Cones and Gauges in Complex Spaces: Spectral Gaps and Complex Perron-Frobenius Theory

ANNALS OF MATHEMATICS

Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory

By Hans Henrik Rugh

SECOND SERIES, VOL. 171, NO. 3 May, 2010

anmaah

Annals of Mathematics, 171 (2010), 1707–1752

Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory

By HANS HENRIK RUGH

Abstract

We introduce complex cones and associated projective gauges, generalizing a real Birkhoff cone and its Hilbert metric to complex vector spaces. We deduce a variety of spectral gap theorems in complex Banach spaces. We prove a dominated complex cone contraction theorem and use it to extend the classical Perron-Frobenius Theorem to complex matrices, Jentzsch’s Theorem to complex integral operators, a Kre˘ın-Rutman Theorem to compact and quasi-compact complex operators and a Ruelle-Perron-Frobenius Theorem to complex transfer operators in dynamical systems. In the simplest case of a complex n by n matrix A Mn.ރ/ we have the following statement: Suppose that 0 < c < is such 2 C1 that Im Aij Amn < c Re Aij Amn for all indices. Then A has a ‘spectral gap’. j x j Ä x 1. Introduction

The Perron-Frobenius Theorem [Per07], [Fro08] asserts that a real square matrix with strictly positive entries has a ‘spectral gap’, i.e. the matrix has a positive simple eigenvalue and all other eigenvalues are strictly smaller in modulus. More generally, let A be a bounded linear operator acting upon a real or complex Banach space and write rsp.A/ for its spectral radius. We say that A has a spectral gap if (1) it has a simple isolated eigenvalue , the modulus of which equals rsp.A/ and (2) the remaining part of the spectrum is contained in a disk centered at zero and of radius strictly smaller than rsp.A/. Jentzsch generalized in [Jen12] the Perron-Frobenius Theorem to integral operators with a strictly positive continuous kernel. Kre˘ın-Rutman [KR48, Th. 6.3] (see also [Rut40] and [Rot44]) gave an abstract setting of this result by considering a punctured, real, closed cone mapped to its interior by a compact operator. Compactness of the operator essentially reduces the problem to ﬁnite dimensions.

1707 1708 HANSHENRIKRUGH

Birkhoff, in a seminal paper [Bir57], developed a more elementary and in- tuitive (at least in our opinion) Perron-Frobenius ‘theory’ by considering the projective contraction of a cone equipped with its associated Hilbert metric. Birkhoff noted that this projective metric satisfies a contraction principle, i.e. any linear map preserving the cone is a contraction for the metric and the contraction is strict and uniform if the image of the cone has finite projective diameter. All these results, or rather their proofs, make use of the ‘lattice’-structure induced by a real cone on a real Banach space (see [Bir67] and also [MN91]). On the other hand, from complex analysis we know that the Poincare´ metric on the unit disk, ބ z ރ z < 1 , and the induced metric on a hyperbolic Riemann surface D f 2 W j j g enjoy properties similar to the Hilbert metric, in particular a contraction principle with respect to conformal maps. More precisely, if U V is a conformal map W ! between hyperbolic Riemann surfaces then its hyperbolic derivative never exceeds one. The map is a strict contraction unless it is a bijection (see e.g. [CG93, Ch. I.4: Ths. 4.1 and 4.2]). By considering analytic images of complex discs Kobayashi [Kob67], [Kob70] (see also [Ves76]) constructed a hyperbolic metric on complex (hyperbolic) manifolds, a tool with many applications also in infinite dimensions (see e.g. [Rug02, App. D]). Given a real cone contraction, perturbation theory allows one on abstract grounds, to consider ‘small’ complex perturbations but uniform estimates are usually hard to obtain. Uniform complex estimates are needed e.g. when (see e.g. [NN87], [BRS05]) proving local limit theorems and refined large deviation theorems for Markov additive processes and also (see e.g. [Rue79], [Rug02]) for studying the regularity of characteristic exponents for time-dependent and/or random dynamical systems (see 10 below). It is desirable to obtain a description of a projective contraction and, in particular, a spectral gap condition for complex operators without the restrictions imposed by perturbation theory. We describe in the following how one may accomplish this goal. In Section 2 we introduce families of ރ-invariant cones in complex Banach spaces and a theory for the projective contraction of such cones. The central idea is simple, namely to use the Poincare´ metric as a ‘gauge’ on two-dimensional affine sections of a complex cone. At first sight, this looks like the Kobayashi construction. A major difference, however, is that we only consider disk images in two-dimensional subspaces. Also we do not take infimum over chains (so as to obtain a triangular inequality, see Appendix A). This adapts well to the study of linear operators and makes computations much easier than for the general Kobayashi metric. Lemma 2.3 shows that this gauge is indeed projective. The contraction principle for the Poincare´ metric translates into a contraction principle for the gauge and, under additional regularity assumptions, developed in Section 3, into a projective contraction, and finally a spectral gap, with respect to the Banach space norm. CONES AND GAUGES IN COMPLEX SPACES 1709

In Sections 4 and 5 we consider real cones and define their canonical complex- n n n ification. For example, ރ u ރ ui uj ui uj ; i; j u ރ C D f 2 W j C j j j 8 g D f 2 W Re ui uj 0; i; j is the canonical complexification of the standard real cone, 8 g ޒn . We show that our complex cone contraction yields a genuine extension of the BirkhoffC cone contraction: A real Birkhoff cone is isometrically embedded into its canonical complexification. It enjoys here the same contraction properties with respect to linear operators. We obtain then in Section 6 one of our main results: When a complex operator is dominated by a sufficiently regular real cone contraction (Assumption 6.1) then (Theorem 6.3) the complex operator has a spectral gap. It is of interest to note that the conditions on the complex operator are expressed in terms of a real cone and, at least in some cases, are very easy to verify. Sections 7–9 thus present a selection of complex analogues of well-known real cone contraction theorems: A Perron-Frobenius Theorem for complex matrices (as stated at the end of the abstract), Jentzsch’s Theorem for complex integral operators, a Kre˘ın- Rutman Theorem for compact and quasi-compact complex operators and a Ruelle- Perron-Frobenius Theorem for complex transfer operators. In Section 10 we prove results on the regularity of characteristic exponents of products of random complex cone contractions. Finally, in Section 11 we discuss how our results compare to those of perturbation theory.

2. Complex cones and gauges

Let ރ ރ denote the Riemann sphere. When U ރ is an open D [ f1g connected subset avoiding at least three points one says that the set is hyperbolic. We write dU for the corresponding hyperbolic metric. We refer to [CG93, Ch. I.4] or [Mil99, Ch. 2] for the properties of the hyperbolic metric which we use in the present paper. As normalization we use ds 2 dz =.1 z 2/ on the unit disk ބ D j j j j and the metric dU on U induced by a Riemann mapping ބ U . One then has: W !

1 z dބ.0; z/ (2.1) dބ.0; z/ log C j j; z tanh : D 1 z j j D 2 j j Let E be a complex topological vector space. We denote by Span x; y f g D x y ; ރ the complex subspace generated by two vectors x and y in E. f C W 2 g Deﬁnition 2.1. (1) We say that a subset Ꮿ E is a closed complex cone if it is closed in E, ރ-invariant (i.e. Ꮿ ރ Ꮿ) and Ꮿ 0 . D ¤ f g (2) We say that the closed complex cone Ꮿ is proper if it contains no complex planes, i.e. if x and y are independent vectors then Span x; y Ꮿ. f g 6 Throughout this paper we will simply refer to a proper closed complex cone as a ރ-cone. 1710 HANSHENRIKRUGH

Let Ꮿ be a ރ-cone. Given a pair of nonzero vectors, x; y Ꮿ Ꮿ 0 , we 2 Á f g consider the subcone: Span x; y Ꮿ. We wish to construct a ‘projective distance’ f g \ between the complex lines ރx and ރy within this subcone. We do this by considering the complex afﬁne plane through 2x and 2y, choosing coordinates as follows (Lemma 2.3 below implies that the choice of afﬁne plane is of no importance):

(2.2) D.x; y/ D.x; y Ꮿ/ ރ .1 /x .1 /y Ꮿ ރ; Á I D f 2 W C C 2 g with the convention that D.x; y/ if and only if x y Ꮿ. The interior of 1 2 2 this “slice” is denoted Do.x; y/ (for the spherical topology on ރ). Continuity of the canonical mapping ރ2 Span x; y implies that D D.x; y/ is a closed ! f g D subset of ރ. As the cone is proper, D ރ is a strict subset so that ރ D is open and nonempty, whence it contains (more than) three points. If, in addition, Do is connected it is a hyperbolic Riemann surface ([CG93, Th. I.3.1]). Deﬁnition 2.2. Given a ރ-cone, we deﬁne the gauge, d Ꮿ Ꮿ Œ0; , Ꮿ W ! C1 between two points x; y Ꮿ as follows: When two vectors are co-linear we set 2 d .x; y/ 0. If they are linearly independent and 1 and 1 belong to the same Ꮿ D connected component U of Do.x; y/ we set:

(2.3) d .x; y/ dU . 1; 1/ > 0: Ꮿ Á In all remaining cases, we set d . Ꮿ D 1 When V Ꮿ is a (sub-)cone of the ރ-cone Ꮿ we write diamᏯ.V / sup dᏯ.x; y/ Œ0; Á x;y V 2 C1 2 for the projective ‘diameter’ of V in Ꮿ. We call it a diameter even though the gauge need not verify the triangular inequality, whence it need not be a metric (see Appendix A for more on this issue).

LEMMA 2.3. Let Ꮿ be a ރ-cone. The gauge on the cone is symmetric and projective, i.e. for x; y Ꮿ and a ރ : 2 2 d .y; x/ d .x; y/ d .ax; y/ d .x; ay/: Ꮿ D Ꮿ D Ꮿ D Ꮿ Proof. For .1 /a .1 / 0 we write C C ¤ .1 /a .1 / .1 /ax .1 /y C C ..1 R/x .1 R/y/ C C D 2 C C with .1 /a .1 / R Ra./ C : D D .1 /a .1 / C C Then Ra extends to a conformal bijection Ra D.ax; y/ Ra./ D.x; y/ W 2 7! 2 (a Mobius¨ transformation of ރ) preserving 1 and 1. The hyperbolic metric is invariant under such transformations so indeed d .x; y/ d .ax; y/ (but both Ꮿ D Ꮿ CONES AND GAUGES IN COMPLEX SPACES 1711 could be inﬁnite). Similarly, the map yields a conformal bijection between 7! the domains D.x; y/ and D.y; x/, interchanging 1 and 1, and the symmetry follows.

LEMMA 2.4. Let T E1 E2 be a complex linear map between topological W ! vector spaces and let Ꮿ1 E1 and Ꮿ2 E2 be ރ-cones for which T.Ꮿ / Ꮿ . 1 2 Then the map, T .Ꮿ ; d / .Ꮿ ; d /; W 1 Ꮿ1 ! 2 Ꮿ2 is a contraction. If the image has ﬁnite diameter, i.e. diam T Ꮿ < , D Ꮿ2 1 1 then the contraction is strict and uniform. More precisely, there is ./ < 1 D (depending on only) for which

d .T x; Ty/ d .x; y/; x; y Ꮿ : Ꮿ2 Ä Ꮿ1 8 2 1 Proof. Let x; y Ꮿ and set D1 D.x; y Ꮿ1/ and D2 D.T x; Ty Ꮿ2/ 2 1 D I D I for which we have 1; 1 D1 D2 ރ: f g Suppose that T x; Ty Ꮿ are linearly independent and that D2 and D1 are hyper- 2 2 bolic (if not, dᏯ2 .T x; Ty/ vanishes and we are through). Since shrinking a domain increases hyperbolic distances, it follows that d .T x; Ty/ d .x; y/ (although Ꮿ2 Ä Ꮿ1 both could be inﬁnite). Suppose now that < . Then 1 and 1 belong to the same connected C1 component, V , of Do.T x; Ty/. We may suppose that 1 and 1 also belong to the same connected component, U , of Do.x; y/ (or else d .x; y/ / and we Ꮿ1 D 1 are through). Our assumptions imply that U V is a strict inclusion and that diamV .U / . Choose U and pick p V U for which dV .; p/ (this Ä 2 2 n Ä is possible as the inclusion U V is strict and the diameter of U did not exceed ). The inclusion U, V p is nonexpanding and the inclusion V p , V is ! f g f g ! a contraction which has hyperbolic derivative uniformly smaller than some D ./ < 1 on the punctured -neighborhood, BV .p; /, of p (see Remark 2.5). In particular, the composed map (see Figure 1) U, V p , V has hyperbolic ! f g ! derivative smaller than ./ at BV .p; / . As U was arbitrary this is true 2 2 at any point along a geodesic joining 1 and 1 in U so that d .T x; Ty/ dV . 1; 1/ dU . 1; 1/ d .x; y/: Ꮿ2 D Ä D Ꮿ1 Remark 2.5. An explicit bound may be given using the expression ds 1 D dz =. z log z / for the metric on the punctured disk at z ބ (see e.g. [Mil99, j j j j j j 2 2t 1 Ex. 2.8]). Denoting, t tanh =2, we obtain the bound, ./ 1 t 2 log t D D D sinh./ log.coth 2 / < 1. Often, however, it is possible to improve this bound. For example, suppose that U is contractible in V (e.g. if V is simply connected) 1712 HANSHENRIKRUGH

V − {p} V pp U λ λ λ −1 +1

Figure 1. The sequence of inclusions U, V p , V in the ! f g ! proof of Lemma 2.4 and that U is contained in a hyperbolic ball of radius 0 < R < . Lifting to the 1 universal cover we may assume that V ބ and that U z ބ z < t with R D D f 2 W j j g 0 < t tanh 2 < 1. The inclusion .U; dU /, .ބ; dބ/ has hyperbolic derivative 1 z D2=t 2 ! R t j j t for z U . We may thus use tanh < 1 for the contraction 1 z 2 Ä 2 D 2 constant.j j Recall that for a real Birkhoff cone [Bir57] one may take tanh (an D 4 open interval in ޒ of diameter is a ball of radius =2 in ޒ).

3. Complex Banach spaces and regularity of ރ-cones

Let X be a complex Banach space and let Ꮿ X be a ރ-cone (Deﬁnition 2.1). We denote by X the dual of X and we write ; for the canonical duality 0 h i X X ރ. We will consider a bounded linear operator T L.X/ which preserves 0 ! 2 Ꮿ and is a strict and uniform contraction with respect to our gauge on Ꮿ. We seek conditions that assure: (1) The presence of an invariant complex line (existence of an eigenvector of nonzero eigenvalue) and (2) A spectral gap. In short, an invariant line appears when the cone is not too ‘wide’ and the spectral gap when, in addition, the cone is not too ‘thin’.

