arXiv:1802.09422v2 [math.SP] 28 Feb 2018

SPECTRAL GAPS FOR HYPERBOUNDED OPERATORS

arXiv:1802.09422v2 [math.SP] 28 Feb 2018

¨ JOCHEN GLUCK Abstract. We consider a positive and contractive linear operator T on Lp over a finite measure space and prove that, if T Lp ⊆ Lq for some q > p, then the essential spectral radius of T is strictly smaller than 1. As a special case, we obtain a recent result of Miclo who proved this assertion for self-adjoint operators in the case p = 2 (under a few additional assumptions). Moreover, we also prove a version of our main theorem on non-finite measure spaces. Our methods are qualitative in nature. They rely on an ultra power argument and on the fact that an infinite dimensional Lp -space cannot by isomorphic to an Lq -space for q 6= p.

1. Introduction Let us consider a linear operator semigroup (Tt )t∈(0,∞) on a Banach space. The asymptotic behaviour of the operators Tt as the time t tends to ∞ is one of the essential questions in the study of such a semigroup. In the present paper we are particularly interested in the question whether the operators Tt converge uniformly (i.e. with respect to the operator norm) as t → ∞. In order to prove this kind of behaviour for a given operator semigroup, one always needs some kind of spectral gap property of the operators Tt . For instance, it is very helpful to know that one of the operators Tt is quasi-compact, i.e. that its essential spectral radius is strictly smaller than 1 (see for instance [8, Section V.3]). One is thus interested in good criteria which ensure quasi-compactness of a linear operator. The aim of the present article is to prove the the Theorems 1.1 and 1.2 below which both give such a criterion for an important class of operators. The first theorem is a special case of the second one. Theorem 1.1. Let 1 ≤ p < q ≤ ∞ and let (Ω, µ) be a finite measure space. Let T be a positive linear operator on Lp (Ω, µ) of norm kT k ≤ 1. If T Lp(Ω, µ) ⊆ Lq (Ω, µ), then the essential spectral radius of T is strictly smaller than 1. Operators on Lp whose range is contained in Lq for some q > p are sometimes called hyperbounded. This explains the title of the paper. A few notes on the terminology used in the theorem are in order. We call an operator on Lp (Ω, µ) positive if T f ≥ 0 for every function f ≥ 0. Moreover, recall that the essential spectral radius of a bounded linear operator T on a Banach space E is defined to be the spectral radius of T in the Calkin algebra L(E)/K(E), where L(E) denotes the Banach algebra of all bounded linear operators on E and K(E) denotes the ideal of all compact operators in L(E). The essential spectral radius of T is strictly smaller than 1 if and only if every spectral value of T which has modulus at least 1 is a pole of the resolvent of T and the corresponding spectral projection has finite-dimensional range. Another equivalent condition is that there exists an Date: March 1, 2018. 2010 Mathematics Subject Classification. Primary 47A10; Secondary: 47B65, 47B38, 47D06, 46E30, 46B08. Key words and phrases. Essential spectral radius; quasi-compactness of positive operators; convergence of semigroups; functions spaces; ultra power techniques; geometry of Banach lattices. 1

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integer n ∈ N and a compact operator K on E such that kT n −Kk < ∞; this is why operators with essential spectral radius < 1 are sometimes called quasi-compact. Our second theorem is a bit more general version of the first one; here we consider Lp and Lq over different, possibly infinite measure spaces and assume that Lq is embedded as a sublattice in Lp . We also allow the case q < p in this general situation. Theorem 1.2. Let p, q ∈ [1, ∞] be two distinct numbers and let (Ω1 , µ1 ) and (Ω2 , µ2 ) be two arbitrary measure spaces. Moreover, let j : Lq := Lq (Ω2 , µ2 ) → Lp := Lp (Ω1 , µ1 ) be an injective lattice homomorphism (i.e. |j(f )| = j(|f |) for all f ∈ Lq ). If T is a positive linear operator on Lp of norm kT k ≤ 1 and if T Lp ⊆ j(Lq ), then the essential spectral radius of T is strictly smaller than 1. We point out that the lattice homomorphism j is of course positive, and thus automatically continuous [29, Theorem II.5.3]. The generality of Theorem 1.2 is perhaps best illustrated by the following corollary for (unbounded) subsets of Rd : Corollary 1.3. Let Ω be a Borel measurable subset of Rd , endowed with the ddimensional Lebesgue measure λ, and let 1 ≤ p < ∞. Let T be a positive linear operator on Lp (Ω, λ) of norm kT k ≤ 1 and assume that there exist numbers ε1 , ε2 > 0 such that Z |T f (x)|p (1 + |x|)ε2 dλ(x) < ∞ Ω Z |T f (x)|p+ε1 (1 + |x|)ε2 dλ(x) < ∞ and Ω

