathcal G\).

*\(A_m\) is closed under composition.* Let \(h,k\in A_m\), let \(a_1,\dots,a_q\in K\) and \(\delta>0\). By Lemma 3.1 choose \(k_\delta\in\operatorname{conv}\mathcal H_m\) with \(\|k_\delta(a_i)-k(a_i)\|<\delta\), and then \(h_\delta\in\operatorname{conv}\mathcal H_m\) with \(\|h_\delta(k(a_i))-h(k(a_i))\|<\delta\). Since \(h_\delta\) is nonexpansive, \(\|h_\delta(k_\delta(a_i))-h(k(a_i))\|<2\delta\). So every basic neighbourhood of \(h\circ k\) in \(K^K\), which restricts finitely many values in the weak topology, contains some \(h_\delta\circ k_\delta\in\operatorname{conv}\mathcal H_m\); thus \(h\circ k\in A_m\). Hence \(A\) is a semigroup.

Right translation by \(k\in A\) is continuous on \(K^K\): the value of \(h\circ k\) at \(a\) is the value of \(h\) at the fixed point \(k(a)\). Lemma 4.1 gives an idempotent \(P\in A\). For any \(a\in K\), \(x=P(a)\) satisfies \(P(x)=x\).

The sets \(B_m\) are nonempty, weakly closed subsets of the weakly compact set \(K\), and decreasing, so \(D_x\) is nonempty and weakly compact. Since \(P\in A_m\), Lemma 3.1 at the single point \(x\) gives, for every \(\delta>0\), some \(g=\sum_r\lambda_rh_r\in\operatorname{conv}\mathcal H_m\) with \(\|g(x)-x\|<\delta\); so \(x\) lies in the norm closure of \(\operatorname{conv}\{h(x):h\in\mathcal H_m\}\subseteq\overline{\operatorname{conv}}\,B_m\). Hence for every \(\ell\in X^*\),
\[
\max_{z\in B_m}\ell(z)\ge\ell(x),
\]
the maximum being attained because \(B_m\) is weakly compact and \(\ell\) weakly continuous. The sets \(B_m\cap\{\ell\ge\ell(x)\}\) are nonempty, weakly compact and decreasing, so some point of \(D_x\) satisfies \(\ell\ge\ell(x)\). If \(x\) were not in \(\overline{\operatorname{conv}}\,D_x\), the strict separation theorem would give \(\ell\in X^*\) with \(\ell(x)>\sup_{D_x}\ell\), a contradiction. \(\square\)

## 6. Choices with prescribed precision

Fix \(x\) as in Proposition 5.1 and a metric \(\varrho\) inducing the weak topology on \(K\) (Lemma 2.1(b) of the first lesson).

**Lemma 6.1** (adaptive selection). Let \(\tau>0\). Consider a procedure that, for \(i=1,2,\dots\), first receives an integer \(m_i\ge1\), which may depend on everything chosen before, and then chooses \(G_i\in\mathcal H_{m_i}\). There is a rule for these choices such that, whatever the \(m_i\), after finitely many steps \(M\) some convex combination satisfies
\[
\begin{gathered}
\Bigl\|\sum_{i=1}^M\beta_iG_i(x)-x\Bigr\|<\tau,\\
\beta_i\ge0,\quad\sum_i\beta_i=1.
\end{gathered}\tag{6.1}
\]

**Proof.** By Proposition 5.1 choose \(d_1,\dots,d_q\in D_x\) and convex coefficients \(\lambda_r\) with \(\|\sum_r\lambda_rd_r-x\|<\frac\tau2\). At step \(i\), let \(r(i)\in\{1,\dots,q\}\) be congruent to \(i\) modulo \(q\). Since \(d_{r(i)}\in B_{m_i}\), there is \(G_i\in\mathcal H_{m_i}\) with \(\varrho(G_i(x),d_{r(i)})<\frac1i\); choose it. Stop at the first \(M\) for which some convex combination of \(G_1(x),\dots,G_M(x)\) lies within \(\tau\) of \(x\).