Definition 3.1. Let Ꮿ X be a closed complex cone in a complex Banach space (in 4 we will use the very same definition for a real cone in a real Banach space). When m X is a nonzero functional, bounded on the vector space gener- 2 0 ated by Ꮿ, we define the aperture of Ꮿ relative to m: m u K.Ꮿ m/ sup k k k k Œ1; : I D u Ꮿ m; u 2 C1 2 jh ij We define the aperture of Ꮿ to be: K.Ꮿ/ inf K.Ꮿ m/ Œ1; . When D m X 0;m 0 I 2 C1 K.Ꮿ/ < we say that Ꮿ is of bounded aperture2 (or¤ of K-bounded aperture with C1 K.Ꮿ/ K < if we want to emphasize a value of the bounding constant). Ä C1 Definition 3.2. (1) We call Ꮿ inner regular if it has nonempty interior in X. CONES AND GAUGES IN COMPLEX SPACES 1713

We say that Ꮿ is reproducing (or generating) if there is a constant g < C1 such that for every x X we may find x1; x2 Ꮿ for which x x1 x2 and 2 2 D C (3.4) x1 x2 g x : k k C k k Ä k k We say that Ꮿ is T -reproducing (or T -generating) if there are constants g < , q ގ 0 and 2 p < such that for every x X and " > 0 there are C1 2 [ f g Ä C1 2 y1; : : : ; yp Ꮿ with 2 y1 y2 yp g x k k C k k C C k k Ä k k and q y1 y2 yp T x < ": k C C C k (2) We say that Ꮿ is outer regular if K.Ꮿ/ < . We say that Ꮿ is of C1 K-bounded sectional aperture or of bounded sectional aperture (with a bounding constant 1 K < ) if and only if for every pair x; y X, the subcone Ä C1 2 Span x; y Ꮿ is of K-bounded aperture, i.e. there is a nonzero linear functional, f g \ m m x;y Span x; y 0, such that D f g 2 f g 1 (3.5) m; u u m ; u Span x; y Ꮿ: jh ij K k k k k 8 2 f g \ (3) We say that Ꮿ is regular if and only if the cone is inner and outer regular. Remarks 3.3. When a cone is of bounded aperture then the cone has a bounded global transverse section not containing the origin. This is often a too strong re- quirement. For example, in L1-spaces this is usually acceptable but not in Lp with 1 < p unless we are in finite dimensions. Being inner regular means Ä C1 containing an open ball and this typically fails in Lp for 1 p < , again with Ä C1 the exemption of the finite-dimensional case. The notions of being (T -)reproducing and of bounded sectional aperture, respectively, are more flexible and may circum- vent the two above-mentioned restrictions. We illustrate this in Example 4.9 and Theorem 7.2. Obviously, ‘inner regular’ reproducing T -reproducing. Also, ) ) ‘outer regular’ bounded sectional aperture. ) It is necessary to create a passage between the cone gauge and the Banach space norm. The regularity properties defined above will enable us to do so in the following two lemmas:

LEMMA 3.4. Let Ꮿ be a closed complex cone of K-bounded sectional aperture. Then Ꮿ is proper, whence a ރ-cone (Deﬁnition 2.1). If x; y Ꮿ and 2 m m x;y is a functional associated to the subcone Span x; y Ꮿ as in equation D f g f g\ (3.5) then: x y 4K d .x; y/ d .x; y/ tanh Ꮿ K Ꮿ : m; x m; y Ä m 4 Ä m h i h i k k k k 1714 HANSHENRIKRUGH

Proof. We normalize the functional so that m K. Then u m; u k k D k k Ä jh ij Ä K u for all u Span x; y Ꮿ. Denote x x and y y and consider, k k 2 f g \ y D m;x y D m;y as a function of ރ, the point u .1 /hx i .1 /y.h Wheni u Ꮿ the 2 D C y C y 2 properties of m show that u m; u .1 / .1 / 2 and therefore, k k Ä jh ij D j C C j Á x y u . x y / 4: j j ky yk Ä k k C kyk C kyk Ä Setting R 4 Œ2; we see that D.x; y/ B.0; R/. The radius R is D x y 2 C1 y y bounded if andky only yk if x and y are independent so the cone is proper. Enlarging a domain decreases hyperbolic distances and so 1 1 1 1 R dᏯ.x; y/ dDo.x;y/. 1; 1/ dB.0;R/. 1; 1/ dބ ; 2 log C 1 : D y y D R R D 1 R x y 1 dᏯ.x;y/ dᏯ.x;y/ Therefore, ky yk tanh , and the stated bound follows. 4 D R Ä 4 Ä 4 LEMMA 3.5. Let Ꮿ be a ރ-cone of a K-bounded sectional aperture and let x Ꮿ , y X. Suppose that there is r > 0 such that x t y Ꮿ for all t ރ 2 2 C 2 2 with t < r. Then j j 2 (3.6) d .x; x s y/ s o. s / as s 0 Ꮿ C Ä r j j C j j ! and K (3.7) y x : k k Ä r k k Proof. Let s < r. Using the scale-invariance of the cone we see that j j D.x; x s y/ ރ .1 /x .1 /.x sy/ Ꮿ C D f 2 W C C C 2 g n s o ރ x .1 / y Ꮿ : D 2 W C 2 2 Our hypothesis implies that D.x; x s y/ contains a disc of radius 2r , centered C s at 1. Shrinking a domain increases hyperbolic distances, whence j j s 1 s =r 2 dᏯ.x; x s y/ dB.1; 2r /. 1; 1/ dބ 0; j j log C j j s o. s /: C Ä s D r D 1 s =r D r j j C j j j j j j Choose m associated to the subcone Span x; y Ꮿ as in equation (3.5), f g \ normalized so that m K. As m is nonzero on the punctured subcone, 0 < k k D m; x t y m; x t m; y for all t < r, and this implies that r m; y jh C ij D jh iC h ij j j jh ij Ä m; x . Possibly after multiplying x and y with complex phases we may assume jh ij that m; x m; ry > 0. But then h i h i 2r y x ry x ry m; x ry x ry 2K x : k k Ä k C k C k k Ä h C C i Ä k k THEOREM 3.6. Let Ꮿ be a ރ-cone of K-bounded sectional aperture. Let T 2 L.X/ be a strict cone contraction, i.e. T Ꮿ Ꮿ with diam T.Ꮿ / < . W ! D Ꮿ 1 Let < 1 be as in Lemma 2.4. Then: CONES AND GAUGES IN COMPLEX SPACES 1715

(1) Ꮿ contains a unique T -invariant complex line, ރh. We define ރ by setting T h h. 2 D (2) There are constants R; C < and a map c Ꮿ ރ so that for any x Ꮿ z C1 W ! 2 and n 1: (3.8) nT nx c.x/h C n 1 x ; k k Ä z k k (3.9) c.x/h R x : k k Ä k k Proof. Let x0 Ꮿ and set e1 T x0= T x0 T.Ꮿ / Ꮿ . We will construct 2 D k k 2 a Cauchy-sequence .en/n ގ recursively. Given en, n 1 choose, as in Definition 2 3.2 (2), a functional mn X normalized so that mn K, associated to the 2 0 k k D subcone Span en; Ten Ꮿ. Set n mn; Ten = mn; en (for which we have the f g \ D h i h i bound 0 < n T K) and define the next element in our recursion: j j Ä k k 1 n Ten n 1 en 1 1 T C Ꮿ: C D Ten 2 k n k Using Lemma 3.4 and then Lemma 2.4 (with a contraction constant < 1) we obtain for n 1: en Ten n n 1 (3.10) dᏯ.en; Ten/ diamT Ꮿ : mn; en mn; Ten Ä Ä Ä h i h i As 1 mn; en K and mn; Ten T K we get: Ä jh ij Ä jh ij Ä k k 1 n 1 n 1 (3.11) en Ten K and nen Ten T K : k n k Ä k k Ä k k Noting that en 1, the first inequality implies: k k D n 1 (3.12) en en 1 2K : k C k Ä The sequence, .en/n ގ, is therefore Cauchy, whence has a limit, 2 (3.13) h lim en Ꮿ; h 1: D n 2 k k D The limit belongs to Ꮿ because the cone was assumed closed. Writing

.n 1 n/en 1 .T n/en .n 1 T /en 1 .T n/.en 1 en/ C C D C C C C C and using the second inequality in (3.11) as well as (3.12) and n T K we j j Ä k k obtain n 1 (3.14) n n 1 .1 .2 2K// T K ; j C j Ä C C C k k so that the limit limn n exists also. But T h h limn Ten nen 0 D k k D k k D shows that T h h Ꮿ which implies that 0, whence that ރh Ꮿ is a D 2 ¤ T -invariant complex line. Suppose also that ރk Ꮿ (with k 0) is T -invariant. ¤ 1716 HANSHENRIKRUGH

Then d .h; k/ d .T h; T k/ d .h; k/ < and this implies d .h; k/ Ꮿ D Ꮿ Ä Ꮿ Ä C1 Ꮿ 0 so that the two vectors must be linearly dependent. Thus, ރh is unique. D n To see the second part, let x Ꮿ and deﬁne for n 1: xn T x. This 2 D time we pick mn X 0 associated to the subcone Span xn; h Ꮿ and set cn n 2 n f g \ D mn; xn = mn; h for which cn K xn . As in (3.10) we get h i h i j j Ä k k xn h K n n (3.15) dᏯ.T x; T h/ mn; xn mn; h Ä mn h i h i k k K n K n 1 diamT Ꮿ : Ä mn Ä mn k k k k n n n 1 Thus, xn cnh K xn and k k Ä k k n 1 n 1 n n xn 1 xn T . xn cnh/ . xn cnh/ k C k D k k 1 n n 1 .1 T /K xn : Ä C k k k k n 1 1 n 1 n Then xn 1 1 .1 T / K xn so we get the fol- k C k Ä C C k k k k lowing uniform bound in n 1: n Y 1 k 1 (3.16) xn 1 .1 T / K x1 k k Ä C C k k k k k 0 K (3.17) exp .1 1T / 1 T x R x : Ä C k k 1 k k k k Á k k n 1 1 n Writing .cn 1 cn/h .cn 1h xn 1/ . T /. xn cnh/ we C D C C C obtain 1 n 1 cn 1 cn cn 1h cnh T K R x : j C j D k C k Ä C k k k k

Therefore, c lim cn ރ exists and by re-summing to , D 2 1 1 T n 1 c cn C k k K R x : j j Ä 1 k k Then also 1 n 1 T n 1 n 1 xn c h C k k K R x C x k k Ä 1 k k Ä z k k which implies that c.x/ c depends only on x (and not on the choice of mn’s). Á n n We also have: c.x/ c.x/h limn xn sup xn R x . j j D k k D k k Ä n k k Ä k k THEOREM 3.7. Let T L.X/ and let Ꮿ be a ރ-cone which is a K-bounded 2 sectional aperture and which, in addition, is reproducing. Suppose that T is a strict cone contraction, i.e. T Ꮿ Ꮿ with diam T.Ꮿ / < . Then T has W ! D Ꮿ 1 a spectral gap. CONES AND GAUGES IN COMPLEX SPACES 1717

Proof. Let x X and let g be the ‘reproducing’ constant from (3.4). Pick 2 x1; x2 Ꮿ with x x1 x2 and x1 x2 g x . We apply the previous 2 D C k k C k k Ä k k theorem to x1 and x2 and set c c.x1/ c.x2/ for which c R x1 D C n n j j Ä nk 1 k C R x2 gR x . In a similar way we obtain T x c h g C x , k k Ä k k k k Ä z k k n 1. Letting n , we see that c.x/ c depends on x but not on the ! 1 Á choice of the decomposition. Linearity of T furthermore implies that the mapping x ; x c.x/ ރ must be linear, and, as a linear functional, X is bounded ! h i Á 2 2 0 in norm by A gR. We have shown that Á (3.18) nT nx h ; x C n 1 x ; x X; n 1; k h ik Ä k k 8 2 with C < . Therefore, is a simple eigenvalue of T corresponding to the eigen- C1 projection, x h ; x and the remainder has spectral radius not exceeding . ! h i j j In the more general case, when Ꮿ is T -reproducing, we let g; p; q be constants from Deﬁnition 3.2. For x X and ﬁxed k 1, set " 1=k and choose: 2 D y1; : : : ; yp Ꮿ with y1 y2 yp g x and y1 y2 yp q 1 2 k k C k k C C k k Ä k k k C C C T x < k . Setting ck c.y1/ c.yp/ we obtain in this way a sequence k nD n q C C n 1 1 n n .ck/k 1 for which T C x ckh gC x k T x , n 1. The k k Ä z k kC k k sequence is bounded so we may extract a convergent subsequence c c and km ! conclude that nT n qx c h gC n 1 x , n 1. Again, ; x c k C k Ä z k k 8 h i Á depends linearly on x and T has a spectral gap. Remark 3.8. Some explicit estimates for the constants in Theorem 3.7 when Ꮿ is reproducing: K (3.19) c A g 1 T exp 1 1 T k k Ä Á k k C k k 1 and K C 1 1 T A: D C k k 1 Note, however, that in this setting there is no a priori lower bound on . In j j particular, to get an explicit bound on 1 T one needs further information on k k the map T and the cone Ꮿ. Example 3.9. Let X be a complex Banach space and consider e X, ` X 2 2 0 with `; e 1. We write P e ` for the associated one-dimensional projection. h i D D ˝ For 0 < < we set C1 n o (3.20) Ꮿ x X .1 P /x P x : D 2 W k k Ä k k e Then B e; 1 .1 k/k P Ꮿ and K.Ꮿ / .1 / P so that Ꮿ is a regu- C C k k Ä C k k lar ރ-cone. Furthermore, if 0 < 1 < < , then a calculation shows that C1 diam Ꮿ < . Ꮿ 1 C1 1718 HANSHENRIKRUGH

Remark 3.10. We have the following characterization of the spectral gap property: A bounded linear operator, T L.X/, has a spectral gap if and only if it is a 2 strict contraction of a regular ރ-cone. Sketch of proof: One direction is the content of Theorem 3.7 (since a regular cone in particular is of uniformly bounded sectional aperture). For the other direction one uses the spectral gap projection P to construct an adapted norm (equivalent to ): x P x P k T k.1 P /x for kk k k D k kC k 0 k k some fixed choice of .; 1/. Using this norm to define the cone family in (3.20), 2 it is not difficult to see that T is a strict and uniform contraction of Ꮿ , > 0.