p

for each f ∈ L (Ω, λ). Then the essential spectral radius of T is strictly smaller than 1. We demonstrate at the end of the introduction how Corollary 1.3 can be derived from Theorem 1.2. Assume for a moment that the Borel measurable set Ω ⊆ Rd is bounded; then Corollary 1.3 becomes a special case of Theorem 1.1; conversely, it is not difficult to deduce Theorem 1.1 from Corollary 1.3 in case that the space Ω in the theorem is a bounded and Borel measurable subset of Rd and µ is the Lebesgue measure. Obviously, one can also apply Theorem 1.2 to operators on ℓp whose range is contained in ℓq for some q < p. In this case, however, the assertion of the theorem is an immediate consequence of a much more general of result of Pitt which asserts that, for 1 ≤ q < p < ∞, every bounded linear operator from ℓp to ℓq is compact (the case p = ∞ is not included in Pitt’s theorem, but this case is rather simple anyway; compare Subsection 2.1). For a thorough discussion of Pitt’s theorem and possible generalisations we refer the reader to the Appendix of [28]. Historical remarks and related literature. For self-adjoint operators on L2 Theorem 1.1 was recently proven – under a few additional assumptions – by Miclo [21, Theorem 1]; this solved a long open conjecture of Simon and Høegh-Krohn [30]; the case of self-adjoint operators on L2 is, of course, of special importance in mathematical physics. Of all the related articles that appeared during the last decades, let us mention the works [1, 16, 17, 13] which impose a strengthened positivity assumption on the operator, the paper [33] which assumes explicit numerical bounds for the operator norm of T from Lp to Lq and the recent article [34] where the technique used by Miclo is further developed. For the general asymptotic theory of operator semigroups we refer to [8, Chapter V], [3, Chapter 5] and to the monograph [32]. For the long term behaviour of, in


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particular, positive semigroups we refer to the [2, 7] and to the recent monograph [4]. In addition to the references mentioned above, some recent contributions to the asymptotic theory of positive semigroups include the article [22] which has its focus on spectral theory and growth fragmentation equations, the paper [10] on so-called lower bound methods for semigroups on L1 , the articles [25, 11, 27, 26, 9] which all deal with semigroups that dominate integral operators, and the work [23] which considers perturbed semigroups on L1 -spaces and which is related to the aforementioned series of articles. Preliminaries. We assume the reader to be familiar with the basic theory of real and complex Banach lattices; standard references for this theory are, for instance, the monographs [29, 36, 20]. Here we only recall the basic terminology that a linear operator T on a Banach lattice E is called positive if T f ≥ 0 for each 0 ≤ f ∈ E. The reader is also assumed to be familiar with standard spectral theory for linear operators on Banach spaces; for a detailed treatment we refer, for instance, to the spectral theory chapters in the monographs [31, 18, 35]. If T is a bounded linear operator on a complex Banach space E and if λ ∈ C is not a spectral value of T , then we denote the resolvent of T at λ by R(λ, T ) := (λ − T )−1 . We make extensive use of ultra power arguments; the most important facts about the construction of ultra powers are briefly recalled at the beginning of Subsection 2.3; for a detailed treatment of ultra powers and ultra products of Banach spaces we refer the reader to the survey article [15]. Throughout the paper, all occurring Banach spaces and Banach lattices are assumed to be defined over the complex scalar field. All measure spaces in the paper are allowed to be non-σ-finite unless otherwise stated. Organisation of the article. In the remaining part of the introduction we prove Corollary 1.3. In Subsection 2.1 we first demonstrate that our main theorems are not particularly surprising if one of the numbers p and q is either 1 or ∞. The rest of Section 2 is then devoted to the proofs of our main results in the non-trivial case p, q ∈ (1, ∞). We briefly discuss the consequences of our results for the long term behaviour of positive operator semigroups in Section 3, and we conclude the paper with an open problem in Section 4. A proof of Corollary 1.3. On this subsection we show how Corollary 1.3 can be derived from Theorem 1.2. Proof of Corollary 1.3. Let us define δ : Ω → R by δ(x) = 1 + |x| for all x ∈ Ω. Choose q ∈ (p, ∞) sufficiently close to p such that q ≤ p + ε1 and d( pq − 1) < ε22 . From p < q ≤ p + ε1 it follows by interpolation that Z |T f |q δ ε2 dλ < ∞ Ω

