Geom. Funct. Anal. Vol. 34 (2024) 486–511
https://doi.org/10.1007/s00039-024-00658-x© The Author(s) 2024

A METRIC FIXED POINT THEOREM AND SOME OF ITSAPPLICATIONS
Anders Karlsson
Abstract. A general ﬁxed point theorem for isometries in terms of metric functionals
is proved under the assumption of the existence of a conical bicombing. It is new for
isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces comefrom the study of diﬀeomorphism groups, birational transformations, and compact
Kähler manifolds. A special case of the ﬁxed point theorem provides a novel mean
ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The
theorem accommodates classically ﬁxed-point-free isometric maps such as those of
Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together
with some geometric arguments of independent interest, one can deduce that every
bounded invertible operator of a Hilbert space admits a nontrivial invariant metricfunctional on the space of positive operators. This is a result in the direction of
the invariant subspace problem although its full meaning is dependent on a future
determination of such metric functionals.

1 Introduction
Brouwer’s ﬁxed point theorem from 1912 as well as the inﬁnite dimensional extensionthat Schauder proved around 1930 have many applications. These theorems requirecompactness, as Kakutani’s 1943 examples of ﬁxed point free (1 + )-Lipschitz mapsof the closed unit ball in Hilbert spaces illustrated. For isometries there is an inﬂuential paper [BM48] by Brodskii and Milman proving conditions for the existence ofﬁxed points. For 1-Lipschitz maps, also called nonexpansive maps, there are ﬁxed
point theorems of Browder, Göhde and Kirk [Bro651, Bro652, Goh65, Kir65] without
compactness assumption but instead requiring certain good properties of the Banach
spaces. When the maps are uniformly strict contractions, there is the simple but
fundamental contraction mapping principle, abstractly formulated in Banach’s thesis from 1920, applicable in any complete metric space and providing a unique ﬁxed
point. On the other hand, for isometric maps Kakutani gave examples of the closed
unit ball in certain Banach spaces without ﬁxed points (see below). An example of
Alspach [Dal81] showed moreover that for isometries even with weak compactness ofthe convex set, there may not be any ﬁxed point. See the handbook [KS01] for moredetails on these topics.
The author was supported in part by the Swedish Research Council grant 104651320 and the
Swiss NSF grants 200020-200400 and 200021-212864.

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The main purpose of this paper is to establish an entirely new type of ﬁxed point
theorem. It could have been discovered decades ago and is in several ways moregeneral than existing ﬁxed point theorems for individual isometries. In particular itis new for isometries of convex sets of Banach spaces. It has wide applicability as is
testiﬁed below with corollaries in ergodic theory, operator theory, geometric grouptheory, dynamical systems, complex analysis, and other topics. In some cases working
out the exact consequences is beyond the scope of this paper and may require further
study.
Let (X, d) be a metric space. A metric space is proper if every closed bounded
set is compact. We call a map T : X → X isometric, or isometric embedding, if it
preserves distances. It is an isometry in case it has an (isometric) inverse, whichamounts to T being a surjective isometric map.
Let x0 be a point of X. We map X into the space Hom(X, R) of nonexpansivefunctions X → R with the topology of pointwise convergence,Φ : X → Hom(X, R)
deﬁned via
x → hx (·) := d(·, x) − d(x0 , x).
This is a continuous injective map and the closure Φ(X) is compact. We call
X := Φ(X) the metric compactiﬁcation of X (following the terminology of [Rie02]
for proper spaces) and we call its elements metric functionals. (The choice of topology here is diﬀerent from the one taken in [Gro78] where it is uniform convergenceon bounded sets. The limit functions in this other topology are here called horofunctions.)
The action of isometries on X extends to an action on X by homeomorphisms,
via(T h)(x) = h(T −1 x) − h(T −1 x0 ).
Alternatively one can consider everything up to additive constants. This is related
to the fact that the compactiﬁcation is independent of x0 .
A bicombing is a selection of a geodesic between every two points (distinguishedgeodesics). More precisely, a weak conical bicombing is a map σ : X × X × [0, 1] → Xsuch that σ(x, y, ·) is a constant speed geodesic from x to y such thatd(σxy (t), σxy (t)) ≤ td(y, y  )for all x, y, y  ∈ X and 0 ≤ t ≤ 1.
Examples of spaces admitting such bicombings include Banach spaces, and convex
subsets thereof, complete CAT(0)-spaces and injective metric spaces. In addition, aparticularly interesting speciﬁc example is Pos, the space of positive bounded linear
operators of a Hilbert space, explained more below, and on which every invertible

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bounded linear operator acts by isometry. The 1- Kantorovich-Wasserstein distanceon spaces of probability measures and the Weil-Petersson metric on Teichmüller
spaces are two further examples.
Theorem 1. Let X be a metric space with a weak conical bicombing. Let T : X → X
be an isometric embedding. Then there is an element h ∈ X such thath(T x) = h(x) − dfor all x and where d = inf x d(x, T x). In the case h = hy for some point y ∈ X thenT y = y . If T is an isometry, thenT h = h.
An inequality for nonexpansive maps was previously known from Gaubert andVigeral’s paper [GV12], and even earlier from [Kar01] with a weaker but more generalstatement. For Banach spaces and in terms of linear functionals an inequality is also
proved in [KN81], but the use of linear functionals instead of metric functionals
gives a strictly weaker result as can be inferred from an example by Mertens in thatsame paper. The simple case of the theorem when d = 0 was observed by Gromov in[Gro78], see also Proposition 13 below. We call h invariant under T when the ﬁrst
equality in Theorem 1, h(T x) = h(x) − d, holds for all x ∈ X, and we say that h is aﬁxed point of T when T acts on the compactiﬁcation X and T h = h. Note that someconvexity or bicombing property is necessary (as is the case for Brouwer, Schauderet al) since a nontrivial rigid rotation of a circle has no ﬁxed point and the circle is acompact space, it means that there are also no ﬁxed metric functional. In contrast:
Corollary 2. Any isometry T of a metric space as in the theorem and withinf x d(x, T x) > 0 must have inﬁnite order, in fact d(x, T n x) → ∞ for any x.
For comparison, Basso proved a ﬁxed point theorem for groups of isometries ofspaces with conical bicombings with a midpoint property, in a more traditional sense
and assuming compactness [Bas18].

Banach spaces. Here is a ﬁrst application. Let U be a norm-preserving linear
operator, that is U v = v for all v in a Banach space X, where as usual all
line segments provide the required bicombing. In this setting the metric ﬁxed point
theorem proves a new mean ergodic theorem:
Corollary 3. (Mean ergodic theorem in Banach spaces) Let U be a normpreserving linear operator of a Banach space X and v ∈ X . Then there exists a
metric functional h such that




n−1

1
U k v = − inf U x + v − x
h
x
n
k=0

for all n ≥ 1.