4. Real cones

Let Xޒ denote a real Banach space. Recall that a subset Ꮿޒ Xޒ is called a (real) proper closed convex1 cone if it is closed and convex and if

(4.21) ޒ Ꮿޒ Ꮿޒ; C D (4.22) Ꮿޒ Ꮿޒ 0 : \ D f g The real cone is said to be reproducing (or generating) provided Ꮿޒ . Ꮿޒ/ C D Xޒ. By Baire’s Theorem and convexity of Ꮿޒ it is not difficult to see that this is equivalent to the existence of g < such that every x Xޒ decomposes into C1 2 x x x with x ; x Ꮿޒ and such that D C C 2 (4.23) x x g x : k Ck C k k Ä k k Remark 4.1. A possible generalization: As in Definition 3.2 (1) we may say that Ꮿޒ is T -reproducing (or T -generating) if there is g < and q ގ so that C1 2 for every x Xޒ and " > 0 there are y ; y Ꮿޒ (Ꮿޒ is convex) with 2 C 2 (4.24) y y g x and y y T qx < ": k Ck C k k Ä k k k C k In the following, we will refer to a real proper closed convex cone as an ޒ-cone. We assume throughout that such a cone is nontrivial, i.e. not reduced to a point. Given an ޒ-cone one associates a projective (Hilbert) metric for which we here give two equivalent definitions (for details we refer to [Bir57], [Bir67]). The first, originally given by Hilbert, uses cross-ratios and is very similar to our complex cone gauge: Let ޒ ޒ denote the extended real line (topologically D [f1g a circle). For x; y Ꮿޒ Ꮿޒ 0 , we write 2 Á f g n o (4.25) `.x; y/ t ޒ .1 t/x .1 t/y Ꮿޒ Ꮿޒ D 2 W C C 2 [ with the convention that `.x; y/ if and only if x y Ꮿޒ Ꮿޒ. Properness 1 2 2 [ of the cone implies that `.x; y/ ޒ if and only if x and y are co-linear. In that case D 1 Convexity of a real cone often is a useful property that a fortiori is lost when dealing with complex cones. CONES AND GAUGES IN COMPLEX SPACES 1719 we set their distance to zero. Otherwise, `.x; y/ is a closed (generalized) segment Œa; b ޒ containing the segment Œ 1 1; see Figure 3 in Section 5. I The logarithm of the cross-ratio of a; 1; 1; b ޒ, 2 a 1 b 1 (4.26) d .x; y/ R.a; 1; 1; b/ log C ; Ꮿޒ D D a 1 b 1 C then yields the Hilbert projective distance between x and y. Birkhoff [Bir57] found an equivalent definition of this distance: For x; y Ꮿޒ Ꮿޒ 0 , one defines 2 Á f g (4.27) ˇ.x; y/ inf > 0 x y Ꮿޒ .0; D f W 2 g 2 C1 in terms of which: (4.28) d .x; y/ log .ˇ.x; y/ˇ.y; x// Œ0; : Ꮿޒ D 2 C1 A simple geometric argument shows that indeed the two definitions are equivalent. Given a linear functional, m X 0 , the image of the cone, m; Ꮿޒ , equals 2 ޒ h i either 0 , ޒ , ޒ or ޒ. One defines the dual cone as f g C 0 0 (4.29) Ꮿޒ m Xޒ m Ꮿޒ 0 : D f 2 W j g Using Mazur’s Theorem, cf. e.g. [Lan93, p. 88], one sees that the ޒ-cone itself may be recovered from:

(4.30) Ꮿޒ x Xޒ m; x 0; m Ꮿ 0 : D f 2 W h i 8 2 ޒg Given an ޒ-cone Ꮿޒ we use Definition 3.1 (replacing ރ by ޒ, complex by real) to define the aperture of Ꮿޒ. It is given as the infimum of K-values for which there exists a linear functional m X 0 satisfying (see Figure 2) 2 ޒ (4.31) u m; u K u ; u Ꮿޒ: k k Ä h i Ä k k 2

LEMMA 4.2. The aperture, K.Ꮿޒ/ Œ1; , of an ޒ-cone, Ꮿޒ Xޒ, is 2 C1 determined by 1 x1 xn (4.32) inf k C C k xi Ꮿޒ; n 1 : K.Ꮿޒ/ D x1 xn W 2 k k C C k k Pn Pn Proof. Let x1; : : : ; xn Ꮿ and note that a xi = xi belongs to 2 ޒ D 1 1 k k A Conv.Ꮿޒ @B.0; 1//, the convex hull of cone elements of norm one. The Á \ reciprocal of the right-hand side in (4.32) therefore equals r inf a a A D fk k W 2 g 2 Œ0; 1. Suppose that r > 0. Then B.0; r/ and Cl A are disjoint convex subsets. The vector difference, Z a b a Cl A; b B.0; r/ , is open, convex and does D f W 2 2 g not contain the origin, whence [Lan93, Lemma 2.2, p. 89] there is ` X whose 2 ޒ0 kernel does not intersect Z. We may normalize ` so that B.0; r/ ` < 1 and Cl A ` 1 : f g f g 1720 HANSHENRIKRUGH

1 K Ꮿޒ 1 m 1 Á

Figure 2. A real cone Ꮿޒ of K-bounded sectional aperture

Then x (4.33) x `; x k k; x Ꮿޒ ; k k Ä h i Ä r 8 2 1 and therefore K.Ꮿޒ/ . To get the converse inequality let m be positive and PÄ r P P P verify (4.31). Then xi m; xi m; xi K xi . k k Ä h i D h i Ä k k LEMMA 4.3. Let Ꮿޒ Xޒ be a d-dimensional ޒ-cone of K-bounded sectional aperture. Then Ꮿޒ itself is of dK-bounded aperture. Proof. Let F ޒd . By a theorem of Caratheodory,´ a point in the convex hull, ConvF , is a fortiori in the convex hull of d 1 points in F (see e.g. [Rud91, C p. 73]). If x @ ConvF , we may even write it as a limit of convex combinations 2 of d points in F . Now, apply this to the set A in the proof of the previous lemma. In the formula, (4.32), it thus suffices to consider d cone elements which we may order decreasingly according to their norm, x1 x2 x . Using k k k k k d k Lemma 4.2 with n 2, the K-bounded sectional aperture implies that D 1 1 1 1 x1 xd x1 x x1 x2 x x1 k k C C k k: k C C d k K k kC K k C C d k K k k K d Thus x1 x 1 k C C d k ; x1 x dK k k C C k d k and in view of Lemma 4.2, we see that Ꮿޒ is of dK bounded aperture. d LEMMA 4.4. Let Ꮿޒ ޒ be an ޒ-cone. Then Ꮿޒ is outer regular. Proof. As in the previous lemma it suffices to look at the supremum in (4.32) over d-tuples. The set A x1; : : : ; x Ꮿޒ x1 x 1 is compact Á f d 2 W k k C C k d k D g CONES AND GAUGES IN COMPLEX SPACES 1721 and x1 xd is continuous and nonvanishing on A, whence has a minimum, k C C k 1 r > 0. It follows that K.Ꮿޒ/ < . Ä r C1 Remark 4.5. In the literature, an ޒ-cone Ꮿޒ is said to be norm-directed (with a constant 1 K < ) if x y K x y , x; y Ꮿޒ. For an ޒ-cone Ä 1 k k Ä k C k 8 2 our notion of uniformly bounded sectional aperture is equivalent (up to a small unavoidable loss in constants) to that of being norm-directed. To see this note that if Ꮿޒ is of K-bounded sectional aperture and ` verifies (4.33) then x; y Ꮿޒ: 8 2 x y x y `; x `; y `; x y K x y ; k k Ä k k C k k Ä h i C h i D h C i Ä k C k which shows that Ꮿޒ is K-norm-directed. Conversely, if Ꮿޒ is K-norm-directed then x y x y x y .1 K/ x y k k C k k Ä k C k C k k Ä C k C k and Lemma 4.2 shows that Ꮿޒ is of .K 1/-bounded sectional aperture. For exam- d C d ple, .ޒ ; 1/ is 1-norm directed and of 1-bounded aperture, whereas .ޒ ; / kk kk1 is 1-normC directed, of 2-bounded sectional aperture but only of d boundedC aperture. Lemma 3.5 is a complex cone analogue of being norm-directed.

THEOREM 4.6. Let A L.Xޒ/ and let Ꮿޒ Xޒ be a reproducing ޒ-cone 2 which is K-norm-directed. Suppose that A is a strict cone contraction, i.e. A W Ꮿޒ Ꮿޒ with A diamᏯޒ A.Ꮿޒ/ < . Then A has a spectral gap. More ! D C1 1 precisely, there are > 0 and a one-dimensional projection P for which A P A has spectral radius not greater than tanh 4 < 1. Proof. The statement of this theorem is very close to Birkhoff’s Theorem 4 in [Bir57]. The proof of that theorem may be adapted to the present case. Al- ternatively, we may here simply use Remark 5.10 below. Our Theorem 4.6 also generalizes easily to the case when ‘reproducing’ is replaced by ‘T-reproducing’, equation (4.24). We omit the proof. d COROLLARY 4.7. Let Ꮿޒ be an ޒ-cone in ޒ , d < and suppose that C1 A L.Xޒ/ veriﬁes A.Ꮿ / Int Ꮿޒ. Then A has a spectral gap. 2 ޒ Proof. Implicitly it is assumed that Ꮿޒ has nonempty interior. Lemma 4.4 shows that Ꮿޒ is outer regular, in particular, norm-directed. As is easily shown, the map x; y Ꮿޒ d .x; y/ is continuous. Compactness of Ꮿޒ x 1 then 2 7! Ꮿޒ \ fj j D g implies that diam A.Ꮿޒ / < so the corollary follows from Theorem 4.6. Ꮿޒ C1 Let Ꮿޒ Xޒ be an ޒ-cone. It is standard to write x y y x Ꮿޒ , 2 for the induced partial ordering of x; y Xޒ. For A; B L.Xޒ/, we also write 2 2 A B x Ꮿޒ A.x/ B.x/. , 8 2 W The following dominated cone contraction theorem is trivial in the context of an ޒ-cone contraction. In Section 6 we show that a similar (nontrivial) result holds in the complex case. 1722 HANSHENRIKRUGH

THEOREM 4.8. Let A; P L.Xޒ/ be contractions of the ޒ-cone Ꮿޒ. Suppose 2 that there are constants 0 < ˛ ˇ < for which ˛P A ˇP . Then Ä C1 ˇ diam A.Ꮿޒ / 2 log diam P.Ꮿޒ /: Ꮿޒ Ä ˛ C Ꮿޒ

Proof. Given x; y Ꮿޒ , suppose that ; > 0 are such that P x Py Ꮿޒ, 2 0 2 Py P x Ꮿޒ. Then also ˇAx ˛Ay Ꮿޒ and ˇAy ˛Ax Ꮿޒ so that 0 2 2 0 2 ˇ d .Ax; Ay/ 2 log log. / Ꮿޒ Ä ˛ C 0 and the claim follows by Birkhoff’s characterization (4.28).

Example 4.9. (1) Let A Mn.ޒ/ and suppose that 0 < ˛ Aij ˇ < 2 Ä Ä C1 for all indices. Setting Pij 1 we see that Á n P .ޒ / .t; : : : ; t/ t > 0 C D f W g which is of zero projective diameter in ޒn . By Theorem 4.8 we recover the standard result: C n ˇ diamޒn A .ޒ / A 2 log : C Ä D ˛ n C The cone ޒ is regular and so Theorem 4.6 applies. If 1 > 0 and 2 denote the j j leading eigenvalueC and the second largest eigenvalue (in absolute value), respec- 2 A ˇ ˛ tively, then j j tanh . 1 Ä 4 D ˇ ˛ (2) The standard Perron-FrobeniusC Theorem generalizes to integral operators; cf. Jentzsch’s Theorem [Jen12] and the generalization given by Birkhoff in [Bir57]. We present a somewhat different generalization: Let .; / be a measure space p p p q and let Xޒ L L .; /, 1 p . Let h L (h > 0, a.e.) and m L D Á Ä Ä C1 2 2 (m > 0, a.e.) with q p=.p 1/ Œ1; being the conjugatedC exponent so thatC R D 2 C1 0 < h m d < . Let kA ޒ be a -measurable map. We C1 W ! C ˝ suppose there are constants 0 < ˛ ˇ < so that for -almost all x; y : Ä C1 2 ˛ h.x/m.y/ kA.x; y/ ˇ h.x/m.y/: Ä Ä R Let A L.Xޒ/ be the integral operator deﬁned by A.x/ kA.x; y/.y/ d.y/. 2 D ˇ ˛ Then A has a spectral gap (again with a contraction rate given by ˇ ˛ ). C p p Proof. We write Ꮿޒ L .; / for the cone of positive L -functions ( 0, D a.e.) and compare the operatorC A with the one-dimensional projection P R R D h m d. Our assumption, h m d > 0 shows that P Ꮿޒ Ꮿޒ and that ˇ W ! P 0. By Theorem 4.8, diamᏯޒ A.Ꮿޒ/ 2 log ˛ . D Ä 1 1=p Ꮿޒ is (trivially) reproducing with a constant g 2 2. To see that D Ä Ꮿޒ is of uniformly bounded sectional aperture let f; g Ꮿޒ be of unit norm in p q 2 L , 1 p < , and pick f ; g L ./ with f q g q 1 (the case C Ä C1 Q Q 2 C k Qk D k Qk D CONES AND GAUGES IN COMPLEX SPACES 1723 p , q 1 should be treated separately; we leave this to the reader) and D 1 D R f f d R g g d 1. The functional m.u/ R .f g/u d then veriﬁes Q D Q D D Q CQ u p m.u/ 2 u p for all u Ꮿޒ Span f; g . Thus Ꮿޒ is of 2-bounded k k Ä Ä k k 2 \ f g sectional aperture.

5. The canonical complexiﬁcation of a real Birkhoff cone

A complex cone yields a genuine extension/generalization of the cone contraction described by Birkhoff [Bir57], [Bir67]. More precisely, we will show that any Birkhoff cone may be isometrically embedded in a complex cone, enjoying qualitatively the same contraction properties. Let Xޒ be a Banach space over the reals. A complexification Xރ of Xޒ is a complex Banach space, equipped with a bounded anti-linear complex involution, 2 J Xރ Xރ, J Id (the identity map), J.x/ J.x/, J.x y/ J.x/ J.y/, W ! D 1 D C D C ރ, x; y Xރ, for which Xޒ .Id J /Xރ is the real part. Then Xރ 2 2 D 2 C D Xޒ iXޒ is a direct sum. (Note that this is not the same as regarding Xރ as a ˚ real Banach space. For example, ރn is a complexification of ޒn for any `p-norm, 1 p , while the real dimension of ރn is 2n.) Ä Ä 1 For simplicity we will assume that J is an isometry on Xރ in which case the canonical projections, Re 1 .Id J/ and Im 1 .Id J/, have norm one. We note D 2 C D 2i that any real Banach space, .Xޒ; ޒ/, admits a complexification, Xރ Xޒ iXޒ kk D ˚ as follows: We adopt the obvious rules for multiplying by complex numbers, set J.x iy/ x iy and introduce a norm e.g. using real functionals, C D x iy ރ sup `; x i `; y ` X 0 ; ` ޒ 1 : k C k D fjh i C h ij W 2 ޒ k k Ä g The latter norm is equivalent (within a factor of 2) to any other conjugation invariant norm on Xރ having as real part the given space .Xޒ; ޒ/. For the rest of this k k section Xޒ will denote the real part of a complex Banach space Xރ. A real linear functional, m X 0 , extends to a complex linear functional by setting m; x iy 2 ޒ h C i D m; x i m; y for x iy Xޒ iXޒ. h i C h i C 2 ˚ Definition 5.1. Given an ޒ-cone Ꮿޒ Xޒ we define its canonical complexification: n o (5.34) Ꮿރ u Xރ Re m; u `; u 0; m; ` Ꮿ 0 : D 2 W h i h i 8 2 ޒ PROPOSITION 5.2. We have the following polarization identity:

(5.35) Ꮿ .x iy/ ރ; x y Ꮿ : ރ D f C W 2 ˙ 2 ޒg Proof. Let u Ꮿ . Our deﬁning condition (5.34) means that m; u (assume 2 ރ h i here it is nonzero) must have an argument that varies within a =2 angle as m 2 Ꮿ 0 varies. Normalizing appropriately, we may write u v with ރ and ޒ D 2 1724 HANSHENRIKRUGH

Arg m; v =4. If we set v x iy with x; y Xޒ then m; y m; x for h i Ä D C 2 jh ij Ä h i all m Ꮿ 0 . Hence, m; x y 0 for all such functionals and by (4.30) this is 2 ޒ h ˙ i equivalent to x y Ꮿޒ. If x y (or x y) then we may write u .1 i/ x ˙ 2 D D D C (or u .1 i/ x) so that we may always assume x y Ꮿ . D ˙ 2 ޒ LEMMA 5.3. Let Ꮿޒ be an ޒ-cone of K-bounded aperture. Then its canonical complexiﬁcation, Ꮿރ, is of 2p2 K-bounded aperture.