p

for each f ∈ L (Ω, λ). Now we choose a real number α which strictly larger than d(1 − pq ) but strictly smaller than d(1 − pq ) + ε22 pq . Then we have p d(1 − ) < α q

and

q α < ε2 , p

R q hence, Ω |T f |q δ p α dλ < ∞ for each f ∈ Lp (Ω, λ). Set r := q/p ∈ (1, ∞) and choose r′ ∈ (1, ∞) such that 1/r + 1/r′ = 1 (i.e. q ′ r = q−p ); moreover, let µ denote the measure on the Borel σ-algebra over Ω which has density δ αr with respect to the Lebesgue measure, i.e. dµ = δ αr dλ.

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For every function f ∈ Lq (Ω, µ) it follows from H¨ older’s inequality that Z 1/r Z 1/r′ Z Z p α 1 p pr αr −αr ′ |f | dλ = |f | δ α dλ ≤ |f | δ dλ δ dλ , δ Ω Ω Ω Ω and hence kf kLp(Ω,λ) ≤ kf kLq (Ω,µ)

Z

Ω

1′ r p ′ δ −αr dλ

d q−p q ,

We chose α to be strictly larger than hence we have αr′ > d. This implies R −αr′ dλ < ∞, so we have checked that Lq (Ω, µ) continuously embeds into that Ω δ p L (Ω, λ) (and obviously, the embedding is a lattice homomorphism). Moreover, we R have already noted above that Ω |T f |q δ αr dλ < ∞ for all f ∈ Lp (Ω, λ), so the range of T is contained in Lq (Ω, µ). Thus, the assertion follows from Theorem 1.2. 2. Main arguments 2.1. The end points of the Lp -scale. We first consider the cases p ∈ {1, ∞} and q ∈ {1, ∞} in Theorem 1.2. In this cases, the theorems are much simpler since we can show that a power of T is compact under the given assumptions (and in fact, we do not even need the contractivity nor the positivity T for that). We need the following observations from Dunford–Pettis theory: let T be a bounded linear operator between two Banach spaces E and F . Recall that T is called weakly compact if it maps the closed unit ball in E to a relatively weakly compact subset of E. Moreover, T is said to be a Dunford–Pettis operator or to be completely continuous if, for every sequence (xn ) in E which converges weakly to a vector x ∈ E, the sequence (T xn ) in F converges in norm to the vector T x; equivalently, T maps weakly compact subsets of E to norm-compact (equivalently: relatively norm-compact) subsets of F . Every compact operator between E and F is a Dunford–Pettis operator, and the converse is true if E is reflexive; thus, Dunford– Pettis operators are particularly interesting on non-reflexive Banach spaces. We point out that, if E, F, G are Banach spaces, T : E → F is weakly compact and S : F → G is a Dunford–Pettis operator, then ST : E → G is compact. We will make repeated use of this simple observation in the proofs of the two subsequent propositions. Now, let (Ω, µ) be an arbitrary measure space. Then the spaces E = L1 (Ω, µ) and E = L∞ (Ω, µ) are so-called Dunford–Pettis spaces, i.e. every weakly compact linear operator from E to any Banach space F is a Dunford–Pettis operator; see [20, Proposition 3.7.9]. The following proposition proves Theorem 1.2 (and much more) if p ∈ {1, ∞} or q ∈ {1, ∞}. Although the result follows from rather standard arguments from Dunford–Pettis theory, we include the proof for the convenience of the reader. Proposition 2.1. Let p, q ∈ [1, ∞] and let (Ω1 , µ1 ) and (Ω2 , µ2 ) be two arbitrary measure spaces. Moreover, let j : Lq := Lq (Ω2 , µ2 ) → Lp := Lp (Ω1 , µ1 ) be an injective lattice homomorphism Let T be a bounded linear operator on Lp and assume that T Lp ⊆ j(Lq ). (a) If q ∈ (1, ∞] and p = 1, then T 2 is compact. (b) If q ∈ [1, ∞) and p = ∞, then T 2 is compact. (c) If p ∈ (1, ∞) and q ∈ {1, ∞}, then T is compact. Proof. We first observe that, in any case, j −1 T : Lp → Lq is continuous due to the closed graph theorem. (a) Let us first show that T : L1 → L1 is weakly compact. Consider the case q 6= ∞ first. Then j −1 T is even weakly compact since Lq is reflexive. Since the