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A METRIC FIXED POINT THEOREM AND SOME OF ITS APPLICATIONS489

In case of a Hilbert space one can deduce the mean ergodic theorem proved by
von Neumann using spectral theory, in a very diﬀerent way:
Corollary 4. (von Neumann, 1931) Let U be a unitary operator of a Hilbert space
H and v ∈ H . Then the strong limit
1 n−1
U k v = πU (v),
n→∞ n
k=0

lim

where πU is the projection onto the U -invariant vectors.
(See Sect. 4 for the proofs.) The mean ergodic theorem can thus be viewed as aﬁxed point theorem. The classical formulation (the convergence part in Corollary 4)
is known to fail for general Banach spaces, such as ∞ (N) and 1 (N) for shift operators, while Corollary 3 always holds true. An abstract characterization for when
the classical version holds is the content of the mean ergodic theorem in Banach
spaces of Yosida-Kakutani in [YK41]. There are many papers on the topic of which
operators are mean ergodic, see [Eis10] for a good exposition. See Sect. 4 for a newmean ergodic theorems for power bounded operators.
Theorem 1 applies in particular to any closed convex set X of a Banach space, andprovides ﬁxed points in a generalized sense. Note that it is known that the MazurUlam theorem fails in general, that is, not every isometry of a convex set is aﬃne,
see the examples of Schechtman in [Sch]. We now relate Theorem 1 to well-known
conventionally ﬁxed-point-free isometric maps.
Example (Kakutani). The map T (x1 , x2 , . . .) = (1, x1 , x2 , . . .) is an isometric ﬁxedpoint-free map of the closed unit ball in the sequence space c0 with the supremum
norm and all sequences tending to 0. The metric functional of the theorem for thismap is h(x1 , x2 , . . .) = supk |xk − 1| − 1.
Example The isometric map T (x1 , x2 , . . .) = (1, x1 , x2 , . . .) of either 1 (N) or 2 (N),clearly has no ﬁxed point in the usual sense. On the other hand, in the 1 (N) case it
leaves the metric functional associated to (1, 1, 1, . . .) invariant ([GK21])h(x1 , x2 , . . .) =∞
|xk − 1| − 1,

k=1

and in the 2 (N) case the trivial metric functional h ≡ 0 is invariant.
Example (Edelstein [Ede64]). A more sophisticated example in the Hilbert spacesetting was given by Edelstein. It is an isometry with unbounded but recurrent orbits.
The trivial metric functional is ﬁxed. This map is an isometry also of 1 (N) in whichcase the metric functional in the previous example is ﬁxed also for this map, as shownin [Pen21]. In this context, note that by [Gut19], the linear functional f (x) ≡ 0 is
not a metric functional of 1 (N).

490

A. KARLSSONGAFAExample (Alspach [Dal81]). This example is essentially what is called the Baker’stransformation in ergodic theory. The convex set X in L1 ([0, 1]) consists of the integrable functions taking values in [0, 2] of integral 1. The isometric map is T f (t) ismin{2, 2f (2t)} for 0 ≤ t ≤ 1/2 and max {0, 2f (2t − 1) − 2} for 1/2 < t ≤ 1. Gutièrrez
noticed an invariant metric functional of this map in [Gut20].
Example (Shift in 1 ). Bader, Gelander, and Monod [BGM12] proved an unexpected ﬁxed point theorem for L1 spaces. In the traditional sense there may not
be any ﬁxed points of isometries preserving a bounded closed convex set, but they
showed that there is a ﬁxed point which may lie outside the set. An example, mentioned in the introduction of [BGM12], is the shift map T on 1 (Z) and the convex
bounded set being all the non-negative functions whose values sum to 1. The only
ﬁxed point of this isometry is 0. From the viewpoint of the present article, given any
two points x and y, applying the shift enough times on one of them, the 1 -distanced(x, T n y) is approaching x + T n y = x + y. This shows that any orbit converges to the metric functional h(x) = x = h0 (x), which is the metric functional of
Theorem 1. Thus T 0 = 0 recovering the obvious ﬁxed point in this case.
Example (Basso [Bas18]). This is an example of an isometry of a bounded complete
convex subset of a strictly convex Banach space without a ﬁxed point. The map isthe shift map T on 1 (Z) which is renormed with an equivalent norm that is strictly
convex and T still an isometry. The closed convex hull C of the vectors which has 1
in one coordinate and 0 elsewhere has T as an isometry. The only ﬁxed point in theusual sense is clearly the zero-vector which lies outside of C. Let x0 be the vector
that is 1 in the 0th coordinate and 0 elsewhere. The metric functional that is thelimit of hT n x0 as n → ∞,
h(x) =(x1 + 1)2 + x22 + 1 − 1

is ﬁxed by T .

CAT(0)-spaces. When X is a complete CAT(0)-space, geodesics are unique andprovide a conical bicombing almost directly from deﬁnitions. Such spaces have astandard bordiﬁcation adding equivalence classes of geodesic rays, sometimes called
the visual bordiﬁcation, or the visual compactiﬁcation in case X is proper, see [BH99].
In Gromov’s choice of topology [Gro78] on Hom(X, R), the closure of Φ(X) above isthe visual bordiﬁcation [Gro78, BH99, Theorem 8.13]. In particular, for nonproper
X the visual boundary may be empty, but when proper it coincides with the metriccompactiﬁcation. Immediately from Theorem 1 we therefore get:Corollary 5. Let X be a complete CAT(0)-space. Then any isometry of X ﬁxes apoint in X . In case X is proper, any isometry ﬁxes a point in the visual compactiﬁcation.
In the proper case this is a standard fact based on the classiﬁcation of isometries,
the elliptic part going back to E. Cartan. This is of fundamental importance for theGAFA

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theory [BGS85]. In the special case of ﬁnite dimensional Cartan-Hadamard manifolds, the compactiﬁcation is a closed ball in this case and one could instead appeal
to Brouwer’s ﬁxed point theorem. The present paper provides a diﬀerent proof ofthis, and in fact the ﬁrst equality in Theorem 1 gives more precise information.
In the nonproper case, as explained in [BH99] there are isometries that have noﬁxed points in the visual bordiﬁcation. For example take 2 (Z) and consider the shift
and adding 1 to the 0th coordinate. Edelstein’s isometry mentioned above has d = 0(for example since the orbit does not tend to inﬁnity, thus τ (f ) = 0, see below for
the deﬁnition). This is no contradiction since for Hilbert spaces the function h ≡ 0
is a metric functional [Kar22, Gut19]. On the other hand, if d > 0 the ﬁxed metricfunctional is nontrivial. See also Theorem 21 below for more information. This is also
the case for CAT(-1)-spaces in general since it is Gromov hyperbolic and for such
spaces it follows from Maher-Tiozzo [MT18] that h ≡ 0 is not a metric functional even
without local compactness. For isometries of CAT(-1)-spaces previous results can befound in [Kar01, KN04, Theorem 5]. Claassens [Cla20] has completely determined
the metric functionals for the inﬁnite dimensional real hyperbolic space.
Important examples of isometric actions on nonproper CAT(0)-space are provided
by the Cremona groups of birational transformations in algebraic geometry [Can11,
LU21].