Proof. Let ` X satisfy x `; x K x ; x Ꮿޒ and extend ` to a 2 ޒ0 k k Ä h i Ä k k 2 complex linear functional. When u Ꮿރ we use polarization, Proposition 5.2, to 2 write u .x iy/ with `; x y 0. Then x y `; x `; y K x y , D C h ˙ i k ˙ 1k Ä h i˙h i Ä k ˙ k from which x `; x and y `; x so that x iy `; x `; x k k Ä h i k k Ä h i 2k C k Ä h i Ä jh i C i `; y . As `; y `; x we also have `; x iy p2 `; x p2 K x h ij jh ij Ä h i1 jh C ij Ä h i Ä k k Ä p2 K x iy . Therefore, u `; u p2 K u and the result follows. k C k 2k k Ä jh ij Ä k k PROPOSITION 5.4. Let Ꮿޒ be an ޒ-cone. If Ꮿޒ is (1) inner regular / (2) reproducing / (3) outer regular/ (4) of bounded sectional aperture, then so is its canonical complexiﬁcation.

Proof. (1) One checks that if Ꮿޒ contains an open ball BXޒ .h; r/ then Ꮿރ contains BXރ .h; r=2/.

(2) Note that when u; v Ꮿޒ then by Proposition 5.2, 2 1 i 1 i u v u v u iv .1 i/ u C v .1 i/ C i Ꮿރ; C D C 2 C 2 D C 2 C 2 2 u v u v because Ꮿޒ. Now, let w u iv Xޒ iXޒ and use that Ꮿޒ is C2 ˙ 2 2 D C 2 ˚ reproducing in Xޒ to write u u u and v v v with u ; u ; v ; v Ꮿޒ D C D C C C 2 and u u g u and v v g v . Then k Ck C k k Ä k k k Ck C k k Ä k k w .u iv / .u iv / Ꮿރ Ꮿރ D C C C C 2 C and u iv u iv u u v v k C C Ck C k C k Ä k Ck C k k C k Ck C k k g. u v / 2g w : Ä k k C k k Ä k k (3) As shown in Lemma 5.3, if Ꮿޒ Xޒ is of K-bounded aperture then Ꮿރ is of 2p2 K-bounded aperture.

(4) Let u1; u2 Ꮿ and write W Span u1; u2 Ꮿރ for the subcone gener- 2 ރ D ރf g\ ated by these two elements. We also write F Span Re u1; Im u1; Re u2; Im u2 D ޒf g and Vޒ F Ꮿޒ which is an at most four- and at least one-dimensional ޒ-subcone D \ of Ꮿޒ. Now, if w W then w .x iy / with x y Ꮿޒ and clearly also 2 D 0 0 C 0 0 ˙ 0 2 x ; y F . But then x y Vޒ so that also w Vރ, with Vރ the complexiﬁcation 0 0 2 0 ˙ 0 2 2 CONES AND GAUGES IN COMPLEX SPACES 1725 of Vޒ. By Lemma 4.3, Vޒ is of 4 K bounded aperture so by Lemma 5.3, Vރ and therefore also W , are of 8p2 K bounded (complex) aperture.

THEOREM 5.5. Let Ꮿޒ be an ޒ-cone and let Ꮿރ denote its canonical complex- iﬁcation (5.34). Then Ꮿރ is a ރ-cone (Deﬁnition 2.1). With dᏯރ for our projective gauge on the complex cone, the natural inclusion,

Ꮿ ; d , Ꮿ ; d ; ޒ Ꮿޒ ! ރ Ꮿރ is an isometric embedding.

Proof. The set Ꮿރ is clearly ރ-invariant. Consider independent vectors, x, y Ꮿޒ . By Lemma 4.4 any finite-dimensional subcone of Ꮿޒ is outer regular, and 2 so is in particular of uniformly bounded sectional aperture. Our previous lemma shows that the corresponding complex cone is of bounded sectional (complex) aperture. But then Ꮿރ must be proper by Lemma 3.4. Regarding the embedding we may normalize the points so that `.x; y/ Œa; b D is a bounded segment in ޒ. Define the ‘boundary’ points, x0 .1 a/x .1 a/y D C C and y0 .1 b/x .1 b/y. For any " > 0 the point "x0 .1 "/y0 is outside D C C C C the closed convex cone Ꮿޒ. By Mazur’s Theorem, [Lan93, p. 88], we may separate this point from Ꮿޒ by a functional ` Ꮿ 0 . For any " > 0 we may then find m; ` Ꮿ 0 2 ޒ 2 ޒ for which

m; x0 `; y0 " and m; y0 `; x0 1: h i D h i D h i D h i D Then u x0 y0 Ꮿރ only if Re ." /." / 0 for any 0 < " 1 D C 2 C C x Ä whence only if Re 0 2 2: x , j C j j j Conversely, when Re 0 and m; ` Ꮿ 0 then x 2 ޒ Re m; u `; u Re . / . m; y0 `; x0 m; x0 `; y0 / 0; h i h i x h i h i C h i h i so this condition is also sufﬁcient. We thus have: D.x0; y0/ ބ. Therefore D D S D D.x; y/ ރ is a generalized disc, symmetric under complex conjugation for which `.x; y/ D.x; y/ ޒ (see Figure 3). In this situation we know explicit formulas D \ for both (4.26) the real and (2.1) the complex hyperbolic metrics in terms of cross- ratios; so we get d .x; y/ d o . 1; 1/ log R.a; 1; 1; b/ d .x; y/. Ꮿރ D D .x;y/ D D Ꮿޒ

COROLLARY 5.6. For n 1 the set n n n ރ u ރ Re ui uj 0; i; j u ރ ui uj ui uj ; i; j C D f 2 W 8 g D f 2 W j C j j j 8 g n n is a regular ރ-cone. The inclusion, ..ޒ / ; d n /, ..ރ / ; d n / is an isometric ޒ ! ރ embedding. C C C C 1726 HANSHENRIKRUGH

−11 −1 1 b a a b

Figure 3. Illustration of two possible conﬁgurations of D.x; y/

Re z2 Im .z1 z2/ C Re z1

Figure 4. An attempt to illustrate the canonical complexiﬁcation 2 2 ރ of ޒ in the coordinate system .Re z1, Re z2, Im .z1 z2// C C C where Im .z1 z2/ 0. We show the part of the cone contained D 2 2 in the region .Re .z1 z2// .Im .z1 z2// 1 The shaded C C C Ä region shows the intersection with the real cone, ޒ2 . C

n Proof. Let ì .ޒ / , i 1; : : : ; n denote the canonical coordinate projections. 2 0 D Then n n ޒ x ޒ ì ; x 0; i and C D f 2 W h i 8 g n n ރ u ރ Re ì ; u `j ; u 0; i; j : C D f 2 W h ih i 8 g Thus, ރn is the canonical complexification of the standard real cone ޒn . See Figure 4C for an illustration of ރ2 . C C CONES AND GAUGES IN COMPLEX SPACES 1727

Below we shall need the following complex polarization:

LEMMA 5.7. Let x y Ꮿ be at a distance d .x y; x y/ < . ˙ 2 ޒ D Ꮿޒ C C1 We may then ﬁnd ˛ ޒ so that the ‘rotated’ vector x iy ei˛.x iy/, or 2 0 C 0 D C equivalently: x x cos ˛ y sin ˛; 0 D y x sin ˛ y cos ˛ 0 D C veriﬁes: x ty Ꮿޒ; t coth : 0 0 2 8 j j Ä 4 In particular, x y Ꮿ and we have for all ` Ꮿ 0 : 0 ˙ 0 2 ޒ 2 ޒ 1 (5.36) `; y0 tanh `; x0 and `; x `; x0 p2 `; x : jh ij Ä 4 h i p2h i Ä h i Ä h i Proof. We have that

`.x y; x y/ t ޒ .1 t/.x y/ .1 t/.x y/ Ꮿޒ Ꮿޒ C D f 2 W C C C 2 [ g t ޒ x ty Ꮿޒ Ꮿޒ : D f 2 W 2 [ g Now write t tan./, ޒ and set D 2 (5.37) ‚.x; y/ ޒ x tan y Ꮿޒ Ꮿޒ D f 2 W 2 [ g ޒ cos x sin y Ꮿޒ Ꮿޒ D f 2 W 2 [ g i ޒ Re e .x iy/ Ꮿޒ Ꮿޒ : D f 2 W C 2 [ g Let Œ1; 2 denote the connected component of ‚.x; y/ containing Œ =4 =4. 3 I Then 1; 2 ; Œ and 2 1 < . When we insert a tan 1 and b tan 2 2 4 4 D D in equation (4.26) (see Figure 5) standard trigonometric formulae show that the projective distance between x y and x y may be written as C sin 2 cos 2 sin. 1/ cos. 1/ d.1; 2/ log C C : Á sin 2 cos 2 sin. 1/ cos. 1/ If we do a complex rotation, x iy ei˛.x iy/ then the last expression 0 C 0 D C in (5.37) shows that ‚.x0; y0/ Œ1 ˛; 2 ˛. Here, the interval J of allowed D ˛-values is such that ‚.x0; y0/ contains Œ =4; =4. Now, J 1 4 ; 1 3 3 D C C Œ 2 ; 2 Œ. The derivative of ˛ J d .x y ; x y / 4 \ 4 4 2 ! Ꮿޒ 0 0 0 C 0 D d.1 ˛; 2 ˛/ equals 2 2 ; cos.2.1 ˛// cos.2.2 ˛// which vanishes precisely at ˛ .2 1/=2. So the minimal distance between D C x y and x y is obtained for this value of ˛ and corresponds to a symmetric 0 0 0 C 0 1728 HANSHENRIKRUGH

x y x y C

1 < 0 2 > 0

a tan.1/ 1 1 b tan.2/ D C D

Figure 5. The subcone Ꮿޒ Span x; y viewed in the x,y- \ f g coordinate system.

2 1 conﬁguration in which `.x0 y0; x0 y0/ Œ L; L with L tan 2 > 1 and L C1 D D d.x0 y0; x0 y0/ 2 log LC1 or equivalently, C D L coth : 4

Thus x ty Ꮿޒ whenever t L and we have obtained the ﬁrst claim. Since 0 0 2 j j Ä `; x ty 0 for all L t L we also get the ﬁrst inequality in (5.36). To h 0 0i Ä Ä see the last inequality note that `; x y 0 implies that `; x cos ˛ `; x h ˙ i h 0i D h i sin ˛ `; y p2 `; x . Similarly `; x cos ˛ `; x sin ˛ `; y p2 `; x h i Ä h i h i D h 0iC h 0i Ä h 0i (because `; x y 0). h 0 ˙ 0i

LEMMA 5.8. Let x1; x2 Ꮿޒ be at a distance d .x1; x2/ < . 2 D Ꮿޒ C1 Through a positive real rescaling, e.g. replacing x1 by tx1 for a suitable t > 0, we may assure that ` Ꮿ 0 : 8 2 ޒ =2 =2 `; x1 e `; x2 ; `; x2 e `; x1 h i Ä h i h i Ä h i and `; x1 x2 tanh `; x1 x2 : jh ij Ä 4 h C i Proof. From the Birkhoff characterization (4.28) of the projective distance we =2 =2 may rescale, say x1, to obtain e x1 x2 Ꮿޒ and e x2 x1 Ꮿޒ. Then =2 =2 2 2 `; x1 e `; x2 and `; x2 e `; x1 . From this we get: h i Ä h i h i Ä h i =2 =2 .e 1/. `; x1 `; x2 / .e 1/. `; x1 `; x2 / h i C h i C h i h i =2 2.e `; x2 `; x1 / 0 D h i h i and similarly with x1 and x2 interchanged. Rearranging terms yields the claim. CONES AND GAUGES IN COMPLEX SPACES 1729

1 PROPOSITION 5.9. Let Ꮿޒ Ꮿޒ be an inclusion of ޒ-cones and denote 1 by Ꮿރ Ꮿރ the inclusion of the corresponding complexified cones. Let ޒ 1 1 D diam .Ꮿ / Œ0; and ރ diam .Ꮿ / Œ0; be the projective Ꮿޒ ޒ 2 C1 D Ꮿރ ރ 2 C1 diameters of the respective inclusions. Then ޒ is finite if and only if ރ is finite.

Proof. From the embedding in Theorem 5.5 we see that ޒ ރ which Ä implies one direction. To see the converse, suppose that tanh ޒ=4 < 1 and 1 D 1 let u1; u2 Ꮿ . We write u1 1.x1 iy1/ with x1 y1 Ꮿ (and similarly 2 ރ D Q C Q Q ˙ Q 2 ޒ for u2). Possibly after applying a real rescaling of e.g. u1 we may by Lemma 5.8 assume that:

ޒ=2 ޒ=2 (5.38) `; x1 e `; x2 and `; x2 e `; x1 : h Q i Ä h Q i h Q i Ä h Q i

Rotating the complex polarization of x1 iy1 and x2 iy2, we may by Q C Q Q C Q Lemma 5.7 assume that u1 x1 iy1 and u2 x2 iy2 with D C D C

(5.39) `; y1 `; x1 and `; y2 `; x2 jh ij Ä h i jh ij Ä h i for all ` Ꮿ 0 . The complex rotation may, however, push x1 and x2 out of the small 2 ޒ cone2 but using the second inequality in (5.36) as well as (5.38) we still have the bound:

ޒ=2 ޒ=2 (5.40) `; x1 2e `; x2 and `; x2 2e `; x1 : h i Ä h i h i Ä h i Proceeding as in the proof of Lemma 5.8 we see that

2eޒ=2 1 (5.41) `; x1 x2 `; x1 x2 with < 1: jh ij Ä h C i Á 2eޒ=2 1 C Let us write u .1 /u1 .1 /u2, ރ, and similarly for x and y . In D C C 2 order to prove our claim it suffices to find a fixed open neighborhood U U.ޒ/ D of the segment Œ 1 1 ރ, depending on ޒ but not on u1 and u2, such that I u Ꮿރ for every U . Let 1 t 1. Then `; yt `; xt and we get 2 2 Ä Ä jh ij Ä h i (with `1; `2 Ꮿ 0 ) (a): 2 ޒ 2 Re `1; ut `2; ut `1; xt `2; xt `1; yt `2; yt .1 / `1; xt `2; xt h ih i D h ih i C h ih i h ih i p 2 as well as (b): `; ut 1 `; xt . jh ij Ä C h i p 2 2 We also obtain the estimates (c): `; u1 u2 `; x1 x2 and jh ij Ä C h C i (d): `; xt .1 t / `; x1 x2 .1 / `; x1 x2 . Let us write t z jh ij j j h C i h C i D C with 1 t 1 and z ރ. Using the expansion `; u `; ut z `; u1 u2 Ä Ä 2 h i D h i C h i

2I am grateful to Loïc Dubois for having pointed this out to me. 1730 HANSHENRIKRUGH and inserting the estimates (a)–(d) we obtain 2 Re `1; u `2; u .1 / `1; xt `2; xt h qih i q h ih i 2 2 2 z 1 . `1; xt `2; x1 x2 `1; x1 x2 `2; xt / j j C C h ih C i C h C ih i 2 2 2 z . / `1; x1 x2 `2; x1 x2 j j C h C ih C i q p 2 2 2! 2 `1; xt `2; xt 2 1 z C : h ih i C C j j 1 p2 p1 2 This remains positive when z r0 where r0 C .1 / > 0 depends j j Ä D p2 2 C upon ޒ only. The set, U , of such t z-values is the r0-neighborhood, of D C the segment Œ 1 1 ރ. Since enlarging a domain decreases hyperbolic distances, I we conclude that ރ dU .1; 1/ < . Ä 1 Remark 5.10. Suppose that T is a real, bounded, linear operator, that Ꮿޒ is a reproducing K-norm-directed cone and that T Ꮿޒ has finite projective diameter in Ꮿޒ. By Theorem 4.6 the operator has a spectral gap. In view of Remark 4.5 and the properties of the canonical complexification shown above, the same conclusion follows when considering the complexified operator acting on the canonically complexified cone. Our complex cone contraction thus contains the real contraction as a special case (but, of course, with a more complicated proof).