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embedding j is continuous, we conclude that T = jj −1 T is weakly compact. Now consider the case q = ∞. Then j maps the unit ball of Lq = L∞ into a subset of L1 of the form J + iJ, where J is an order interval in L1 . But order intervals in J are weakly compact as L1 has order continuous norm (see [20, Theorem 2.4.2(i) and (vi)]), so j is weakly compact and hence, so is T = jj −1 T . We have thus proved that T is weakly compact. In particular, T is a Dunford– Pettis operator as L1 is a Dunford–Pettis space and hence, T 2 is compact (as a composition of a weakly compact operator with a Dunford–Pettis operator). (b) The mapping j −1 T : L∞ → Lq maps the unit ball of L∞ into a subset of q L of the form J + iJ, where J is an order interval in Lq . Since Lq has order continuous norm, we again conclude that order intervals in Lq are weakly compact [20, Theorem 2.4.2(i) and (vi)]. Hence, j −1 T is weakly compact and thus, so is T = jj −1 T . As L∞ is a Dunford–Pettis space, we conclude that T is a Dunford– Pettis operator, so T 2 is compact (as a composition of a weakly compact operator with a Dunford–Pettis operators). (c) As Lq is reflexive, the mapping j −1 T : Lp → Lq is weakly compact and hence a Dunford–Pettis operator, as Lp is a Dunford–Pettis space (since p ∈ {1, ∞}). Moreover, the reflexivity of Lq also implies that the embedding j : Lq → Lp is weakly compact. Hence, T = jj −1 T is compact. The proof of assertion (c) in the above proposition is actually a special case of the more general (and well-known) observation that a bounded linear operator on a Dunford–Pettis space which factorises through a reflexive space is compact. The arguments from Dunford–Pettis theory used in the above proofs can be put in a more general context if one considers so-called principle ideals in Banach lattices; this is explained in detail in [6, Section 2] and in [5, Section 2]. 2.2. Dimension of the fixed space. Our first ingredient to the proof of Theorem 1.2 in the case p, q ∈ (1, ∞) is the following proposition. It says that, in the situation of the theorem, the fixed space of T is finite dimensional; by the fixed space of T we mean the closed vector subspace of E given by fix T := ker(1 − T ). Proposition 2.2. Let p, q ∈ (1, ∞) be two distinct numbers and let (Ω1 , µ1 ) and (Ω2 , µ2 ) be two arbitrary measure spaces. Moreover, let j : Lq := Lq (Ω2 , µ2 ) → Lp := Lp (Ω1 , µ1 ) be an injective lattice homomorphism. If T is a positive linear operator on Lp of norm kT k ≤ 1 and if T Lp ⊆ j(Lq ), then fix T is finite dimensional. For the case of self-adjoint operators on L2 , a version of Proposition 2.2 (under the assumption that (Ω1 , µ1 ) = (Ω2 , µ2 ) is a finite measure space) has already been proved by Gross as a part Theorem 1 in [14]; in this reference, an explicit bound of the dimension of fix T in terms of the operator norm of T as an operator from L2 to Lq is given. For the proof of Proposition 2.2 we need the following simple lemma. We call a vector subspace F of a Banach lattice E a sublattice of E if |f | ∈ F for each f ∈ F . Lemma 2.3. Let 1 < p < ∞, let (Ω, µ) be an arbitrary measure space and let T be a positive linear operator on Lp (Ω, µ) of norm at most 1. Then fix T is a sublattice of Lp (Ω, µ). Proof. Let f ∈ fix T . Then we have T |f | ≥ |T f | = |f |. On the other hand, the norm of T |f | is not larger than the norm of |f | since T is contractive. Thus, T |f | = |f |. At the end of the above argument we used the fact that the norm on Lp is strictly monotone, i.e. if we have 0 ≤ f ≤ g for two distinct vectors f and g in Lp , then