Injective metric space. Metric space that are hyperconvex, or equivalently injective,
have properties which generalizes L∞ and trees. One important feature of such spaces
is that they admit a conical bicombing as shown in [Lan13, Prop. 3.8]. A result
from 1979 by Sine and Soardi shows that every nonexpansive self-map of a bounded
injective metric space has a ﬁxed point, see [KS01, Ch. 13]. Baillon asked whether
the boundedness could be relaxed to just assuming that the orbit is bounded, but acounterexample was found by Prus [KS01].
Example (Prus). Consider the hyperconvex Banach space X = ∞ (N) and the map
T : X → X deﬁned by T ((yn )) = (1 + lim yn , y0 , y1 , . . .) where the limit is an ultraﬁlter
or Banach limit. This map is isometric with bounded orbits but no ﬁxed point in
X. This is in contrast with Theorem 1 that provides an invariant metric functional
for T . Moreover it must be nontrivial, since as shown below in Proposition 22, no
metric functional for ∞ (N) is trivial, that is, the linear functional f (x) ≡ 0 is not a
metric functional.
Once again, Theorem 1 provides the missing ﬁxed point as it were:
Corollary 6. Let X be an injective metric space. Then every isometry of X ﬁxes
a point in X . In case X is proper, any isometry ﬁxes a point in the visual compactiﬁcation.
Details for how to deduce the second assertion of the corollary, which is very easy,
will be given below. A new topic in geometric group theory is the subject of Hellygroups, which are groups admitting geometric isometric actions on injective metric492

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spaces [HO21]. Note that, although similar to the CAT(0)-case above, the boundaries(metric vs visual) are now diﬀerent even for proper spaces, the case of ∞ illustrates
this as is shown below. As Basso informed me there is a ﬁxed point theorem by U.
Lang for isometry groups of bounded orbits in [Lan13]. For Busemann spaces one
should mention Navas paper [Nav13] in this context.
Every metric space X has an (essentially) unique injective hull E(X) into which it
is isometrically embedded [Isb64], and every isometry f of X extends to an isometryof E(X). This is immediate from the deﬁning properties, see [Lan13, Prop. 3.7].
Therefore we may conclude:
Theorem 7 (Isometry ﬁxed point theorem). Let f be an isometry of a metric space
X . Then f has a ﬁxed point in E(X).
In this way, even a nontrivial rigid rotation of the circle has an associated ﬁxed
point.

Diﬀeomorphims and biholomorphisms. It is well-known that the Weil-Petersson
metric of the Teichmüller space of a closed orientable surface has nonpositive curvature, is not complete but is geodesically convex. Therefore:
Corollary 8. Any element in the mapping class group of a genus g surface ﬁxes
a metric functional of the corresponding Teichmüller space equipped with the WeilPetersson metric.
Fixed points of mapping classes were already well-understood from Thurston’scompactiﬁcation of Teichmüller spaces, using Brouwer’s theorem.
The space of all Riemannian metrics Met(M ) of a manifold M is a space ﬁrst
introduced by DeWitt in general relativity and studied early on by Ebin in the 1960s.
The diﬀeomorphisms group acts on it by isometry.
Corollary 9. Let f be a diﬀeomorphism of a closed orientable compact manifold
M . Then f ﬁxes a metric functional of Met(M ).
Proof. It is known since some time that this space has features of nonpositive curvature with a good description of what the geodesics are. The version that applies
best for us here is the theorem in [Cla13] that asserts that the metric completion isa CAT(0)-space. From this the main ﬁxed point theorem applies directly taking into
account Lemma 15 below.

A certain space of Kähler potentials of a connected compact Kähler manifold
was introduced by Calabi in the 1950s. It has a Weil-Petersson type metric and thefollowing holds for the same reasons as the previous corollary:
Corollary 10. Any complex automorphism of a compact Kähler manifold ﬁxes a
metric functional of the associated space of Kähler potentials.
The completed space which is a CAT(0)-space was used for progress on a conjecture of Donaldson on the asymptotic behavior of the Calabi ﬂow, see [Bac] for aGAFA

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discussion and references on this topic. The Calabi ﬂow is in fact nonexpansive inthis metric and when there is no constant scalar curvature metric on the underlying
Kähler manifold, the ﬂow may diverge. Note that the theorems in [KM99, Kar01](compare with Theorem 21 below) apply to (the time-one map of) the Calabi ﬂow.
These results provide a geodesic ray or metric functional and may thus be relevant
for Donaldson’s conjecture.
Holomorphic maps are always nonexpansive in the Kobayashi pseudo-metric andthus biholomorphisms are isometries. The ﬁxed point theorem in this case will betreated in a forthcoming paper since some additional arguments are needed in thiscase.
In these examples, from algebraic geometry, topology and complex geometry, themetric ﬁxed point theorem provides something new and nontrivial, at least in thecase that d > 0, and the metric compactiﬁcation in these cases deserves to be studied.

The invariant subspace problem. The invariant subspace problem, open since the1950s, asks whether every bounded linear operator of a separable inﬁnite dimensional
complex Hilbert space has a nontrivial closed invariant subspace. In ﬁnite dimensions
the statement follows for example from the fundamental theorem of algebra, and von
Neumman, who is one of the ﬁrst to have considered this problem, proved it in thecase of compact operators. Without loss of generality one may assume that theoperator is invertible otherwise one can add a multiple of the identity map, which
does not change the invariant subspaces.
The set of bounded positive operators Pos of a Hilbert space H has a natural
metric which admits a weak conical bicombing. Moreover every invertible boundedlinear operator g : H → H acts by isometry on Pos via p → gpg ∗ . Applying Theorem 1
we get:Theorem 11. Every bounded, invertible linear operator g of a Hilbert space has aninvariant nonconstant metric functional h : Pos → R,
gh = h.

Moreover, the following equality holds:(pg ∗ v, g ∗ v) 1

inf log sup = lim |log g n g ∗n | .
n→∞p∈Pos 
(pv,
v)
n
v=0

This presents a new approach to the invariant subspace problem and a novel
viewpoint of operator theory more generally. In ﬁnite dimensions the statement above
implies the existence of a nontrivial invariant subspace as follows, either the operator
is unitary with respect to some scalar product or from the explicit determination of
the metric functionals in Lemmens’ paper [Lem] there is an associated closed linear
subspace (the kernel of a semipositive operator), which is nontrivial in case thedimension is at least two and that is invariant.