The procedure stops. Otherwise let \(C_0\) be the norm-closed convex hull of all \(G_i(x)\). For each \(r\), the \(G_i(x)\) with \(r(i)=r\) converge weakly to \(d_r\), and \(C_0\) is weakly closed by Mazur's theorem; so \(d_r\in C_0\) for all \(r\), and \(\sum_r\lambda_rd_r\in C_0\). Hence a finite convex combination of the \(G_i(x)\) lies within \(\frac\tau2\) of \(\sum_r\lambda_rd_r\), and so within \(\tau\) of \(x\), and the procedure would have stopped at the largest index occurring in it. \(\square\)

By (2.2), the precision \(m_i\) prescribes that \(G_i(x)\) is a \(\frac1{m_i}\)-approximate fixed point. Removing the zero coefficients from (6.1) leaves a nonempty list with positive coefficients in the order of choice.

**Lemma 6.2.** The points \(b_j=R_j(x)\), \(j\ge1\), satisfy \(\|Tb_j-b_j\|\le\frac1j\) and \(\|G(b_j)-b_j\|\le\frac{c_G}j\) for every \(G\in\mathcal G\).

**Proof.** (1.2) and (2.1). \(\square\)

So finitely many fixed maps of \(\mathcal G\) move a common late point \(b_j\) arbitrarily little.

## 7. Exercises

**Exercise 7.1** (easy). Show that the constant \(c_G\) in Lemma 2.1 depends on the way \(G\) is built from the generators, and that the estimate (2.1) is all that is used.

**Exercise 7.2** (easy). Let \(S=\{a,b\}\) with the discrete topology and the product \(st=t\) for all \(s,t\). Show that \(S\) is a compact semigroup with continuous right translations, that both elements are idempotents, and that \(S\) has no identity element.

**Exercise 7.3** (medium). In \(\mathbb R^2\) with the maximum norm let \(Q=[-1,1]^2\) and \(T(a,b)=(a,|a|)\). Show that \(T\) is a nonexpansive self-map of \(Q\) whose fixed points \((-1,1)\) and \((1,1)\) have a midpoint that is not an approximate fixed point at all. Explain why, in Proposition 5.1, the values \(h(x)\), \(h\in\mathcal H_m\), are approximate fixed points while \(x=P(x)\) need not be one.

## 8. Solutions

**7.1.** For example \(R_2\circ R_2\) gets the constant \(2\) from the composition rule, although the composite might satisfy (2.1) with a smaller constant. The arguments use only some constant with (2.1), fixed once the representation of \(G\) is fixed.

**7.2.** \((st)u=u=s(tu)\), so the product is associative; a finite discrete space is compact Hausdorff and every map on it is continuous. \(ss=s\) for both elements. An identity \(e\) would satisfy \(s=se=e\) for every \(s\), impossible with two elements.

**7.3.**
\[
\begin{aligned}
&\|T(a,b)-T(a',b')\|_\infty\\
&\quad=\max\{|a-a'|,||a|-|a'||\}\\
&\quad=|a-a'|\le\|(a,b)-(a',b')\|_\infty,
\end{aligned}
\]
and \(T(Q)\subseteq Q\). The fixed points are the points \((a,|a|)\). The midpoint \((0,1)\) of \((-1,1)\) and \((1,1)\) has \(T(0,1)=(0,0)\), at distance \(1\). In Proposition 5.1, every \(h\in\mathcal H_m\) has values that are \(\frac1m\)-approximate fixed points by (2.2), but \(P\) is a pointwise weak limit of convex combinations of such maps, and neither convex combinations nor weak limits preserve the property of being an approximate fixed point. Only the individual values \(G_i(x)\) in Lemma 6.1 are approximate fixed points; the convex combinations return to \(x\).

## References

- [El] R. Ellis, *Distal transformation groups*, Pacific J. Math. 8 (1958), 401–405. https://doi.org/10.2140/pjm.1958.8.401
- [OpenAI-K] OpenAI, *Fixed points of nonexpansive maps in reflexive Banach spaces*, OpenAI Math Release preprint, 24 September 2026, Section 3. https://github.com/openai/math/blob/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026