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kf k < kgk. This is also true for p = 1, so the above lemma remains true on L1 (but not on L∞ ). Now we can prove Proposition 2.2. Proof of Proposition 2.2. First note that j −1 T : Lp → Lq is continuous by the closed graph theorem. In particular, the operator S := j −1 T j : Lq → Lq is continuous. Moreover, a vector f ∈ Lq is in the fixed space of S if and only if j(f ) is in the fixed space of T , i.e. fix S = j −1 (fix T ). The spaces fix T and fix S are closed in Lp and Lq , respectively. Moreover, it follows from Lemma 2.3 that fix T is a sublattice of Lp . Since j is a lattice homomorphism, this implies that fix S is a sublattice of Lq . Now it follows from Kakutani’s representation theorem for abstract Lp -spaces [20, Theorem 2.7.1] that fix T , with the norm induced by Lp , is itself isometrically lattice isomorphic to an Lp -space over some measure space, and likewise it follows that fix S, with the norm induced by Lq , is isometrically lattice isomorphic to an Lq -space over some measure space. Yet, the mapping j|fix S : fix S → fix T is bijective and a lattice homomorphism, hence a lattice isomorphism. As p 6= q, Proposition A.1 in the Appendix shows that this can only be true of fix T is finite dimensional. 2.3. Ultra powers. Let us briefly recall the concept of an ultra power of a Banach space E. Fix a free ultra filter U on N, endow the E-valued sequence space ℓ∞ (E) with its canonical norm kzk := supn∈N kzn kE for z = (zn )n∈N ∈ ℓ∞ (E), and define c0,U (E) := {z ∈ ℓ∞ (E) : lim kzn kE = 0}. n→U

Then ℓ∞ (E) is a Banach space and c0,U (E) is a vector subspace of it. The quotient space E U := ℓ∞ (E)/c0,U (E) is called the ultra power of E with respect to the ultra filter U. For each z = (zn )n∈N ∈ ℓ∞ (E) we use the notation z U for the equivalence class of z in E U . Moreover, for x ∈ E we use the notation xU for the equivalence class of the constant sequence (x)n∈N ∈ ℓ∞ (E) in E U . The mapping E ∋ x 7→ xU ∈ E U is isometric, and via this mapping we may consider E as a closed subspace of E U . For every z ∈ ℓ∞ (E) we can compute the norm of z U in E U by means of the formula kz U k = limn→U kzn kE . If E is a Banach lattice, then so is ℓ∞ (E), and then the space c0,U (E) is a closed ideal in ℓ∞ (E). Thus, the ultra power E U is a Banach lattice, too, and the embedding E ∋ x 7→ xU ∈ E U is an isometric lattice homomorphism in this case. The formula kz U k = limn→U kzn kE for z ∈ ℓ∞ (E) implies that, if E is an Lp space over some measure space for p ∈ [1, ∞), then E U is an abstract Lp -space and thus isometrically lattice isomorphic to a concrete Lp -space by means of Kakutani’s representation theorem [20, Theorem 2.7.1]. Let E, F be Banach spaces. Every bounded linear operator T : E → F can be canonically extended to a bounded linear operator T U : E U → F U which is given by T U z U = (T zn )U for each z = (zn ) ∈ ℓ∞ (E). If E = F , then the mapping T 7→ T U is an isometric and unital Banach algebra homomorphism from the space of all bounded linear operators on E into the space of all bounded linear operators on E U . If E and F are Banach lattices, then T is positive if and only if T U is positive; likewise, T is a lattice homomorphism if and only if T U is a lattice homomorphism. Ultra products are an important tool in operator theory. One of their most useful properties is that that lifting T 7→ T U improves the behaviour of certain parts of the spectrum of T without changing the spectrum as a whole; see for instance [20, Theorem 4.1.6]. Another useful property is that information about the operator T U