494

A. KARLSSONGAFAThe fact that the metric functional is nontrivial comes from the following statement of independent interest. This is in contrast with linear functionals and weaklimits that can be just 0.
Proposition 12. Every metric functional of the space Pos is an unbounded function.
In conclusion, Theorem 11 shows that a general bounded linear operator does
have some structure. Carleson once speculated that the invariant subspace problem
in the Hilbert space case perhaps is a ﬁxed point theorem on some Grassmannian[CJ02]. Even though this is not achieved in the present paper, the author believes
that the metric perspective on operator theory is of future interest more generally.
One suggestion could be to investigate the validity of an extension of the RyllNardzewski ﬁxed point theorem for aﬃne isometry groups and see how it relates to
Dixmier’s unitarization problem. As Monod pointed out to me years ago, a uniformly
bounded representation of a group corresponds to an isometric action with boundedorbits in Pos and the equivalence to a unitary representation is precisely the existenceof a ﬁxed point in Pos (that ﬁxed points may not exist, in contrast to CAT(0)-spaces,is also coherent with Basso’s example mentioned above). It follows therefore from[dN47] that if g is an invertible, bounded linear operator with bounded orbits in
Pos, then it ﬁxes a point in Pos. Some groups are non-unitarizable, so once again
the metric functionals are potentially an appropriate device. This topics connects tothe power-bounded mean ergodic theorem in Sect. 4, since the latter corresponds touniformly bounded representations of Z (or N).
One area where spaces of positive operators have already been useful is multiplicative ergodic theorems for operators. The result [KM99, Corollary 7.1] is a substantial
and surprising strengthening of an important theorem of Ruelle [Rue82] under an extra Hilbert-Schmidt condition that allows using a submanifold of Pos that is CAT(0).
A general result is [GK20, Theorem 1.7] and in [BHL21] several new multiplicative
ergodic theorems are obtained using ﬁnite traces and a study of the corresponding
spaces of positive operators (again CAT(0) here).

2 Metric preliminaries
We consider the metric category, that is, objects are metric spaces and morphims
are nonexpansive maps, which means thatd(f (x), f (y)) ≤ d(x, y)
for all x and y. The composition of nonexpansive maps remains nonexpansive. Isometries are the isomorphisms in this category.
One deﬁnes the minimal displacementd = df = inf d(x, f (x))
x

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and the translation numberτ = τf = lim d(x, f n (x))/n.
n→∞These numbers are analogs of the operator norm and spectral radius, respectively, inparticular note that τ ≤ d. To see this, since τ is independent of the point x we may
take a point close to the inﬁmal displacement in the sense that d(x, f (x)) < d + . By
the triangle inequality, d(x, f n (x)) < n(d + ), so τ < d +  for any  > 0 and hence
the inequality. Rigid rotations of a circle show that the inequality can be strict. On
the other hand, note that when Theorem 1 applies then necessarily τ = d, since by
the theorem and in view of that metric functionals are nonexpansive,d(x, T n x) ≥ |h(x) − h(T n x)| ≥ nd.
This was already known from [GV12] and in the nonpositive curvature case forisometries from Gromov.
Let Hom(X, R) be the space of morphisms X → R (fonctionnelles in [Ban55])
equipped with the topology of pointwise convergence. Given a base point x0 of themetric space X, letΦ : X → Hom(X, R)
be deﬁned via
x → hx (·) := d(·, x) − d(x0 , x).
This is a continuous injective map and the closure Φ(X) is compact, say the ArzelaAscoli or Tychonoﬀ’s theorems. This is the metric space analog of the Banach–
Alaoglu theorem. We call X := Φ(X) the metric compactiﬁcation of X and its elements metric functionals. These are nonexpansive and satisfy a metric Hahn-Banachtheorem [Kar211]. Note that h(x0 ) = 0 for any h ∈ X. There is a growing list of studied examples [Wal18, KL18, H+, MT18, Gut19, Cla20, LP23, AK22] to cite a few
recent papers. It is easy to see that for Banach spaces metric functionals are convex
functions. In [B+22] the same topology is used as here for nonproper spaces, andthat paper contains a careful discussion of this construction.
Busemann observed in the early part of the 20th century that any geodesic ray
γ deﬁnes a metric functional: bγ (y) = limt→∞ d(y, γ(t)) − t, called the Busemannfunction of γ. This is akin to how a vector deﬁnes a linear functional via a scalar
product. Busemann functions have been useful for a long time in the theories ofspaces with nonpositive respectively nonnegative curvature. The usefulness of these
metric notions without curvature conditions is illustrated by the work described in[KL11].
Note the eﬀect of changing base point x0 to y0 :d(y, x) − d(x0 , x) = d(y, x) − d(y0 , x) + d(y0 , x) − d(x0 , x).
It amounts in other words to just adding the constant hx (y0 ).

496

A. KARLSSONGAFAIsometries act by homeomorphisms of X via(T h)(x) = h(T −1 x) − h(T −1 x0 ).
Note that if for some metric functional h one has that h(T x) = h(x) + c (or whichis the same h(T −1 x) = h(x) − c) for all x ∈ X with a constant c, then T h = h.
Conversely, having a ﬁxed point, h(x) = (T h)(x) = h(T −1 x) − h(T −1 x0 ) for all x,then the same equality holds with constant c = h(T −1 x0 ). As discussed in mostreferences on the topic, from [Gro78] to [B+22], one could consider functionals up toan additive constant, projectively, removing the role of any ﬁxed based point. Somethings are more clear from this point of view, but functionals would then just be
deﬁned up to an additive constant.
Let G be a group that ﬁxes h, note that T (g) := −h(gx0 ) = −dg then deﬁnes agroup homomorphism T : G → R.
Bas Lemmens and I observed the following easy fact during a discussion (essentially as recorded in [Kar211]), for isometries this was pointed out by Gromov longtime ago [Gro78]:
Proposition 13. Let T be a nonexpansive map of a metric space X . Suppose thatd(T ) = 0. Then there is a metric functional h ∈ X such thath(T x) ≤ h(x)for all x ∈ X . In case T is an isometric embedding then h(T x) = h(x) for all x, andwhen T is an isometry T h = h.
Proof. For any  > 0, take y such that d(y , T y ) < . Take x ∈ X and considerd(T x, y ) − d(x0 , y ) ≤ d(T x, T y ) + d(T y , y ) − d(x0 , y ) ≤  + d(x, y ) − d(x0 , y ).
This shows that any limit point h of hy as  approaches 0, which exists by compactness, has the property thath(T x) ≤ h(x).
Now let us assume that T is isometric, thend(x, y ) − d(x0 , y ) = d(T x, T y ) − d(x0 , y ) ≤ d(T x, y ) + d(y , T y ) − d(x0 , y )≤  + d(T x, y ) − d(x0 , y ),which in the limit shows that h(x) ≤ h(T x). Thus h(T x) = h(x) and T h = h if T is
an isometry thus actually acting on X.
Let us spell out a special case of Theorem 1 (cf. [Kar212, Theorem 2.2]):
Proposition 14. Let T be an aﬃne isometry of a bounded convex set C of a reﬂexive Banach space. Then there is a point x ∈ C such that T x = x.