6. Dominated complex cone contractions

A real operator P which contracts a real cone Ꮿޒ contracts a fortiori the corresponding complexiﬁed cone Ꮿރ (easy). It is then natural to ask if this complex contraction may be preserved when adding an imaginary part to the operator. Several of our applications below are cast over this idea and, has led us to state an abstract assumption for the action upon Ꮿޒ and a corresponding complex contraction theorem for complexiﬁed cones:

Assumption 6.1. Let P L.Xޒ/ be a contraction of an ޒ-cone Ꮿޒ. Let M 2 2 L.Xރ/ be an operator acting upon the corresponding complex Banach space. We say that M is dominated by P with constants 0 < ˛ ˇ < provided that Ä Ä C1 for all `1; `2 Ꮿ 0 and x1; x2 Ꮿޒ: 2 ޒ 2 (6.42) Re `1; M x1 `2; M x2 ˛ `1; P x1 `2; P x2 ; h ih i h ih i (6.43) Re `1; M x1 `2; M x2 ˇ `1; P x1 `2; P x2 ; h ih i Ä h ih i (6.44) Im `1; M x1 `2; M x2 `1; P x1 `2; P x2 : j h ih ij Ä h ih i Remark 6.2. The above conditions are ޒ -invariant and also stable when taking convex combinations. It thus sufﬁces to verifyC that these conditions hold for CONES AND GAUGES IN COMPLEX SPACES 1731 subsets, V Ꮿޒ and W Ꮿ 0 which are generating for the cone and the dual cone, ޒ respectively, i.e. for which:

Ꮿޒ Cl Conv.ޒ V/ x Xޒ `; x 0; ` W : D C D f 2 W h i 8 2 g When 0 an operator M verifying the above assumption is essentially real. D Possibly after multiplication with a complex phase the operator maps Ꮿޒ into Ꮿޒ itself. The above condition then reduces to the real cone dominated condition of Theorem 4.8. Our goal is here to show that the conclusion of that theorem also applies when M is allowed to have a nontrivial imaginary part. It turns out that the allowed ‘amount’ of imaginary part depends on the rate of contraction of P .

THEOREM 6.3. Let Ꮿޒ Xޒ be a proper convex cone and let P Ꮿ W ޒ ! Ꮿ be a strict cone contraction; i.e. P diam P.Ꮿ / < . Write Ꮿރ for ޒ D Ꮿޒ ޒ C1 the canonical complexiﬁcation of Ꮿޒ. Suppose that M L.Xރ/ is P -dominated 2 (Assumption 6.1) with constants that satisfy: (6.45) cosh P < ˛: 2

Then M Ꮿ Ꮿ and diam M.Ꮿ / < . If , in addition, Ꮿޒ is reproducing W ރ ! ރ Ꮿރ ރ C1 and of uniformly bounded sectional aperture then M has a spectral gap.

P Proof. Let u Ꮿ and `1; `2 Ꮿ 0 . We write tanh < . The 2 ރ 2 ޒ D 4 C1 ﬁrst step is to establish the following inequality (which, in particular, implies that M Ꮿ Ꮿ ): W ރ ! ރ ˛ (6.46) Re `1; M u `2; M u `1; P u `2; P u : h ih i cosh.P =2/ jh ijjh ij

We will use polarization twice to achieve this. First, write u ei .x iy/ with D C ޒ and x y Ꮿ . Then 2 ˙ 2 ޒ

`1; M u `2; M u `1; M.x iy/ `2; M .x iy/ h ih i D h C ih i `1; M x `2; M x `1; My `2; M y D h ih i C h ih i i `1; My `2; M x `1; M x `2; M y C h ih i h ih i 1 `1; M.x y/ `2; M .x y/ `1; M.x y/ `2; M .x y/ D 2 h C ih C iCh ih i i `1; M.x y/ `2; M .x y/ `1; M.x y/ `2; M .x y/ C 2 h C ih ih ih C i 1 i ŒA ŒB: Á 2 C 2 1732 HANSHENRIKRUGH

Since x y Ꮿޒ we may use inequality (6.42) of our assumption to deduce: ˙ 2 Re A ˛ `1; P.x y/ `2; P.x y/ ˛ `1; P.x y/ `2; P.x y/ h C ih C i C h ih i 2˛ `1; P x `2; P x ˛ `1; Py `2; Py D h ih i C h ih i 2˛ Re `1; P.x iy/ `2; P.x iy/ D h C ih i 2˛ Re `1; P u `2;P u : D h ih i For the second term we have by (6.44)

Im B `1; P.x y/ `2; P.x y/ `1; P.x y/ `2; P.x y/ j j Ä h C ih i C h ih C i 2 . `1; P x `2; P x `1; Py `2; Py / D h ih i h ih i 2 `1; P.x iy/ `2; P.x iy/ Ä jh C ij jh ij 2 `1; P u `2; P u ; D jh ij jh ij where for the last inequality we used Schwarz’ inequality. From these two estimates we get:

(6.47) Re `1; M u `2; M u ˛ Re `1; P u `2;P u `1; P u `2; P u : h ih i h ih i jh ij jh ij We note that (6.47) is here independent of the choice of polarization. Since x y Ꮿ we see that the elements P.x y/ Ꮿ and P.x y/ Ꮿ are at a ˙ 2 ޒ C 2 ޒ 2 ޒ projective distance not exceeding P . We may then use Lemma 5.7 to rotate the polarization again and write P u ei˛.x iy / where `; y `; x for all D 0 C 0 jh 0ij Ä h 0i ` Ꮿ 0 . But then 2 ޒ 2 Re `1; P u `2;P u `1; x `2; x `1; y `2; y .1 / `1; x `2; x : h ih i h 0ih 0iCh 0ih 0i h 0ih 0i p p We also obtain `; P u `; x 2 `; y 2 1 2 `; x so that jh ij D h 0i C h 0i Ä C h 0i 1 2 Re `1; P u `2;P u `1; P u `2; P u h ih i 1 2 jh ij jh ij C 1 P cosh `1; P u `2; P u : D 2 jh ij jh ij Together with (6.47) this establishes (6.46).

In order to obtain an estimate for diamᏯރ M.Ꮿރ/ we also need the following inequality: p (6.48) `; M u ˇ `; P u ; ` Ꮿ 0 ; u Ꮿރ: jh ij Ä C jh ij 8 2 ޒ 2 This follows by setting `1 `2 ` in the expression for A and B above and using D D the upper bounds (6.43) and (6.44) of our assumption: A ˇ. `; P.x y/ 2 `; P.x y/ 2/ 2ˇ. `; P x 2 `; Py 2/ 2ˇ `; P u 2 Ä h C i Ch i D h i Ch i D jh ij and the bound B ImB 2 `; P u 2 as before. j j D j j Ä jh ij CONES AND GAUGES IN COMPLEX SPACES 1733

Consider u1; u2 Ꮿ . Using the polarization identity, equation (5.35), we 2 ރ may assume that u1 x1 iy1 with x1 y1 Ꮿ so that `; Py1 `; P x1 . D C ˙ 2 ޒ jh j Ä h i Then also `; P x1 `; P u1 p2 `; P x1 and with the same bounds for h i Ä jh ij Ä h i u2 x2 iy2. Through a real rescaling, Lemma 5.8, we may also assume that D C `; P.x1 x2/ `; P.x1 x2/ . We also write u .1 /u1 .1 /u2 jh ij Ä h C i D C C with t z, 1 t 1 (and similarly for x and y ). By the choice of D C Ä Ä polarization, xt yt Ꮿޒ so that ut Ꮿރ, i.e. belongs to the complex cone for ˙ 2 2 all 1 t 1. We want to show that when z is small enough the same is true Ä Ä j j for M ut z. C First note that `; P xt .1 t / `; P.x1 x2/ . Applying the inequality h i pj j h C i (6.48) we deduce that `; M ut 2 .ˇ / `; P xt and `; M.u1 u2/ p jh ijp Ä C `;Ph xt i jh ij Ä 2 .ˇ / `; P.x1 x2/ 2 .ˇ / h1 t i . Using (6.46) on ut and the C h C i Ä C j j expansion `; M ut z `; M ut z `; M.u1 u2/ , we obtain the inequality h C i D h i C h i (6.49) Re `1; M ut z `2; M ut z Re `1; M ut `2; M ut h C i h C i h i h i z `1; M ut `2; M.u1 u2/ `2; M ut `1; M.u1 u2/ j j jh ij jh ij C jh ij jh ij 2 z `1; M.u1 u2/ `1; M.u1 u2/ j j jh ij jh ij `1; P xt `2P xt h i h i ! ˛ z 2 2.ˇ / 2.ˇ / 1 j j : cosh.P =2/ C C C C 1 t j j When condition (6.45) is satisﬁed, the set of t z values, 1 t 1 for which C Ä Ä the quantity (6.49) is nonnegative contains an open neighborhood, ˇ (6.50) U U ; ; P D ˛ ˛ of the segment Œ 1 1 ރ. It follows that diam M.Ꮿ / dU . 1; 1/ < . I Ꮿރ ރ Ä C1 When Ꮿޒ is reproducing and of uniformly bounded sectional aperture then so is Ꮿރ by Proposition 5.4. Thus, the conclusion follows from our spectral gap theorem, Theorem 3.7.

7. Applications The most striking application is also the simplest. A complex Perron-Frobenius Theorem (see also Appendix 11):

THEOREM 7.1. Let A Mn.ރ/ and suppose there is 0 < c < for which 2 C1 Im Aij Amn < c Re Aij Amn for all indices. Then A has a spectral gap. j x j Ä x n n n Proof. The cone Ꮿޒ ޒ is regular in ޒ . By Corollary 5.6, Ꮿރ ރ is n D C D C regular in ޒ . We will compare M with the constant matrix Pij 1 with respect n Á to the real cone ޒ . As in Example 4.9(1), P 0. The canonical basis and its C D 1734 HANSHENRIKRUGH dual generates the cone and its dual, respectively; cf. Remark 6.2. The constants from Assumption 6.1 then become (sups and infs over all indices) (a) ˛ ˛.A/ D D inf ReAij A , (b) ˇ ˇ.A/ sup ReAij A and (c) .A/ sup ImAij A . xkl D D xkl D D j xkl j Our spectral gap condition of Theorem 6.3 simply reads < ˛ and by ﬁniteness of n this is equivalent to the stated assumptions on A. We also note that the ‘contraction constant’ for the spectral gap, .ˇ=˛; =˛/ < 1 from Lemma 2.4, cf. equations D (6.49) and (6.50), only depends on the ratios ˇ=˛ 1 and =˛ < 1. In the following, denote by osc.h/ ess sup.h/ ess inf.h/ the essential oscil- D lation of a real valued function h on a measured space. Theorem 7.1 may (almost) be viewed as a special case of the following complex version of a result of Jentzsch [Jen12]:

THEOREM 7.2. Let .; / be a measure space and let X Lp.; /, with D 1 p . Let h Lp, h > 0 a.e. and m Lq, m > 0 a.e. with 1 1 1 so Ä Ä C1 2 2 p C q D that 0 < R h m d < ; cf. Example 4.9(2). Given g L . / we deﬁne C1 2 1 the integral operator, Mg L.X/: 2 Z g.x;y/ (7.51) Mg .x/ h.x/ e .y/ m.y/ .dy/: D Set osc.Im g/ and ƒ osc.Re g/. Suppose that < =4 and that tan < D D exp. 2ƒ/. Then Mg has a spectral gap.

Proof. As in Example 4.9(2) we consider the ޒ-cone Ꮿޒ Xޒ 0.a:e:/ R Df 2 W g and we compare with P h m d. We have that P Ꮿ Ꮿ and P 0. D W ޒ ! ޒ D We obtain the following estimate for the constants

g.x;y/ g.x ;y / 2ess inf Re g Re e 0 0 ˛ e cos C Á and g.x;y/ g.x ;y / 2ess sup Re g Im e 0 0 e sin : C Ä Á The cone Ꮿޒ is reproducing. As shown in Example 4.9 (2) the real cone has uniformly bounded sectional aperture. The spectral gap condition of Theorem 6.3 then translates into the stated condition on and ƒ.

8. A complex Kre˘ın-Rutman Theorem

Let X be a complex Banach space. We denote by Gr2.X/ the set of complex planes in X, i.e. subsets of the form ރx ރy with x and y independent vectors C in X. If we write S.X/ for the unit sphere in X then d2.F; F / distH F S.X/; F S.X/ ;F;F Gr2.X/ 0 D \ 0 \ 0 2 CONES AND GAUGES IN COMPLEX SPACES 1735

n deﬁnes a metric on Gr2.X/. In the following let us ﬁx a norm on ރ . The choice may affect the constants below but is otherwise immaterial. The space n .Gr2.ރ /; d2/ is then a compact metric space.

LEMMA 8.1. Let V X be a ރ-cone and let F Gr2.X/. Suppose there is 2 u u F , r > 0 such that B.u; r/ V ∅. Then VF F V has at most 1 k k 2 \ D D \ C r bounded aperture.

Proof. Let m .F /0 be a linear functional with u ker m and m 1. 2 2 a k k D Choose x F for which m; x x . If ax bu VF then u b x B.u; r/ 2 a jh ij D k k C 2 C … so that b j j x and therefore, j j Ä r k k u u ax bu a x 1 k k m; ax bu 1 k k : k C k Ä j j k k C r D jh C ij C r u The two-dimensional space F is spanned by u and x so K.VF / 1 k k . Ä C r LEMMA 8.2. Let V ރn be a ރ-cone. Then there is K < so that V is of 1 K-bounded sectional aperture.