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can sometimes be used to deduce stronger information about the original operator T . Here is an example of this phenomenon: Proposition 2.4. Let E be a reflexive Banach space, let T be a power-bounded linear operator on E (i.e. supn∈N0 kT n k < ∞), and let U be a free ultra filter on N. Assume that fix(T U ) is finite dimensional. Then fix T is finite dimensional, too, and the number 1 is a pole of the resolvent R( · , T ) of order at most 1. Here we use the convention that a number λ0 ∈ C is a pole of R( · , T ) of order 0 iff λ0 is not contained in the spectrum of T . Proof of Proposition 2.4. Clearly, every fixed vector of T is mapped to a fixed vector of T U by the injective mapping E ∋ x 7→ xU ∈ E U , so fix T is finite dimensional, too. As E is reflexive and T is power bounded, the Ces`aro means of the powers of T converge strongly to a projection P on E which commutes with T and whose range coincides with the fixed space of T . In particular, T leaves the range and the kernel of P invariant, and it acts as the identity mapping on the range of P . In order to show that the number 1 is a zero or first order pole of the resolvent R( · , T ), it thus suffices to show that 1 is not a spectral value of the restricted operator T |ker P . So assume for a contradiction that 1 is a spectral value of T |ker P . We first note that 1 is not an eigenvalue of T |ker P . However, as T |ker P is power bounded, its spectral radius cannot be larger than 1, so 1 is contained in the topological boundary of the spectrum of T |ker P . In particular, 1 is an approximate eigenvalue of T |ker P , i.e. there exists a sequence (xn )n∈N in ker P such that kxn k = 1 for all indices n and such that (1 − T |ker P )xn = (1 − T )xn → 0 as n → ∞. Now we use an argument taken from [2, Lemma C-III-3.10]: If a subsequence of (xn )n∈N converges to a vector x, then x is obviously an eigenvector of T |ker P for the eigenvalue 1; hence, the sequence (xn )n∈N has no convergent subsequence. In particular, the set {xn : ∈ N} ⊆ ker P is not precompact, so there exists an ε > 0 such that this set cannot be covered by finitely many balls of radius ε. Therefore, we can find a subsequence (yn )n∈N of (xn )n∈N such that kyn − ym kE ≥ ε for all distinct m, n ∈ N. U For each k ∈ N0 we define y (k) := (yn+k )n∈N ∈ E U . Then y (k) is a fixed U (k) vector of T and has norm ky k = 1. However, for j 6= k we obtain ky (k) − y (j) k = lim kyn+k − yn+j kE ≥ ε. n→U

Thus, the sequence (y k) )k∈N0 in the unit ball of fix(T U ) has no convergent subsequence, so fix(T U ) is infinite dimensional. 2.4. Proofs of the main results. Now we can prove Theorem 1.2 (and thus also Theorem 1.1). The case where at least one of the numbers p and q is 1 or ∞ has already been treated in Subsection 2.1, so we are only left to deal with the case p, q ∈ (1, ∞) here. Proof of Theorem 1.2 in the case p, q ∈ (1, ∞). If the spectral radius of T is strictly smaller than 1, there is nothing to prove, so we assume throughout the proof that T has spectral radius 1. In particular, 1 is a spectral value of T since T is positive [29, Proposition V.4.1]. Fix a free ultra filter U on N. We first show that the fixed space of the operator T U on (Lp )U is finite dimensional. To this end, we are going to employ Proposition 2.2. The space (Lp )U is itself (isometrically lattice isomorphic to) an Lp -space over some measure space, and we have kT U k = kT k ≤ 1, so we have to find an appropriate Lq -space to make the proposition work.