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Proof. From Theorem 1 we have d = τ = 0. Consider the sets
{x : x − T x < } .
They are clearly convex when T is aﬃne and by reﬂexivity the intersection of all
 > 0 is nonempty (Smulian). The points in this intersection must be ﬁxed points ofT.

A referee pointed out that instead of the reﬂexivity assumption one could merely
assume that C is weakly compact. Another extension would be to complete CAT(0)spaces, or more generally, complete uniformly convex metric spaces, which also are
well-known to have the required non-empty intersection property, see e.g. [Kar00,
Prop. 2.2] or [Kel14]. Incidentally, these two references suggest diﬀerent notions ofreﬂexivity of metric spaces.
If one does not assume that the isometry is aﬃne there is a celebrated result ofMaurey [Mau81], that states that in a weakly compact, convex set of a superreﬂexive
Banach space, every isometry has a ﬁxed point.

3 Proof of the metric ﬁxed point theorem
Fix x0 ∈ X. We deﬁne the following family of contractions rs : X → X, 0 ≤ s ≤ 1 byrs (y) = σx0 y (s). Note that by the bicombing deﬁnitiond(rs (x), rs (y)) ≤ sd(x, y)and d(rs (y), y) = (1 − s)d(x0 , y) from the constant speed geodesic requirement.
Lemma 15. Let X be a metric space and Y its metric completion. Then X is canonically homeomorphic to Y .
Proof. We can consider X as a subset of Y in the natural way and select a basepointx0 ∈ X for both spaces. First note that in a canonical way Hom(X, R) → Hom(Y, R)
since any element h ∈ Hom(X, R) has uniquely deﬁned values on Y . Indeed, for any
sequence xn → y,|h(xn ) − h(xm )| ≤ d(xn , xm )which makes h(xn ) a Cauchy sequence of real numbers, hence converging, which is
the thus well-deﬁned value h(y). Similarly, by a 3-argument, it follows that every
limit point in Hom(X, R) is also a limit point in Hom(Y, R). This shows that X → Y .
Second, every element of Hom(Y, R) is also an element of Hom(X, R) by restriction. This also implies that any limit point h in Y is also a metric functional in X
by approximation and restriction. Indeed, say that h is a limit point of a set {hy },then we can approximate each y by a Cauchy sequence in x. This shows the opposite
continuous inclusion and the lemma is established.
498

A. KARLSSONGAFAIn view of the lemma, and that we can also extend our maps rs for the same
reasons since they are also nonexpansive maps, we may for simplicity assume that
X is complete.
The map T rs = T ◦ rs is a uniformly strict contraction if s < 1 since T preserves
distances. By the contraction mapping principle we thus have a unique ﬁxed pointys ∈ X for this map.
Here is a lemma that applies even with T is merely nonexpansive:
Lemma 16. Let ys be the unique ﬁxed point of T ◦ rs for 0 ≤ s < 1. Thend(x0 , ys ) ≤d(x0 , T x0 ).
1−sProof. Let D = d(x0 , T x0 ). Consider the ball of radius R with center z
BR (z) = {x ∈ X : d(x, z) ≤ R} .
Note that from the bicombingrs (BR (x0 )) = BsR (x0 ).
From the isometric property of T obviously T (BR (x0 )) ⊆ BR (T x0 ) for any R. Hence
T rs (BR (x0 )) ⊆ BsR (T x0 ).
Note that BR (T x0 ) ⊆ BR+D (x0 ) by the triangle inequality. This means that if wetake R ≥ D/(1 − s) so that sR + D ≤ R, thenT rs (BR (x0 )) ⊆ BsR (T x0 ) ⊆ BsR+D (x0 ) ⊆ BR (x0 ).
From the contraction mapping principle, the iterates of the map T rs converge towards
the unique ﬁxed point ys . That is, for any x, (T rs )n x → ys as n → ∞. Since BR (x0 )is an invariant set for R = D/(1 − s) we must have that ys ∈ BR (x0 ) as required. To get the ﬁrst inequality we follow the proof of Gaubert-Vigeral [GV12] whicheven applies to nonexpansive maps T . For any x
hys (x) − hys (T x)= d(x, ys ) − d(T x, ys ) = d(x, ys ) − d(T x, T (rs ys )) ≥ d(x, ys ) − d(x, rs ys )≥ d(x, ys ) − d(x, rs x) − d(rs x, rs ys )≥ d(x, ys ) − d(x, rs x) − sd(x, ys ) = (1 − s)d(x, ys ) − d(x, rs x)≥ (1 − s)(d(x0 , ys ) − d(x0 , x)) − d(x, rs x)= d(rs ys , ys ) − (1 − s)d(x0 , x) − d(x, rs x)≥ d(T rs ys , T ys ) − (1 − s)d(x0 , x) − d(x, rs x)= d(ys , T ys ) − (1 − s)d(x0 , x) − d(x, rs x)

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499≥ d(T ) − (1 − s)d(x0 , x) − d(x, rs x).
The last two terms go to 0 as s → 1. By compactness there is a limit point of hys as
s approaches 1 and for any such limit point h the above inequality shows thath(x) − h(T x) ≥ d.
So we have obtained h with the property that h(T x) ≤ h(x) − d for all x ∈ X.
On the other hand, to get the other inequality we use that T is isometric,hys (x) = d(x, ys ) − d(x0 , ys ) = d(T x, T ys ) − d(x0 , ys )≤ d(T x, ys ) + d(ys , T ys ) − d(x0 , ys ) = d(T x, ys ) + d(T rs ys , T ys ) − d(x0 , ys )= hys (T x) + d(rs ys , ys ) = hys (T x) + (1 − s)d(x0 , ys )≤ hys (T x) + (1 − s)d(x0 , T x0 )/(1 − s)= hys (T x) + d(x0 , T x0 ),where the last inequality comes from Lemma 16. This means that for any  > 0 we
can ﬁnd a metric functional h (using x0 such that d(x0 , T x0 ) < d + ) such thath(x) ≤ h(T x) + d + together with h(x) ≤ h(T x) − d both inequalities holding for all x. This applies toany limit point h.
We can now argue as follows, ﬁx a sequence sn 1 as n → ∞. Take a non-principal
ultraﬁlter U on the natural numbers. This deﬁnes the limit
h = lim hysnU =
0 , we have the limit hand for metric functionals with another base point xlimU hysn . Recall the above discussion about changing the base point, that it does
not inﬂuence diﬀerences h(x) − h(T x). Indeed, for any yhy (x) − hy (T x) = h̃y (x) − h̃y (T x)
for any x, since on each side the terms with the base points cancel. Therefore
h(x) − h(T x) = lim hysn (x) − hysn (T x) = lim h̃ysn (x) − h̃ysn (T x)UU= h̃(x) − h̃(T x) ≤ d(x̃0 , T x̃0 ),where the last inequality comes from the above calculation. Now since the base pointx̃0 is arbitrary, we conclude that h(x) − h(T x) ≤ d.
In summary we have for this metric functional that for all x ∈ X
h(x) − d ≤ h(T x) ≤ h(x) − d,