Proof. Suppose that this is not the case. Then we may ﬁnd a sequence Fn of planes for which the aperture K.V Fn/ diverges. Taking a subsequence we may \ assume that Fn converges in Gr2.X/ to a plane F . As V is proper, V F is a \ strict subset of F . Thus there is u F V . But V is closed in ރn so there is r > 0 2 so that B.u; r/ is disjoint from V as well. Given another complex plane, F 0, we may ﬁnd u F for which u u u d2.F; F /. When F and F are close 0 2 0 k 0k Ä k k 0 0 enough, r r u d2.F; F / > 0 and B.u ; r / is also disjoint from V . By our 0 D k k 0 0 0 previous lemma, V F is of aperture not exceeding 1 u =.r u d2.F; F //. \ 0 C k k k k 0 But this contradicts the divergence of K.V Fn/ as Fn F . \ ! LEMMA 8.3. Let V ރn be a ރ-cone and let W V be a closed complex subcone with W Int V . Then there is .W; V / < such that for D C1 x; y W : 2 dW .x; y/ < dV .x; y/ : C1 ) Ä n n 1 Proof. We denote by ރ 0 ރP the canonical projection to W f g !n 1 complex projective space. We consider ރP as a metric space with the metric d n 1 as in equation (A.82). The projected image, .W /, is compact in the open ރP set .Int V / ރP n 1 so there is " ".W; V / > 0 for which the "-neighborhood D of .W / is contained in .V /. Let x; y W be linearly independent and suppose that dW .x; y/ < . Let n 2 C1 F Gr2.ރ / be the complex plane containing x and y. Denote by C the connected 2 set in .F / containing x and y. Let i C , i J , be an "=3-maximally separated " 2 2 S " set in C . Thus, the balls B.i ; 6 /, i J , are all disjoint and i J B.i ; 3 / n 1 2 2 D ރP . The cardinality of J is bounded by a constant depending on " only. Then 1736 HANSHENRIKRUGH

2" " B.i ; 3 / .V /, i J , so by Lemma A.1(2) each B.xi ; 3 /, i J is of radius 1 21=2 2 " not greater than log 1C1=2 log 3 for the dV -metric. Also i J B.; 3 / contains D [ 2 C which is connected. It follows that dV .x; y/ does not exceed 2 log 3 Card.J /. which is bounded by a constant depending on " only.

THEOREM 8.4. Let V ރn be a closed subset which is ރ-invariant and contains no complex planes (in terms of Definition 2.1, V is a ރ-cone). Suppose that A ރn ރn is a linear map for which A.V / Int V . Then A has a spectral gap. W ! Proof. We write W A.V / for the image of V and use the notation and D constants from the two previous lemmas. First note that for x; y V , 2 dV .x; y/ < dV .Ax; Ay/ dV .x; y/ and dV .Ax; Ay/ : 1 ) Ä Ä To see this note that when dV .x; y/ < then dV .Ax; Ay/ < so by Lemma 8.3, 1 1 dV .Ax; Ay/ . If Ꮿ denotes the connected component of F V containing Ä \ x and y then also diamV A.Ꮿ/ . By Lemma 2.4, dV .Ax; Ay/ dᏯ.x; y/ Ä n Än 1 Ä dV .x; y/. Iterating this argument we see that diamV A .Ꮿ / , n 1. By Ä Lemma 8.2, V is of K-bounded sectional aperture, so Lemma A.1(1) assures that n n 1 n1 1 diamރP n 1 A .Ꮿ/ 2K . Fix n1 < so that 2K "=3. Ä C1 Ä Now let i , i J , be an "=3-maximally separated set in W . Setting Vi 1 2 n D B.i ; "/ with i J we see that diamރP n 1 A Vi "=3, n n1. It follows 2 n Ä 1 that there is a map, J J so that A V W B. ; 2"=3/, n n1. W ! i .i/ Á .i/ Since J is of finite cardinality, must have a cycle. Thus, there are i1 J and n1 2 n1 < for which A .Vi1 / Wi1 . The cone Wi1 is regular (easy) and of C1 n1 bounded diameter in Vi1 so that A has a spectral gap and therefore also A. When the operator is sufficiently regular one may weaken the assumptions on the contraction and the outer regularity of the cone. This is illustrated by the following complex version of a theorem of Kre˘ın and Rutman [KR48, Th. 6.3]:

THEOREM 8.5. Let Ꮿ Xރ be an inner regular ރ-cone in the Banach space Xރ. Let A L.Xރ/ be a quasi-compact operator or a compact operator of strictly 2 positive spectral radius and suppose that A Ꮿ Int Ꮿ. Then A has a spectral gap. Proof. Let P be the spectral projection associated with eigenvalues on the spectral radius circle, ރ rsp.A/ . By hypothesis imP is finite-dimensional f 2 Wj jD g and we may find ޒ such that 2 rsp.A.1 P // < < rsp.A/: n n We claim that Ꮿ imP is nonempty: Let x Ꮿ and define en A x= A x \ 2 D k k 2 Ꮿ; n ގ. 2 n n Suppose first that P x 0. Then limn A .1 P /x = A P x 0 so ¤ !1 k k k k D that the distance between en and imP tends to zero. Since im P is locally compact CONES AND GAUGES IN COMPLEX SPACES 1737 and en is bounded we may extract a convergent subsequence e lim en imP D k 2 \ Ꮿ . Suppose instead that P x 0; then Ax Int Ꮿ so there is r > 0 for which D 2 B.Ax; r/ Ꮿ. We may then replace x by Ax u where u imP , u < r and 2 C 2 k k we are back in the first case. Thus Ꮿ Ꮿ imP ∅. Now, P D \ ¤ A Ꮿ .A Ꮿ / imP Int Ꮿ imP Int ᏯP ; W P ! \ \ D the latter for the topology in imP . In particular, Int ᏯP is nonempty so that ᏯP is an inner regular ރ-cone in a finite-dimensional space and A Ꮿ Int ᏯP . W P ! We may then apply the finite-dimensional contraction theorem, Theorem 8.4, to AP A imP L.imP/. It follows that AP , whence also our original operator A D j 2 has a spectral gap. Remark 8.6. In the real cone version (replacing ރ by ޒ) of Theorem 8.5 it is not necessary to assume that the spectral radius of A is strictly positive. This forms part of the conclusion. To see this pick x Ꮿ of norm one. Then Ax Int Ꮿ so 2 2 that there is > 0 for which B.Ax; / Ꮿ. Therefore, Ax x Ꮿ and then 2 also B.A2x; 2/ A.Ax x/ B.Ax; / Ꮿ by the properties of an ޒ-cone. D C More generally, B.Anx; n/ Ꮿ. As 0 @Ꮿ it follows that 2 pn n rsp.A/ lim sup A x > 0: j j The fact that this conclusion is nontrivial is illustrated e.g. by the operator, A.t/ s D R .s/ ds, 0 t 1, which is compact when acting upon X C 0.Œ0; 1/. 0 Ä Ä 2 D It contracts (but not strictly) the cone of positive elements but has spectral radius zero. In the complex setup, if one assumes that Ꮿ is of K-bounded sectional aperture then strict positivity of rsp.A/ also comes for free: Suppose that x Ꮿ, x 1 and n 12 j jn D B.Ax; r/ Ꮿ, r > 0. Then Ax x Ꮿ, < r and also A C x A x Ꮿ C 2 8j j n 1 r n C 2 for such -values. By Lemma 3.5. we see that A C x K A x > 0 from which r j j j j rsp.A/ > 0. K 9. A complex Ruelle-Perron-Frobenius Theorem

The Ruelle-Perron-Frobenius Theorem, [Rue68], [Rue69], [Rue78] (see also [Bow75]), ensures a spectral gap for certain classes of real, positive operators with applications in statistical mechanics and dynamical systems. Ferrero and Schmitt [FS79], [FS88] used Birkhoff’s Theorem on cone contraction to give a conceptually new proof of the Ruelle-Perron-Frobenius Theorem. See also [Liv95] and [Bal00] for further applications in dynamical systems. We present here a generalization to a complex setup. Let .; d/ be a metric space of ﬁnite diameter, D < . We write C 0./ C1 for the Banach space of (real- or complex-valued) continuous functions on under 1738 HANSHENRIKRUGH the supremum norm, 0. When ޒ (or ރ) we write j j W ! Lip./ sup x y =d.x; y/ Œ0; D x y j j 2 C1 ¤ for the associated Lipschitz constant. Then

Xޒ ޒ 0 Lip./ < D f W ! j k k Á j j C C1g (and similarly for Xރ) is a Banach algebra. Let U and let f U be an unramified covering map of which W ! is uniformly expanding. For simplicity, we will take it to be of finite degree (it is an instructive exercise to extend Theorem 9.1 below to maps of countable degree). More precisely, we assume that there are 0 < < 1 and a finite index set J so that for every couple y; y0 we have a pairing ᏼ.y; y0/ .xj ; xj0 / j J of 1 2 1 D f W 2 g the pre-images, f .y/ xj j J U and f .y0/ xj0 j J U , for which D f g 2 D f g 2 d.xj ; xj0 / d.y; y0/, j J . Ä 2 0 Fix an element g Xރ and define for C ./ (or Xރ): 2 2 2 X g.x/ Mg .y/ e .x/; y : D 2 x f .x/ y W D 0 The norm of Mg when acting upon C ./ (in the uniform norm) is given by

X Re g.x/ Mg 0 sup e ; jjj jjj D y x f .x/ y 2 W D and a straightforward calculation shows that Mg L.Xރ/ with 2 M M 0.1 Lip g/: k k Ä jjj jjj C THEOREM 9.1. Denote a Lip Re g, b Lip Im g and osc Im g. Suppose D D D that 2 2 D b 1 4 exp 1 C D a < 1: C 1 2 D a C 1 1 C Then Mg L.Xރ/ has a spectral gap. 2 Proof. We will compare Mg with the real operator P MRe g . For >0 the set, D (9.52) d.y;y / Ꮿ;ޒ ޒ `y;y ; .y/ e 0 .y0/ 0; y; y0 ; D f W ! C j h 0 i Á 8 2 g deﬁnes a proper convex cone in Xޒ which in addition is regular. Inner regularity: Let 1.x/ 1, x and h Xޒ. Then 1 h Ꮿ;ޒ provided Lip h=.1 h 0/ , Á 2 2 C 2 j j Ä whence B.1; min.; 1// Ꮿ;ޒ. Outer regularity: Pick x0 and set `0./ 2 D .x0/. For Ꮿ;ޒ we have Lip 0 so that 2 Ä j j D .1 / 0 .1 /e `0./; k k Ä C j j Ä C and this shows outer regularity. CONES AND GAUGES IN COMPLEX SPACES 1739

Let 0 < 0 < and 1; 2 Ꮿ ;ޒ. As in (4.27) let ˇ .1; 2/ inf > 0 2 0 D f W 1 2 Ꮿ;ޒ . A calculation using the deﬁning properties of the cone family 2 g yields:

1 exp. . 0/d/ 2.y/ 0 2.y/ ˇ .1; 2/ sup C sup C sup ; Ä d>0 1 exp. . 0/d/ y 1.y/ Ä 0 y 1.y/ 2 2 and we get the following bound for the diameter ޒ diamᏯ;ޒ Ꮿ ;ޒ (cf. (4.28)): D 0 0 2.y/ 1.y0/ 0 ޒ 2 log C sup log 2 log C 2 D 0 < : Ä 0 C y;y 1.y/ 2.y0/ Ä 0 C C1 02 The injection Ꮿ ;ޒ , Ꮿ;ޒ is thus a uniform contraction for the respective projec- 0 ! tive metrics. Given Ꮿ;ޒ and using the pairing ᏼ.y; y / we get for the operator 2 0 P MReg : D X P .y/ eRe g.x/.x/ D x f .x/ y W XD eRe g.x0/ .a /d.x;x0/.x / e .a /d.y;y0/P .y /: C 0 C 0 x f .x / y 0W 0 D 0 This implies that P Ꮿ;ޒ Ꮿ ;ޒ with .a /. If we choose > a=.1 / W ! 0 0 D C then P becomes a strict cone contraction of the regular cone Ꮿ;ޒ. We also get the estimate (to obtain an a priori estimate for the contraction one may here try to optimize for the value of ):

P . a/ (9.53) log C C D . a/: 2 Ä . a/ C C C By Theorem 4.6, P L.Xޒ/ has a spectral gap (see [Rue68], [FS79] and also 2 [Liv95]). Returning to the complex operator, Mg , we ﬁx y; y and the correspond- 0 2 ing pairing of pre-images ᏼ.y; y0/ as described above. Let Ꮿ ;ޒ and write P 2 0 `y;y ;Mg j .g/; with h 0 i D j h i g.xj / d.y;y / g.x0 / j .g/; e .xj / e 0 C j .x /; j J: h i Á j0 2 In order to compare with the real operator, we deﬁne complex numbers wj , j J , 2 through the relation

i Im g.xj / j .g/; e wj j .Re g/; : h i D h i Equivalently (when the denominator is nonzero):

g.xj / d.y;y0/ g.xj0 / e .xj / e C .x0 / i Im g.xj / j e wj : Re g.x / d.y;y / Re g.x0 / D e j .xj / e 0 C j .x / j0 1740 HANSHENRIKRUGH

We may apply Lemma 9.3 below with the bounds

Re.z1 z2/ . . a// d.y; y / C 0 and Im.z1 z2/ b d.y; y / to deduce that j j Ä 0 b 2 2 (9.54) Arg wj s0 : and 1 wj 1 s : j j Ä Á . a/ Ä j j Ä C 0 C Given i; j J and 1; 2 Ꮿ ;ޒ we obtain: 2 2 0

j .g/; 1 i .g/; 2 h ih i i.Im g.xj / Im g.xi // e wj wi j .Re g/; i .Re g/; : D h i h i The two last factors are real and nonnegative (because . a/ > 0) and the C complex pre-factor belongs to the set iu 2 A re 1 r 1 s ; u 2s0 : D f W Ä Ä C 0 j j Ä C g Summing over all indices we therefore obtain

`y;y ;Mg 1 `w;w ; M g 2 Z `y;y ; P 1 `w;w ; P 2 ; h 0 ih 0 i D h 0 ih 0 i in which Z is an average of numbers in A and thus belongs to Conv.A/, the convex hull of A. When 2s0 < =4 we conclude that the bounds in Assumption 6.1 are C 2 2 veriﬁed for the constants ˛ cos. 2s0/, .1 s / sin. 2s0/ and ˇ 1 s . D C D C 0 C D C 0 The spectral gap condition in Theorem 6.3 then reads as follows:

2 P (9.55) 1 s tan. 2s0/ cosh < 1: C 0 C 2 Now, in order to get a more tractable and explicit formula we make the following (not optimal) choice for : 2a 1 : D 1 C D 1 Then .a / C so that . .a //=. .a // 0 D C Ä 2 C C C Ä .3 /=.1 /. Using (9.53) we obtain C P 3 1 cosh eP =2 C exp 1 2a D C : 2 Ä D 1 C 1 4 2 3 One also checks that . 2s0/ < 1 implies that .1 s / tan. 2s0/ C C 1 C 0 C 1 < 1 so we may replace (9.55) by the stronger condition

1 4 . 2 s0/ exp 1 C D a < 1: C C 1 1 CONES AND GAUGES IN COMPLEX SPACES 1741

2 2 Finally inserting s0 D b=.1 D a/ we obtain the claimed condition D C which thus sufﬁces to ensure a spectral gap. Remark 9.2. In the literature, one often includes a statement on Gibbs mea- sures as well. If we let h denote the leading spectral projection of P MRe g , ˝ D then positivity of P implies that the ‘state’ Xޒ ./ .h/ is uniformly 2 7! D bounded with respect to 0. By continuity, extends to a linear functional on j j C 0./. If, in addition, we assume compact, then by Riesz, this functional de- ﬁnes a Borel probability measure d on . The measure is invariant and strongly mixing for f . It is known as a Gibbs measure for f and the weight g. This part of the theorem, however, needs the partial ordering induced by the cone of positive continuous functions and does not extend to a complex setup (in general, it is even false there). In the proof we made use of the following complex estimate:

LEMMA 9.3. Let z1; z2 ރ be such that Re z1 > Re z2 and deﬁne w ރ 2 2 through ez1 ez2 ei Im z1 w : Á eRe z1 eRe z2 Then 2 Im .z1 z2/ Im .z1 z2/ Arg w j j and 1 w2 1 : j j Ä Re .z1 z2/ Ä j j Ä C Re .z1 z2/ Proof. Writing t Re .z1 z2/ > 0 and s Im .z1 z2/ we have: D D t is 1 e w t : D 1 e t t 1 e cos s e sin s Taking real and imaginary parts, Re w 1 e t and Im w 1 e t , we get D D sin2.s=2/ t t w 2 1 1 . s /2. Also @ log w 1 @w e e 1 2 t @s w @s 1 e t is 1 e t t j j D C sinh .t=2/ Ä C j jDj jD Ä Ä s j j so that Arg w j j . j j Ä t 10. Random products of cone contractions

A conceptual difference between standard perturbation theory and cone contractions is the behavior under compositions. Composing a sequence of operators that uniformly contracts the same cone, one obtains again a contraction, even with a sub-multiplicative bound for the contraction rate. This is extremely useful when studying time-dependent and/or random products of such operators as it allows for the use of an implicit function theorem. In [Rue79], Ruelle showed the real-analytic behavior of the characteristic exponent of a product of random positive matrices. He did not use Birkhoff’s cone contractions (which would have simplified some estimates and avoided some unnecessary assumptions) but the central part of his 1742 HANSHENRIKRUGH proof may still be viewed as an argument based upon real ‘cone contractions’. We will here show how results similar to [Rue79] hold for complex cone contractions. The resulting theorems and examples of this section may not be deduced from either real cone contractions, nor from standard analytic perturbation theory. In the following we will assume that the ރ-cone Ꮿ is regular (Definition 3.2). This is convenient (if not necessary). In particular, for > 0 sufficiently small the (closed) subcone (10.56) Ꮿ./ Ꮿ B.; / Ꮿ Á f 2 W k k g is nontrivial (not reduced to 0 ). We fix such a value of > 0 in the following. f g Also let < be arbitrary but fixed. We write ./ < 1 for the contraction C1 D constant from Lemma 2.4. Definition 10.1. Let ᏹ ᏹ.; / L.X/ be the (nonempty) family of cone D contractions: M L.X/, M Ꮿ Ꮿ subject to the following uniform bounds: 2 W ! diam M.Ꮿ / and M.Ꮿ/ Ꮿ./. Ꮿ Ä Let .; / be a probability-space and a -ergodic transformation. W ! We denote by ޅ an average taken with respect to . When A is a subset of some Banach space Y we write Ᏹ.; A/ for the set of Bochner-measurable maps from into A (the image of a set of full measure has a countable dense subset). We write Ꮾ.; Y / Ᏹ.; Y / for the Banach space of (-essentially) bounded measurable maps equipped with the (-essential) uniform norm of Y . For M Ᏹ.; ᏹ/ we .n/ 2 write M ! M! M n 1! for the product of operators along the -orbit of D ! . It is again an element of Ᏹ.; ᏹ/. Our goal here is to show the following: 2 THEOREM 10.2. Let t ބ M.t/ Ᏹ.; ᏹ/ be a map for which 2 7! 2 (1) .t; !/ ބ M !.t/ L.X/ is measurable and ! : t M!.t/ is 2 7! 2 8 2 7! analytic. n d o (2) sup M !.t/ = M !.t/ ! ; t ބ < . dt k k W 2 2 C1 ˇ ˇ (3) ޅ ˇ log M !.0/ ˇ < . k k C1 Then for each t ބ the following limit exists -a.s. and is -a.s. independent 2 of ! : 2 1 (10.57) .t/ lim log M.n/.t/ : D n ! The function t ބ .t/ ޒ is real-analytic and harmonic. 2 7! 2 We first use our theory for complex cone contractions to get some necessary uniform bounds. Using outer regularity we find (and fix throughout) m X of 2 0 norm one and K < , such that C1 1 (10.58) u m; u u ; u Ꮿ: k k jh ij K k k 8 2 CONES AND GAUGES IN COMPLEX SPACES 1743

For M Ꮿ Ꮿ a (linear) cone contraction, we write W ! M u (10.59) .u/ ; u Ꮿ ; M D m; M u 2 h i for the natural projection of M u onto the (bounded) subset:

Ꮿm 1 Ꮿ m; u 1 : D Á \ fh i D g .n/ LEMMA 10.3. Given a sequence of matrices, .Mn/n ގ ᏹ, let M .n/ 2 D M1 Mn and write .n/ . Suppose that h Ꮿ./ and Á M D M1 ı ı Mn 2 m; h 1. Then for 1; 2 Ꮿ and X: jh ij D 2 2 .n/ .n/ n 1 (10.60) .1/ .2/ K k k Ä I 2 (10.61) .n/.h / .n/.h/ K n o. / ; as 0: k C k Ä k k C k k ! Proof. Using Lemma 3.4 and then Lemma 2.4 we see that the left hand side .n/ .n/ n 1 in (10.60) is bounded by K d .M 1;M 2/ K . For the second Ꮿ Ä inequality, equation (10.58) and our assumption on h show that h 1 and then k k that B.h; / Ꮿ. The ﬁrst part of Lemma 3.5 implies that d .h ; h/ 2 Ꮿ C Ä k k C o. /. We then use Lemmas 3.4 and 2.4 as before. k k

LEMMA 10.4. For any M ᏹ and u ᏯŒ , 2 2 ˇ ˇ 2 ˇ ˇ 1 ˇ m; M u ˇ K ˇ m; M u ˇ (10.62) ˇh iˇ M ˇh iˇ : K ˇ m; u ˇ Ä k k Ä ˇ m; u ˇ h i h i Proof. Let X . The assumption on u implies that u t Ꮿ when 2 C 2 t < u , i.e. for t < r u = . Then also M u tM Ꮿ whenever j j k k k k j j D k Kk k k K M uC 2K2 ˇ m;M u ˇ t < r. By Lemma 3.5, M M u k k ˇ h i ˇ j j k k Ä r k k D k k u Ä k k ˇ m;u ˇ where the last inequality is a consequence of the propertiesk k in (10.58) of hm. Wei also have: m; M u M u M m; u K. jh ij Ä k k k k Ä k k jh ij

LEMMA 10.5. Let M ᏹ and h Ꮿ with m; h 1. Suppose that U L.X/ 2 2 h i D 2 and X verify U = M and . Then 2 k k k k Ä 4K3 k k Ä 4K2 (10.63) m; .M U /.h / M jh C C ij k k 4K2 and 16K3 (10.64) M U .h / : k C C k Ä 1744 HANSHENRIKRUGH

Proof. We need to show that the denominator in (10.59) stays uniformly bounded away from zero: m; .M U /.h / m; M h M U . h / jh C C ij jh ij k k k k k k k k C k k 1 1 M .1 .1 1// M k k K2 4 4 C k k 4K2 (we have used: h K m; h K and =4K2 K). Then: k k Ä jh ij D k k Ä Ä M 2 2K 16K3 .h / k k : k M U C k Ä M .=4K2/ D C k k We deﬁne for t ބ the (measurable) map t Ᏹ.; Ꮿ/ Ᏹ.; Ꮿm 1/ 2 W ! D through:

(10.65) . t .h//! .h!/; ! ; h Ᏹ.; Ꮿ /: D M! .t/ 2 2 LEMMA 10.6. For each t ބ the map t has a unique fixed point h .t/ in 2 Ᏹ.; Ꮿm 1/ Ꮾ.; X/. D n Proof. The subsets t .Ᏹ.; Ꮿm 1//, n 1 form a decreasing sequence in D n Ᏹ.; Ꮿm 1/ Ꮾ.; X/. By (10.60) the diameters verify: diam t .Ᏹ.; Ꮿm 1// n 1D 0 0 0 D K . Pick h Ꮿm 1 and define .h .t//! h , ! . The sequence, Än 1 n 2 D D 2 h .t/ t .h .t// Ꮾ.; X/, n 0, is thus Cauchy so the map has a (clearly C D 2 unique) fixed point h.t/ t .h.t// Ᏹ.; Ꮿm 1/. D 2 D Recall that m; h!.t/ 1 for all ! . We define the map p t ބ h i D 2 W 2 ! Ᏹ.; ރ/ by m; M !.t/h!.t/ (10.66) p!.t/ h i m; M !.t/h!.t/ ;! : D m; h!.t/ D h i 2 h i R LEMMA 10.7. We have for every t ބ: .t/ log p!.t/ d.!/. 2 D j j Proof. By Lemma 10.4, p!.t/ is equivalent to M !.t/ (within uniformly j j k k bounded constants). Assumption (2) in Theorem 10.2 implies that it is also equivalent to M !.0/ which, by Assumption (3) of that theorem, is log integrable. It k k 1 follows that .! log p!.t/ / L .; /. Our uniform bounds in Lemma 2 7! j j 2 10.4 show that for every ! : 2ˇ .n/ ˇ 1 .n/ 1 ˇ m; M ! .t/h n!.t/ ˇ 1 log M ! .t/ log ˇh iˇ O n D n ˇ m; h!.t/ ˇ C n ˇ h i ˇ n 1 1 X 1 log p k .t/ O : D n j ! j C n k 0 D The latter sum is a ‘Birkhoff’-average of the L1-function log p.t/ so the almost- j j sure convergence follows from Birkhoff’s Ergodic Theorem. This a.s. limit is (a.s.) independent of ! since was supposed ergodic. 2 CONES AND GAUGES IN COMPLEX SPACES 1745

LEMMA 10.8. The map t ބ h .t/ Ꮾ.; X/ is analytic. 2 7! 2 Proof. Pick t0 ބ. It suffices to show that h .t/ is analytic in a neighborhood 2 of t0. Using Assumption (2) of Theorem 10.2 we may find ı > 0 so that for t t0 < ı: j j (10.67) M !.t/ M !.t0/ M !.t0/ ;! : k k Ä 4K3 k k 2 By Lemma 10.5 the map, (10.68) .t; h/ B.t0; ı/ B h .t0/; t .h/ Ꮾ.; X/ 2 4K2 7! 2 is analytic (because it is fiber-wise analytic and uniformly bounded). We denote by T0 Dh t .h .t0// L.Ꮾ.; X// the derivative of t at the fixed point D 0 2 0 h.t0/. By linearization of the uniform bound in (10.61) we see that for each n 1, .n/ n n n Dh .h .t0// T L.Ꮾ.; X// verifies: T .2K=/ . It follows that t0 D 0 2 k 0 k Ä rsp.T0/ < 1. The derivative, 1 T0 L.Ꮾ.; X// of h h t .h/ at the Ä 2 7! 0 fixed point h.t0/ is therefore invertible. We may apply the implicit function theorem and conclude that there is 0 < ı1 < ı and an analytic function (10.69) t B.t0; ı1/ h .t/ B h .t0/; Ꮾ.; X/ 2 7! 2 4K2 for which h .t/ t .h .t// 0 Ꮾ.; X/ for all t t0 < ı1. D 2 j j Proof of Theorem 10.2. For fixed ! the map t ބ p!.t/ ރ is holo- 2 2 7! 2 morphic (being a continuous bilinear form composed with analytic functions). The difficulties here are that the images need not be uniformly bounded (with respect to !) and that we want to define a complex logarithm in a consistent way. We proceed as follows: For t B.t0; ı1/, 2 p!.t/ p!.t0/ M !.t/ M !.t0/ h!.t/ j j Ä k k k k M !.t0/ h!.t/ h!.t0/ Ck k k k K M !.t0/ Ä 4K3 C 4K2 k k M !.t0/ : D 2K2 k k Lemma 10.4 shows that p!.t0/ M !.t0/ so we conclude that j j K2 k k ˇ ˇ ˇ p!.t/ ˇ 1 (10.70) ˇ 1ˇ ;;! ; t t0 < ı1: ˇp!.t0/ ˇ Ä 2 2 j j The difference, Z ˇ ˇ ˇ p!.t/ ˇ (10.71) .t/ .t0/ log ˇ ˇ d.!/; D ˇp!.t0/ˇ 1746 HANSHENRIKRUGH is thus the real part of the following holomorphic function (with the usual logarithm on ރ ޒ ): Z p!.t/ (10.72) H.t/ log d.!/; t t0 < ı1: D p!.t0/ j j Therefore, .t/ is harmonic, whence real-analytic.

Theorem 10.2 applies to certain classes of dominated complex cone contractions as deﬁned in Section 6. We need to impose a further: Assumption 10.9.

(1) We assume that Ꮿޒ Xޒ is a regular cone in a real Banach space. (The real-subcones Ꮿޒ./, > 0, are then deﬁned analogously to (10.56)).

(2) Let P L.Xޒ/, P Ꮿ Ꮿ . We assume that there is 0 > 0 such that 2 W ޒ ! ޒ P.Ꮿޒ/ Ꮿޒ.0/. (And also that P diam P.Ꮿ / < ). D Ꮿޒ ޒ C1 Given a complex operator M L.Xރ/, which is dominated by P , we write 2 ˛.M /, ˇ.M / and .M / for the optimal constants in Assumption 6.1. Now, let 0 < < 1 and deﬁne the following subset of complex operators: ˛.M / .M / P (10.73) ᏹ M L.Xރ/ 1 < ; 0 cosh < : D 2 W ˇ.M / Ä ˛.M / 2

THEOREM 10.10. Suppose that Assumption 10.9 holds. The class of operators ᏹ L.Xރ/ defined in (10.73) verifies the uniform bound in Definition 10.1 (for suitable values of and ). So Theorem 10.2 applies when ᏹ ᏹ . D Proof. The bound (6.50) in the proof of Theorem 6.3 shows that there is .; P / < (depending upon and P only) so that for any M ᏹ : D 1 2 diam M.Ꮿ / . We still need to show that M ᏹ maps Ꮿރ uniformly into its Ꮿރ ރ Ä 2 interior: Ꮿޒ is assumed regular. Proposition 5.4 shows that Ꮿރ is then also regular. We may assume that we have found m Ꮿ 0 , extended to m X which verifies 2 ޒ 2 ރ0 (10.58). By the assumption on P we have for any u Ꮿޒ: B.P u; 0 P u / Ꮿޒ. 2 k k When ` Ꮿޒ is of norm one this implies: `; P u 0 P u . Using the bound 2 0 h i k k (6.48) we then get for x Ꮿޒ: 2 p p K ˇ M x K m; M x K ˇ P x C `; P x : k k Ä jh ij Ä C k k Ä 0 h i Let u Ꮿ and (by Proposition 5.2) write u ei .x iy/ with ޒ and x y Ꮿ . 2 ރ D C 2 ˙ 2 ޒ Then M u M.x y/ M.x y/ k k Ä k C k C k k Kpˇ 2Kpˇ C `; P.2x/ C `; P u : Ä 0 h i Ä 0 jh ij CONES AND GAUGES IN COMPLEX SPACES 1747

2 ˛ 0 Denote . P /=.ˇ / 0; 1Œ and let `1; `2 Ꮿޒ be of norm 1. D cosh 2 C 2 2 Given Xރ we use (6.46) to obtain: 2

Re `1; M u `2; M u h C ih C i ˛ 2 `1; P u `2; P u 2 `; M u P jh ijjh ij k k jh ij k k cosh 2 2 2 M u 2 2 M u 2: 4K2 k k k k k k k k

q 2 This is nonnegative when = M u 1 2 0 1. Thus, k k k k Ä Á C 4K2

B.M u; M u / Ꮿރ: k k 2 COROLLARY 10.11. In the case of ﬁnite-dimensional matrices Assumption 10.9 is indeed veriﬁed. The class of matrices in (10.73) then reduces to (see Theo- rem 7.1): ( ) inf ReAij Akl sup ImAij Akl ᏹ A Mn.ރ/ 1 x < ; 0 j x j < : D 2 W sup ReA A Ä inf ReA A ij xkl ij xkl