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The space (Lq )U is (isometrically lattice isomorphic to) an Lq -space over some measure space, and the mapping j U : (Lq )U → (Lp )U is a lattice homomorphism, but it is not necessarily injective. However, its kernel ker(j U ) is a closed ideal in (Lq )U and thus, as (Lq )U has order continuous norm, even a band in (Lq )U . The quotient space (Lq )U / ker(j U ) is isometrically lattice isomorphic to the complementary band and thus to an Lq -space. Moreover, j U induces an injective lattice homomorphism J : (Lq )U / ker(j U ) → (Lp )U which has the same range as j U . Next we note that the range of T U is contained in the range of j U (and thus in the range of J). Indeed, let f U ∈ (Lp )U where f = (fn )n∈N ∈ ℓ∞ (Lp ). It follows from the closed graph theorem that the mapping j −1 T : Lp → Lq is continuous, so the sequence (j −1 T fn )n∈N is an element of ℓ∞ (Lq ). We thus obtain U U U T U f U = (T fn )n∈N = (jj −1 T fn )n∈N = j U (j −1 T fn)n∈N ,

so T U f U is in the range of j U (and thus in the range of J). Therefore, the assumptions of Proposition 2.2 are fulfilled, and we conclude that the fixed space of T U is finite dimensional. Now we can apply Proposition 2.4 which tells us that the fixed space of T is also finite dimensional and that the number 1 is a pole of the resolvent of T of order 1. Note that the range of the corresponding spectral projection Q coincides with the fixed space of T since the order of the pole equals 1. Next we employ a theorem which goes originally back to Niiro and Sawashima [24, Theorem 9.2] and which can, in the version that we use here, be found in [29, Theorem V.5.5]. The theorem says that, as the spectral radius of our operator T is a pole of the resolvent and as the corresponding spectral projection has finitedimensional range, every spectral value of T of maximal modulus is a pole of the resolvent. Hence, T has only finitely many spectral values on the unit circle, and each such spectral value is a pole of the resolvent. Moreover, it readily follows from the Neumann series representation of the resolvent that the order of each such pole is dominated by the order of the pole 1; hence, all unimodular spectral values of T are first order poles of R( · , T ). It only remains to show that the eigenspace of each unimodular spectral value of T is finite dimensional. This follows, for instance, from the dimension estimate in [12, Theorem 5.5] which asserts that dim ker(λ − T ) ≤ dim ker(λn − T ) for each number λ on the complex unit circle and for each integer n ∈ Z; plugging in n = 0 we obtain dim ker(λ − T ) ≤ dim ker(1 − T ) (note that the assumptions of [12, Theorem 5.5] are fulfilled here since every power bounded operator on a reflexive Banach space is weakly almost periodic). 3. Operator semigroups Now we briefly explain how our main results can be applied to obtain operator norm convergence of positive semigroups. Let E be a Banach lattice. An operator semigroup on E is a family (Tt )t∈(0,∞) of bounded linear operators on E such that the so-called semigroup law Ts+t = Ts Tt is fulfilled for all s, t ∈ (0, ∞). For what follows, we do not need to impose any regularity assumption with respect to the time parameter t on the semigroup. The semigroup (Tt )t∈(0,∞) is called bounded if supt∈(0,∞) kTt k < ∞ and it is called contractive if kTt k ≤ 1 for all t ∈ (0, ∞); it is called positive if Tt is a positive operator for each t ∈ (0, ∞). Let (Tt )t∈(0,∞) be a positive and bounded operator semigroup on a Banach lattice E. It was proved by Lotz in [19, Theorem 4 on p. 153] that, if for some t0 ∈ (0, ∞) the essential spectral radius of Tt0 is strictly smaller than 1, then Tt converges with respect to the operator norm as t → ∞. Hence, we obtain the following corollary of Theorem 1.2.