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hence the desired equality. In case T is an isometry, previous remarks show thatT h = h.
Suppose that h = hy for some point y ∈ X. Then hy (T x) = hy (x) − d, that is,d(y, T x) − d(y, x) = −dfor all x. Applying it to x = y, forces d = 0 and d(y, T y) = 0, since distances are
positive or 0. By the standard axiom for metric spaces it means that T y = y.
With all this, Theorem 1 is proved.

4

Mean ergodic theorems

In 1931 von Neumann proved the ﬁrst ergodic theorem after being inspired by remarks of Koopman and Weil. Around the same time Carleman had similar ideas,
and, as Eisner pointed out to me, von Neumann subsequently published a note analyzing Carleman’s work on the topic. For references and a good general discussion on
this subject can be found in the book by Eisner [Eis10]. The mean ergodic theorem
is proved in virtually every book on ergodic theory. It is also well-known that theoperator version of the mean ergodic theorem fails in general Banach spaces suchas 1 and ∞ . From the metric ﬁxed point theorem, we deduce a new result that incontrast holds in general (cf. [Kar212, Theorem 2.1] for another version):
Corollary 17 (Mean ergodic theorem in Banach spaces). Let U be a normpreserving linear operator of a Banach space X and v ∈ X . Then there exists a
metric functional h such that




n−1

1
U k v = − inf U x + v − x
h
x
n
k=0

for all n ≥ 1.
Proof. The line segments in the Banach space provide the required bicombing. From
the assumption it follows that x → U x + v is an isometric map and the sum in thestatement is the application of this map n times to x = 0. The result now follows
directly from Theorem 1 by iterating the map.

In case of a Hilbert space one can deduce the usual mean ergodic theorem in away quite diﬀerent from the standard proofs:
Corollary 18. Let U be a unitary operator of a Hilbert space H and v ∈ H . Thenthe strong limit

1 n−1
lim
U k v = πU (v),
n→∞ n
k=0

where πU is the projection onto the U -invariant vectors.

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Proof. Either the limit exists equalling 0, or τ = d > 0 (see Sect. 2) and Corollary 17
gives a nontrivial metric functional, which in this case for a Hilbert space must be
linear, see [Kar22]. Hence there is a unit vector w such that for all n ≥ 1 the scalar
product





1 n−1
U k v, w = d,n k=0where d = inf x U x + v − x. Considering that U is norm-preserving, τ = d, andw = 1 it is a simple Hilbert space fact that this implies that the sum in thestatement converges strongly to d · w (see [Kar22]). From Theorem 1 we in addition
have for any x that (x − U x − v, w) = inf x x − U x − v = d. Since the closed linear
span of vectors {x − U x} is orthogonal to the U -invariant vectors, the projection ofv onto the latter subspace is d · w.

A linear operator A of a Banach space is power bounded if
sup Ak < ∞.
k>0Mean ergodic theorems have been considered for such operators for example in theworks of Kakutani and Yosida. Parallel to the above discussion, the usual mean
ergodic formulation fails in general but there is now instead a metric replacement:
Corollary 19. (Power bounded ergodic theorem) Let A be a power bounded operator of a Banach space X . Letx1 := sup Ak x
k>0

for x ∈ X . Then for any v ∈ X there is a metric functional h of X with the norm
·1 such that
h

n−1




A v ≤ −nτkk=0

for all n > 0 and where τ = inf x Ax + v − x1 .
Proof. Note that ·1 is a norm and that x → Ax + v is nonexpansive in the corresponding metric. The statement then follows from the ﬁrst part of the proof of themetric ﬁxed point theorem.


5 Some remarks on CAT(0), injective spaces, and ∞
The following remarks are related to the main topic of this paper, and althoughperhaps not new, some of them seems not to be stated in the literature.
CAT(0)-spaces Let X be a proper CAT(0)-space. It is shown in [Gro78, BH99,
Theorem 8.13] that the metric compactiﬁcation is topologically equivalent to the502

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visual compactiﬁcation. Let g be an isometry of X with ﬁxed point h from Theorem 1.
In case h is associated with a bounded sequence, then h = hy for some point y ∈ X
and then as shown above gy = y (the usual argument uses a circumcenter of thebounded orbit, see [BH99, Corollary 2.8]). In the case h ∈ ∂X then h corresponds
precisely to a equivalence class of geodesic rays, and hence this is the ﬁxed point
again in the traditional sense. Although this latter case is a simple fact in view ofthe following argument: If the orbit of g is unbounded, let g nk x0 be a convergent
sequence to a boundary point γ. Note then that for any isometry g  commuting withg (for example g itself) ﬁxes this boundary point:g  (γ) = g  (lim g nk x0 ) = lim g  g nk (x0 ) = lim g nk (g  (x0 )) = γ,kkkwhere the last equality comes from the fact that any two sequences staying on
bounded distance from each other (in the present case d(g  x0 , x0 )) have the same
accumulation points. Thus:
Proposition 20. Let X be a proper CAT(0)-space and g an isometry with unbounded orbits. The centralizer of g ﬁxes a point in the visual boundary of X .
As noted previously, without properness, the metric compactiﬁcation is diﬀerent from the visual bordiﬁcation (which can be empty even for unbounded spaces).
Therefore Corollary 5 is of interest for groups of isometries of nonproper spaces. Important examples of isometric actions on nonproper CAT(0)-space are provided by
the fundamental Cremona groups of birational transformations of varieties. In thetwo variable case this action was found by Manin and Zariski, and greatly exploitedfor example by Cantat in [Can11]. More recently, actions by the other Cremonagroups by isometry on certain CAT(0)-cube complexes were constructed by Lonjou
and Urech [LU21]. Fixed points of individual isometries play a crucial role for thesestudies and the translation length are related to algebraic notions.
In this context, it may be worthwhile to point out the following:
Theorem 21. Let g be an isometry of a complete CAT(0)-space. If inf x d(x, gx) > 0,then g has a ﬁxed point in the visual bordiﬁcation. The metric functional corresponding to this point is ﬁxed and is a Busemann horofunction. It is moreover the uniquehorofunction such thath(gx) = h(x) − dg ,and for the geodesic ray γ corresponding to h it holds that1d(g n x0 , γ(dg n)) → 0.
n
Proof. First, it is known from [BGS85], but also reproved here in a diﬀerent way thatτg = dg . Therefore τg > 0, and a special case of the main result in [KM99] then showsthat g n x0 converges to a visual boundary point γ. In fact the last assertion of thepresent theorem is proven and which is a ﬁner notion of convergence to the boundary.