Theorem 10.2 thus applies when ᏹ ᏹ. D

Example 10.12. Let .n/n ގ be a sequence of independent and identically distributed random variables with2 values in ބ. For t ބ deﬁne: 2 8 t 7 n Mn.t/ C : D 7 in 6 it C

One checks that (see e.g. the calculation leading to (11.78) below) each Mn.t/ 1 2 ᏹ 0:75, t ބ. By the previous corollary, a.s. .t/ lim log M1.t/ Mn.t/ D 2 D n k k exists and deﬁnes a harmonic function of t ބ. 2 For a closely related result we mention [Rue79] in which Ruelle was able to show that the characteristic exponent of a product of random real matrices (under additional assumptions) behaves real-analytically with respect to the matrices. Us- ing Corollary 10.11 above, it is possible to recover the result of Ruelle together with an explicit estimate for the analytic extension. It seems plausible that a result similar to Theorem 10.2 should hold for integral operators of the type given in our Theorem 7.2 (a complex generalization of Jentzsch’s Theorem), but our proofs are insufﬁcient as the cones needed in Theorem 10.2 are not regular. 1748 HANSHENRIKRUGH

11. Complex cone contraction versus perturbation theory

The applications in Sections 7 and 9 were based upon Theorem 6.3 on dominated complex cone contractions. In Theorem 6.3 one may view the complex operator M as a ‘perturbation’ of a real operator P . It is natural to ask what are then the benefits from our above-mentioned applications relative to applying standard analytic perturbation theory when looking for a spectral gap for one fixed (real) operator. We address this question here. We consider the case of an operator T contracting a regular real cone so that T P R where P is a projection of rank 1, > 0 and the residual operator R D C verifies RP PR 0 and nRn C n 1, n 1 for some C < and D D k k Ä C1 0 < 1. Now, recall what can be obtained within the framework of perturbation Ä theory; see e.g. Kato [Kat95, III.6.4 and IV.3.1]: If we perturb T then the spectral gap persists provided that the perturbation is ‘small’ compared to the resolvent on a suitable separating circle. To be more precise, for z sp.T / we have: … (11.74) R.z; T / .z T/ 1 .z / 1P .z R/ 1.1 P /: D D C When > z > , one has: j j X (11.75) R.z; T / .z / 1P z 1.1 P/ z n 1Rn: D C C n 1 Using the estimate for the residual operator and re-summing we get: P 1 P C (11.76) R.z; T / k k k k : k k Ä z C z C z . z / j j j j j j j j Consider now the closed curve (our not necessarily optimal choice for a separating 1 circle), z C z C . For z : D f 2 W j j D 2 g 2 2 P 2 1 P 4C 1 (11.77) R.z; T / k k k k : k k Ä .1 / C .1 / C .1 2/ Á C When S < , the von Neumann series k k .z T S/ 1 R.z; T / R.z; T /SR.z; T / D C C converges normally on . It follows that [Kat95, II,1.3 and IV.3.1] the spectral projections on the two components of C depend analytically on S. In particular, the algebraic dimension of the spectral projection on the unbounded component stays constant, i.e. equals one. In other words the spectral gap persists. To be more concrete, consider then the case of a real matrix T M .ޒ/ and 2 d constants 0 < m M < such that m Tij M for all indices. Let us perturb Ä C1 Ä Ä by a purely imaginary matrix iS with S Md .ޒ/ and such that Sij < r 2 2j j2 for all indices. The matrix A T iS then verifies Re Aij A m r and D C kl CONES AND GAUGES IN COMPLEX SPACES 1749

Im Aij A 2M r. Thus, the condition in Theorem 7.1 simply reads: 2rM kl Ä Ä m2 r2. Consequently if m2 (11.78) r ; Ä M pM 2 m2 C C then A has a spectral gap. For example if m 3 Tij M 4 then Sij < D Ä Ä D j j r 9 1 sufﬁces to proved a spectral gap. Also, one cannot take r to be D p25 4 D C 3 3i 3 3i bigger than 3 or else 3C3i 33i provides a counter-example of a matrix without a spectral gap. C Perturbation theory works well in a special case, namely when T is itself of rank one. For example, suppose that Tij 1, so that (by the Euclidean norm) D m M 1, P 1 P 1 and d. In this case C 0 so that setting D D k k D k k D D D D z d=2 we obtain from (11.76): j j D P 1 P 4 1 (11.79) R.z; T / k k k k : k k Ä z C z Ä d Á j j j j Thus, if we add S M .ޒ/ with S < d=4 then T iS has a spectral gap. 2 d k k D C In particular, when Sij < r then S rd so one needs r < 1=4 in order to apply j j k k Ä this result. By comparison, the bound obtained from the complex cone contraction (11.78) is r < 1 . On the other hand, and in favor of the perturbation result, 1 p2 note that it appliesC to some perturbations which are not immediately seen by the complex cone contraction, e.g. when only one element of Sij is nonzero, and this element is strictly smaller than d=4. Consider now the case when the original matrix T is not of rank one. In order to make a computation within perturbation theory note that the matrix contracts the real standard cone ޒn . Given the constants 0 < m M < from above, one has C Ä C1 n n M M m (see Example 4.9): diamޒ .T ޒ / 2 log n . From this, tanh 4 M m C C Ä D D D and 1 2m . Using T Md and md we also see that 1 T C D M m k k Ä k k Ä M=m. One alsoC has P M . In order to get a bound on C we may use the k k Ä m constants in Remark 3.8, equation (3.19), which were obtained for the complex cone but apply equally well to the real case. With the Euclidean norm on ޒn one has K g p2 and D D (11.80) M M .p2/ 2 log M M .p2/ 2 log M A p2 exp 1 m ;C 1 m A: m m 2m m 2m D C M m D C M m C C 1 When z C we have the bound: j j D 2 M.M m/ .M m/2 C.M m/2 1 (11.81) R.z; T / C3 C 2 C 2 : k k Ä dm C dM m C dM m Á 1750 HANSHENRIKRUGH

When m 3 and M 4 we obtain from these estimates that r =d D D D D 0:01835 : : : ensures a spectral gap. This is within two orders of magnitude to r 1 D which we obtained above from equation (11.78). Increasing, however, the ratio of M to m substantially deteriorates the pertur- bative bounds. When e.g. m 10 and M 100 we obtain r 2 10 175 (!!) D D from (11.80) and (11.81). This should be compared to the bound r 0:4987 : : : D obtained from equation (11.78) when using the complex cone contraction.

Appendix A. Projective space

Let X be a complex Banach space. Given nonzero elements x; y X 2 Á X 0 we write x y if and only if ރx ރy. Let X X = denote the f g D W ! quotient map and write Œx ރ x for the equivalence class of x X . We equip D 2 the quotient space .X / with the following metric

(A.82) d.X /.Œx; Œy/ distH .ރx S; ރy S/ D \ \ x y inf ; ރ ; x; y X D x y W 2 2 k k k k in which distH is the Hausdorff distance between nonempty sets and S S.X/ is D the unit-sphere.

LEMMA A.1. (1) Let Ꮿ X be a ރ-cone of K-bounded sectional aperture. Then for all x; y Ꮿ : 2 d.X /.Œx; Œy/ 2KdᏯ.x; y/: Ä 1 (2) Let x X , 0 < r < 1 and set V B.X /.Œx; r/. Then for all y V 2 D 2 r d .Œx; Œy/ .X / dV .x; y/ log C : Ä r d.X /.Œx; Œy/ Proof. Using the inequality x y 1 1 (A.83) 2 x y min ; ; x; y X ; x y Ä k k x y 2 k k k k k k k k we obtain from Lemma 3.4:

x= m; x y= m; y x y h i h i 2 m 2KdᏯ.x; y/ x= m; x y= m; y Ä k k m; x m; y Ä k h ik k h ik h i h i and the ﬁrst conclusion follows. For the second claim, normalize so that d .Œx; Œy/ x y < r and .X / 1 1 1 D k uk x y 1. Let u C x y x .y x/. By (A.83), x k kDk kD D 2 C 2 D C 2 k u kÄ 1 2r k k j j y x which remains smaller than r when 1 < 2R .2; . 2 k k j j x y Á 2 C1 1 R 1k k Then dV .x; y/ dBރ.1;2R/. 1; 1/ dބ.0; R / log RC1 . Ä D D CONES AND GAUGES IN COMPLEX SPACES 1751

Given any two points x; y Ꮿ we may follow Kobayashi [Kob67], [Kob70] 2 and define a projective pseudo-distance between x and y through: X dᏯ.x; y/ inf dᏯ.xi ; xi 1/ x0 x; x1; : : : ; xn y Ꮿ : Q D f C W D D 2 g Since d is a (projective) metric, the previous lemma implies that .X / THEOREM A.2. Suppose that Ꮿ is of K-bounded sectional aperture in X. Then the inclusion map, .Ꮿ ; dᏯ/ .Ꮿ ; d.X // is 2K-Lipschitz. Q ! In other words, this new distance does not degenerate when taking the inf over finite chains, so distinct complex lines in Ꮿ have a nonzero dQᏯ-distance. This is conceptually very nice, but, in our context, not particularly useful. The reason is that even if T L.X/ maps Ꮿ into a subset of finite diameter in Ꮿ for the metric 2 dQ, this does not seem to imply a uniform contraction of T , i.e. no spectral gap. We leave a further study of this metric to the interested reader. Acknowledgments. I am grateful to A. Douady for a key suggestion in the proof of Lemma 2.4, to Oscar Bandtlow for suggesting the use of ‘reproducing’ complex cones, to Lo¨ıc Dubois for discovering errors in previous versions of the paper and to anonymous referees for several valuable suggestions and corrections.

References

[Bal00] V. BALADI, Positive Transfer Operators and Decay of Correlations, Advanced Series in Nonlinear Dynamics 16, World Scientiﬁc Publishing Co., River Edge, NJ, 2000. MR 2001k:37035 Zbl 1012.37015 [BRS05] J.D.BIGGINS and A.RAHIMZADEH SANI, Convergence results on multitype, multivari- ate branching random walks, Adv. in Appl. Probab. 37 (2005), 681–705. MR 2006f:60082 Zbl 1076.60072 [Bir57] G.BIRKHOFF, Extensions of Jentzsch’s theorem, Trans. Amer. Math. Soc. 85 (1957), 219– 227. MR 19,296a Zbl 0079.13502 [Bir67] , Lattice Theory, Third edition. Amer. Math. Soc. Colloq. Publ. XXV, Amer. Math. Soc., Providence, R.I., 1967. MR 37 #2638 Zbl 0153.02501 [Bow75] R.BOWEN, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lec- ture Notes in Math. 470, Springer-Verlag, New York, 1975. MR 56 #1364 Zbl 0308.28010 [CG93] L.CARLESON and T. W. GAMELIN, Complex Dynamics, Universitext: Tracts in Math., Springer-Verlag, New York, 1993. MR 94h:30033 Zbl 0782.30022 [FS79] P. FERRERO and B.SCHMITT, Ruelle’s Perron-Frobenius theorem and projective metrics, Colloq. Math. Soc. János Bolyai 27 (1979), 333–336. [FS88] , Produits aléatoires d’opérateurs matrices de transfert, Probab. Theory Related Fields 79 (1988), 227–248. MR 90e:47006 Zbl 0633.60014 [Fro08] G.FROBENIUS, Über Matrizen aus nicht negativen Elementen, S.-B. Preuss. Akad. Wiss. (1908), 471–476. [Jen12] R.JENTZSCH, Über Integralgleichungen mit positivem Kern., J. Reine Angew. Math. 141 (1912), 235–244. 1752 HANSHENRIKRUGH

[Kat95] T. KATO, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer- Verlag, New York, 1995, Reprint of the 1980 edition. MR 96a:47025 Zbl 0836.47009 [Kob67] S.KOBAYASHI, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460–480. MR 38 #736 Zbl 0158.33201 [Kob70] , Hyperbolic Manifolds and Holomorphic Mappings, Pure and Appl. Math. 2, Mar- cel Dekker, New York, 1970. MR 43 #3503 Zbl 0207.37902 [KR48] M.G.KREIN˘ and M.A.RUTMAN, Linear operators leaving invariant a cone in a Banach space, Uspekhi Matem. Nauk. 3 (1948), 3–95, in Russian; translated in Amer. Math. Soc. Transl. 26, 199–325 (1950). MR 12,341b [Lan93] S.LANG, Real and Functional Analysis, third ed., Grad. Texts Math. 142, Springer-Verlag, New York, 1993. MR 94b:00005 Zbl 0831.46001 [Liv95] C.LIVERANI, Decay of correlations, Ann. of Math. 142 (1995), 239–301. MR 96e:58090 Zbl 0871.58059 [MN91] P. MEYER-NIEBERG, Banach Lattices, Universitext, Springer-Verlag, New York, 1991. MR 93f:46025 Zbl 0743.46015 [Mil99] J.MILNOR, Dynamics in One Complex Variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. MR 2002i:37057 Zbl 0946.30013 [NN87] P. NEY and E.NUMELLIN, Markov additive processes, I and II, Ann. Probab. 15 (1987), 561–592. [Per07] O.PERRON, Zur Theorie der Matrices, Math. Ann. 64 (1907), 248–263. MR 1511438 JFM 38.0202.01 [Rot44] E.ROTHE, On non-negative functional transformations, Amer. J. Math. 66 (1944), 245– 254. MR 6,71b Zbl 0060.27703 [Rud91] W. RUDIN, Functional Analysis, second ed., Internat. Ser. Pure and Appl. Math., McGraw- Hill, Inc., New York, 1991. MR 92k:46001 Zbl 0867.46001 [Rue68] D.RUELLE, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys. 9 (1968), 267–278. MR 38 #3013 Zbl 0165.29102 [Rue69] , Statistical Mechanics: Rigorous Results, W. A. Benjamin, New York, 1969. MR 44 #6279 Zbl 0177.57301 [Rue78] , Thermodynamic Formalism, Encyclopedia Math. Appl. 5, Addison-Wesley Publ. Co., Reading, MA, 1978. MR 80g:82017 Zbl 0401.28016 [Rue79] , Analycity properties of the characteristic exponents of random matrix products, Adv. in Math. 32 (1979), 68–80. MR 80e:58035 Zbl 0426.58018 [Rug02] H.H.RUGH, Coupled maps and analytic function spaces, Ann. Sci. École Norm. Sup. 35 (2002), 489–535. MR 2004b:37039 Zbl 1020.37007 [Rut40] M.M.RUTMAN, Sur les opérateurs totalement continus linéaires laissant invariant un certain cône, Rec. Math. [Mat. Sbornik] N. S. 8 (50) (1940), 77–96. MR 2,104b Zbl 0023.33002 [Ves76] E.VESENTINI, Invariant metrics on convex cones, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3 (1976), 671–696. MR 55 #6206 Zbl 0357.46011

(Received April 25, 2006) (Revised November 23, 2007)

E-mail address: [email protected] UNIVERSITÉDE CERGY-PONTOISE,DÉPARTEMENT DE MATHÉMATIQUES, UMR CNRS 8088, F-95800 CERGY-PONTOISE,FRANCE http://www.u-cergy.fr/rech/pages/rugh/