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Corollary 3.1. Let p, q ∈ [1, ∞] be distinct numbers and let (Ω1 , µ1 ) and (Ω2 , µ2 ) be arbitrary measure spaces. Moreover, let j : Lq := Lq (Ω2 , µ2 ) → Lp := Lp (Ω1 , µ1 ) be an injective lattice homomorphism. Let (Tt )t∈(0,∞) be a positive and contractive operator semigroup on Lp and assume that Tt0 Lp ⊆ j(Lq ) for at least one time t0 ∈ (0, ∞). Then Tt converges with respect to the operator norm as t → ∞. Note that the corollary remains valid if we replace the assumption that the operator semigroup be contractive with the assumption that the operator semigroup is bounded and that merely the operator Tt0 is contractive. 4. Concluding remarks We conclude the paper with the following open problem: Open Problem 4.1. Do Theorems 1.1 and 1.2 remain valid if we replace the assumption kT k ≤ 1 with the weaker assumption that T be power-bounded (i.e. supn∈N0 kT n k < ∞)? If we merely assume that T is power-bounded instead of contractive, there is only one point where our proof of Theorem 1.2 fails: we can no longer use the argument from Lemma 2.3 to conclude that the fixed space of T is a sublattice of Lp (Ω1 , µ1 ). However, it is still possible to show that fix T is a lattice subspace of Lp (Ω1 , µ1 ), i.e. a vector lattice in its own right with respect to the order inherited from Lp (Ω1 , µ1 ) (but with possibly different lattice operations). Indeed, if f ∈ fix T , then the limit g := limn→∞ T n |f | exists with respect to the norm on Lp (Ω1 , µ1 ) and yields the modulus of f in the space fix T (this argument is taken from the proof of [2, Corollary C-III-4.3(a)]). Still, since the lattice operations in a lattice subspace can differ from the lattice operations in Lp (Ω1 , µ1 ) itself, we cannot simply conclude that the norm on fix T is p-additive, and this is where our argument fails. Acknowledgements. I am indebted to Delio Mugnolo for several very helpful discussions and comments; he told me about L. Miclo’s article [21], which was the motivation for writing the present paper, he brought Pitt’s theorem (mentioned after Corollary 1.3) to my attention and he suggested to generalise the result in Theorem 1.1 to Theorem 1.2. Appendix A. Lattice isomorphisms between Lp - and Lq -spaces The fact that, for instance, Lp ([0, 1]) and Lq ([0, 1]) are not isomorphic as Banach spaces for p 6= q is usually shown by techniques from the geometric theory of Banach spaces. For our purposes, though, we only need the much simpler fact that an infinite dimensional Lp -space is never lattice isomorphic to an Lq -space for p 6= q. In the following proposition we give an elementary proof of this fact. Proposition A.1. Let p, q ∈ [1, ∞) be two distinct numbers, let (Ω1 , µ1 ) and (Ω2 , µ2 ) be arbitrary measure spaces and assume that Lp := Lp (Ω1 , µ1 ) and Lq := Lq (Ω2 , µ2 ) are isomorphic as Banach lattices (i.e. there exists a lattice isomorphism Lq → Lp ). Then Lq (and hence Lp ) has finite dimension. For the proof we need the simple observation that, in every infinite dimensional Banach lattice E, there exists a sequence (xk )k∈N ⊆ E+ of normalised and pairwise disjoint vectors (i.e. kxk k = 1 for all indices k and xj ∧ xk = 0 whenever j 6= k).

¨ JOCHEN GLUCK

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Proof of Proposition A.1. We may assume that p < q. Assume for a contradiction that Lq is infinite-dimensional. Then there exists a sequence (fk )k∈N of normalised P∞ k and pairwise disjoint vectors 0 ≤ fk ∈ Lq . The series k=1 kf1/p converges in Lq since, for 1 ≤ m ≤ n, we have n n n

X X X kfk kqq 1 fk

q

= → 0 as m, n → ∞.

=

1/p q q/p q/p k k k k=m k=m k=m

Now, let J : Lq → Lp be a lattice isomorphism. Then the vectors gk := Jfk ∈ Lp P∞ are also pairwise disjoint. As J is continuous, the series k=1 kgkp converges in Lp . However, the mapping J −1 is continuous, too, so we have kgk kp ≥ ckfk kq = c for a constant c > 0 and all indices k. This implies that, for 1 ≤ m ≤ n, n n n

X X X kgk kpp gk cp

p = ≥ 6→ 0 as m, n → ∞.

1/p k k k p k=m

k=m

k=m

This is a contradiction.

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