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As shown above, we then must have g(γ) = γ. This bordiﬁcation is equivalent toGromov’s horofunction bordiﬁcation, thus g ﬁxes h, the metric functional (which inthis case is a Busemann horofunction bγ ) associated to the geodesic ray emanatingfrom x0 representing γ. As shown in [KL11] the last assertion in the theorem is
equivalent to1− h(g n x0 ) → τgnfor h = bγ . Comparing Theorem 1 this is the metric functional in that statement sothere is an equality (which actually is clear from remarks in Sect. 2).
Note that as alluded to above, shown for example in [BH99], in the case of
inf x d(x, gx) = 0, there may not exist any ﬁxed point in the visual bordiﬁcation. In
case of CAT(0) cube complexes there is however a more precise version (no parabolic
isometries) available due to Haglund [Hag07].
Some spaces are not complete but still have a conical bicombing, for example theWeil-Petersson metric on the Teichmüller spaces since it is geodesically convex with
nonpositive curvature.
Injective spaces A visual compactiﬁcation was deﬁned in [DL15] for consistentbicombings of proper injective spaces (although since the choice of bicombing is
not unique it is apparently unclear how unique the visual compactiﬁcation is, while
the metric compactiﬁcation of course is unique), and another one in [Bas24] which
work without the consistency condition. We need to justify the second assertion ofCorollary 6: From Theorem 1 we have that either there is a point y ∈ X that is
ﬁxed be the isometry, or the isometry has unbounded orbits in which case the smallargument above for CAT(0)-spaces applies, since any sequence staying on boundeddistance from a convergent unbounded sequence must converge to the same point asfollows from the basic properties of the visual compactiﬁcation as in [Bas24]. This
proves that any isometry of a proper injective metric space has a ﬁxed point in avisual compactiﬁcation.
The ∞ -spaces The ∞ -space is one of the standard injective spaces. Let us ﬁrst
consider it on a ﬁnite set X := ∞ ({1, 2, . . . , N } = C({1, 2, . . . , N }). In this case thevisual compactiﬁcation (as deﬁned from the bicombing of lines) is not equivalent to
the metric one. To see this, ﬁrst notice that the rays can be represented by a unit
vector v and γ(t) = tv. Now for the metric functionals, which are already known
from papers by Walsh, Gutiérrez and others. Take x0 = 0. Since we can argue withsequences we take a sequence yn such that yn  → ∞. By taking subsequences wemay divide the index set into A and B such that the coordinates yni ≥ 0 for all n
and i ∈ A and ynj < 0 for all n and j ∈ B. Moreover we can assume that the following
limits exist
yni − yn  → ai ∈ [−∞, 0]
for i ∈ A and for j ∈ B−ynj − yn  → bj ∈ [−∞, 0].

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Denote by Af and Bf those indices with a ﬁnite limiting value. Since the norm needs
to be realized on some coordinate we have that Af ∪ Bf is non-empty, indeed at least
one of these values is 0. The corresponding metric functional which is the limit of
hyn is nowh(x) = max



sup a − x , sup b + x
i

i∈Af

i

j

j

.

j∈Bf

These are the metric functionals of X in addition to hy . For a ray yn = nv deﬁnedby a unit vector v notice that Af = i : v i = 1 and Bf = j : v j = −1 . This meansthat the other coordinates of v play no role for its Busemann function. This shows
the diﬀerence between the two compactiﬁcations.
Determining the metric compactiﬁcation of ∞ in inﬁnite dimensions seems not
yet have been done, although there is an interesting paper by Walsh [Wal18] determining the so-called Busemann points for Banach spaces (which lead him to a direct
alternative proof of the Mazur-Ulam theorem). Maybe one should keep in mind thatthe determination of the linear dual space of ∞ in functional analysis leads to signed
ﬁnitely additive measures.
Here is one interesting observation, which is in contrast to Hilbert spaces but
similar to a phenomenon Gutiérrez discovered for 1 spaces [Gut19] (also true for
hyperbolic spaces [MT18, Cla20, B+22]). Actually, as Lytchak pointed out to me, thisfeature was established for CAT(0)-spaces with ﬁnite telescoping dimension alreadyin [CL10, Lemma 4.9].
Proposition 22. For ∞ -spaces the linear functional f ≡ 0 is not a metric functional. In fact all metric functionals are unbounded functions.
Proof. Let S be a set and consider X = ∞ (S). The ﬁnite dimensional case wastreated already. Any accumulation point of a bounded set cannot be identically 0
by evaluating on a point x which lies outside a ball centered at 0 containing thebounded set, showing that they all are unbounded functions.
For the case of accumulation points of unbounded sets of metric functionals,
consider the set C of y ∈ X with y > 2. Take the subset A of X for which there isa coordinate y s0 such that y s0 > y − 1/2 > 0. In the complement of A in C there
is instead a coordinate such that −y s0 > y − 1/2 > 0.
Look at x ∈ X such that −c for every s ∈ S, and some c > 1. Then for any y ∈ A
hy (x) = sup |y s − xs | − y ≥ y s0 − (−c) − y > −1/2 + c > 1/2.
s

For y in the complement of A in C we instead look at z ∈ X such that z s = c for alls, some c > 1, and have similarlyhy (z) = sup |y s − z s | − y ≥ −y s0 + c − y > −1/2 + c > 1/2.
sNote that for any metric functional |h(z)| ≤ d(0, z) = 1. The two sets {f : f (x) < 1/2}and {f : f (z) < 1/2} are neighborhoods of f ≡ 0 in Hom(X, R). The above two in-GAFA

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equalities show that every cluster point of Φ(C) stays outside the intersection ofthese two neighborhoods, thus the functional identically equal to 0 is not in X. Indeed, we see that each h takes values larger than c − 1/2 but c is arbitrarily large,
hence all metric functionals are unbounded.


6 The space Pos
Let H be a Hilbert space and Pos = Pos(H) the set of positive deﬁnite symmetric(self-adjoint) bounded linear operators on H. This is an open cone in the Banach
space of all symmetric operators equipped with the operator norm (which by thespectral theorem equals the spectral radius). This is the prototype space for spaces of
metrics discussed above concerning diﬀeomorphisms and biholomorphisms. Therefore
I expect that some things discussed below will be relevant for the other contexts.
One can put a complete metric on the space Pos, the Thompson metric, forp, q ∈ P osd(p, q) =supv∈H,v=1log (qv, v)  ,(pv, v) which is also a Finsler metric [CPR94]. It is known from this reference and subsequent
papers that it has a weak conical bicombing which follows from Segal’s inequality:exp(u + v) ≤ exp(u/2) exp(v) exp(u/2)
for self-adjoint bounded operators and the operator norm. The bicombing does notcome from the lines in the cone even though these lines are also geodesics, but insteadthe distinguished geodesics are the image of lines in the tangent space Sym under the
exponential map (note that the exponential map in the sense of algebra and analysis,
coincides for this space with the notion in diﬀerential geometry). We consider Sym
with the distance that the operator norm gives. The inequality shows that the exponential map is distance preserving on lines, and metric noncontracting in general[CPR94]. Note that this discussion centered around the point x = I but since Pos
is a homogeneous space the same applies to any base point. Instead of considering
the full algebra of bounded linear operators one can do the same constructions forC ∗ -algebras, see for example [KM99].
Invertible bounded linear operators g act on Pos by isometry p → gpg ∗ . Thisaction is transitive. If g ﬁxes a point p it means it is a unitary operator in thecorresponding scalar product.
In ﬁnite dimensions there are several studies about the metric compactiﬁcation in
the literature, in the standard Riemannian metric (of less interest for us) [BH99] andfor various Finsler metrics [KL18, H+, Lem]. For the application we have in mind,
we need to go to inﬁnite dimensions, in this setting there is basically only [Wal18].
Here we now establish the following, which remedies a vexing property of weak
topologies in general in functional analysis that a weak limit could just be 0:

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Proposition 23. The function identically 0 is not a metric functional of Pos andthus every metric functional is nonconstant, in fact unbounded.
Proof. First note that since h(I) = 0 for any metric functional h, being constant is
the same as being 0 everywhere. (Here we are taking the identity operator to be thebase point of Pos.)
Secondly, any cluster point of a bounded set of hp is clearly an unbounded function: since taking q with large norm much bigger than the radius of the ball aroundI that contains the set of p under consideration, thenhp (q) = d(q, p) − d(I, p) ≥ d(q, I) − 2d(I, p)  0,
and it shows the function is unbounded in q. (This applies to any metric space.)
We now consider the set of p outside the ball of a certain suﬃciently large radius. We will show that this does not intersect the following neighborhood of 0 in
Hom(Pos, R):
N (0) = {f : |f (a)| < 1/2 and |f (b)| < 1/2}
where a = exp(I) and b = exp(−I).
To see this take p = exp(Y ) ∈ Pos with Y  > C > 2. Consider:
hp (x) = d(x, p) − d(I, p)
which is greater or equal to hY (log x) = log x − Y  − Y  by the metric propertiesof the exponential map recalled above. For a constant  with 0 <  < 1/2, we takea unit vector v in H such that |(Y v, v)| > Y  −  in view of a standard fact ofsymmetric operators. Now, in the case (Y v, v) > Y  − , we thus have
hp (b) ≥ hY (−I) = −I − Y  − Y  ≥ |(−Iv − Y v, v)| − Y  = |−1 − (Y v, v)| − Y 
and assuming C > 1 +  this expression equals
1 + (Y v, v) − Y  ≥ 1 −  > 1/2.
In the opposite case −(Y v, v) > Y  − , we instead look at
hp (a) ≥ hY (I) = I − Y  − Y  ≥ |(Iv − Y v, v)| − Y  = |1 − (Y v, v)| − Y 
and assuming C > 1 +  this expression equals
1 − (Y v, v) − Y  ≥ 1 −  > 1/2.
In total this shows that the set of all functionals hp with p > exp(2), does notintersect N (0). It follows that f ≡ 0 cannot be a cluster point of the set Φ(Pos) andthus not a metric functional.
Similarly to the case with ∞ one sees that we can conclude that every metricfunctional of this space is not only nonconstant but in fact unbounded.


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Application to bounded linear operators

Recall ﬁrst that we already noticed one application to norm-preserving linear operators, showing a new result that deﬁes the standard counterexamples, and strong
enough to deduce the most classical version of the mean ergodic theorem.
The Invariant Subspace Problem (ISP) asks: does every bounded linear operator
T of a separable complex Hilbert space of dimension at least two, admit a nontrivial
invariant closed subspace? This is a famous and longstanding open problem, see forexample the book [CP11] devoted to it. For the purpose of investigating invariant
linear subspaces, we may add a multiple of the identity map or multiply the operator
by a scalar, since these operation do not change any invariant subspace. Therefore
we may assume that the operator g = T is invertible.
Note that by applying the operator g iteratively on the basepoint x0 = Id we get






τ (g) = lim log g n g ∗n 1/n n→∞this is semipositive and ﬁnite (since g is an invertible bounded operator). It is 0 if
and only if the spectrum of g is contained in the unit circle (for example when theoperator is unitary). As remarked in Sect. 2 above d(g) = τ (g) (note that d makes
sense in operator theory but is an invariant perhaps not much considered). In viewof Proposition 23 we thus get from Theorem 1 a nontrivial metric functional h of
Pos which is ﬁxed by gg(h) = h.
This proves Theorem 11. It is clearly a statement in the direction of the ISP, but
how does it actually relate to closed invariant subspaces? As already remarked when
g ﬁxes a point inside Pos it is unitary and thus the spectral theorem applies andsettles the issue of invariant subspaces. If it ﬁxes a point s in the natural boundary of
Pos consisting of bounded semipositive operators, then g ∗ must preserve its kernel.
Indeed, take v ∈ ker s then0 = gs(v) = s(g ∗ v)
showing that g ∗ (ker s) ⊆ ker s. This is a closed subspace and its closed orthogonal
complement is invariant under g and is a nontrivial subspace. In general, one conceivable path forward is to reﬁne the theorem in this special case, and deduce that the
metric functional must be a Busemann point, and that as such it has an associatedclosed linear subspace that must be invariant.
Another discussion of the invariant subspace problem from a metric perspective
without the space Pos can be found in [GK21] which establishes some cases when
there exist metric or linear functionals f of the Banach space such that f (g n (0)) ≤ 0
for all n > 0 or f (g(x)) ≤ f (x) for all vectors x.

508

A. KARLSSON

GAFA

Acknowledgements
I thank Giuliano Basso, Tanja Eisner, Armando Gutiérrez, Bas Lemmens, Alexander
Lytchak, Leonid Potyagailo, and a referee for several helpful comments.

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Open access funding provided by University of Geneva.
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Anders Karlsson
Section de Mathématiques, Université de Genève, Case postale 64, 1211, Genève,
Switzerland
andMathematics Department, Uppsala University, Box 256, 751 05, Uppsala, Sweden.
Anders.Karlsson@unige.ch
Received: 18 January 2023
Revised: 6 September 2023
Accepted: 12 September 2023