What does a rate in a mean ergodic theorem imply?
aa r X i v : . [ m a t h . F A ] N ov WHAT DOES A RATE IN A MEAN ERGODIC THEOREMIMPLY?
ALEXANDER GOMILKO AND YURI TOMILOV
Abstract.
We develop a general framework for the inverse mean ergodic the-orems with rates for operator semigroups thus completing a construction ofthe theory initiated in [16] and [17]. Introduction
In this paper we are concerned with the rates of convergence of Ces´aro means(1.1) C t ( A ) := 1 t Z t T ( s ) ds, t > , for a bounded C -semigroup ( T ( t )) t ≥ with generator − A on a (complex) Banachspace X. Recall that in general { x ∈ X : C t ( A ) x strongly converges } = ker( A ) ⊕ ran( A ) . (1.2)Moreover, C t ( A ) x converges to zero if and only if x ∈ ran( A ).A mean ergodic theorem provides conditions under which the means in (1.1)converge strongly on the whole of X . One of the most well-known mean ergodictheorems says that if X is reflexive then X = ker( A ) ⊕ ran( A ) , hence C t ( A ) , t > , are strongly convergent. Mean ergodic theorems is a classicalchapter of the ergodic theory and for its basic results one may consult [26].If a mean ergodic theorem holds then it is natural to try to equip it with acertain convergence rate. After a simple normalization, one can assume withoutloss of generality, that C t ( A ) , t > , converge to zero as t → ∞ . Thus we will studythe decay rates of k C t ( A ) x k , x ∈ X. (See the introduction in [17] for a more detaileddiscussion.)The rates in ergodic theorems were studied in many settings and backgrounds.For some of the achievements in this area one may consult the survey papers [24],[3] (and references therein) and also [1], [6]–[14], [25], [28] and [32]. However nosystematic approach to characterizing rates in mean ergodic theorems was proposeduntil very recent time. The present paper provides one more step towards such acharacterization. It is a companion to our previous articles [17] and [16] where thetheory of rates in mean ergodic theorems was developed by methods of functionalcalculus. It was our initial idea that a functional calculus approach might produce Date : October 9, 2018.2010
Mathematics Subject Classification.
Primary 47A60, 47A35; Secondary 47D03.
Key words and phrases. mean ergodic theorem, rate of convergence, functional calculus, C -semigroup.The authors were partially supported by the NCN grant DEC-2011/03/B/ST1/00407. certain rates of decay of Ces´aro means in a canonical way and thus would allowus to quantify their convergence properties. This idea appeared to be fruitful andopened a door to many tools from outside of ergodic theory. Taking advantage ofthese new tools we are now able to introduce and study in details abstract inversetheorems on decay rates, the main subject of this paper.To set the scene, let us first recall certain direct theorems on rates obtained in[16] and [17]. The direct problem in the study of rates for Ces´aro means can beformulated as follows. Direct Problem:
Given x from the range (or the domain) of a function of A finda rate of decay (if any) for C t ( A ) x and prove its optimality. (Of course, we shouldspecify what we mean by ‘function’ and ‘optimality’ and that will be clear fromfurther considerations.)Theorems answering the direct problem will be called direct mean ergodic theo-rems with rates. Motivated by probabilistic applications, the problem of obtainingvarious direct theorems with rates in has attracted considerable attention last years.We note the foundational paper [15] and then the subsequent papers [1], [6]–[14],[25].Recently, we proposed in [16] and [17] an abstract framework which allowedus to encompass many partial results and to solve certain open problems on therates of decay of Ces´aro means. In particular, we proved in [17, Theorem 3.4 andProposition 4.2] that if ( T ( t )) t ≥ is a bounded C -semigroup on X and f is aBernstein function, lim t → f ( t ) = 0 , then(1.3) x ∈ ran( f ( A )) = ⇒ k C t ( A ) x k = O( f ( t − )) , t → ∞ . As corollaries, we obtained rates of decay of Ces´aro means on the ranges of polyno-mial and logarithmic functions thus extending and sharpening known results. Ourresults were proved to be optimal in a natural sense.To understand the limitations of direct mean ergodic theorems with rates it isnatural to ask whether the implication (1.3) can be reversed. Examples show thatone cannot in general expect the implication opposite to (1.3) to be true (see e.g.Section 4 of the present paper and [15, Example, p. 121] concerning the discretesetting). Thus we are interested in the best possible conditions on the decay of themeans implying the converses of (1.3), and our abstract inverse problem reads asfollows.
Inverse Problem:
Given the rate of decay of C t ( A ) x for an element x ∈ X provethat x is in an appropriate range (or domain) of a function of A and show optimalityof the result.Statements of that form will be called inverse mean ergodic theorems with rates .The first inverse theorems were proved in the discrete setting by Browder [2] andButzer and Westphal [4]. They showed (indirectly in the first case) that if X isreflexive and T is a power bounded operator on X , then (cid:13)(cid:13)(cid:13) /n P n − k =0 T k x (cid:13)(cid:13)(cid:13) = O(1 /n )implies that x ∈ ran( I − T ) . It was also noted in [4] that one cannot produce betterrates than 1 /n, since (cid:13)(cid:13)(cid:13) /n P n − k =0 T k x (cid:13)(cid:13)(cid:13) = o(1 /n ) implies x = 0 . Thus one has todeal with rates between 1 /n and o(1) , and the same is true for the continuous timemeans C t ( A ) x when the rate 1 /n is replaced by 1 /t - see [17] for a discussion.This complicates the study of rates since many plausible conditions involving ratesappear to be too strong in view of the extremal 1 /n (or 1 /t ) property. See e.g. ourAppendix. HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 3
Various partial situations (mostly of polynomial rates and mostly in the discreteframework) were considered in [11], [12], [5], [15], and [25]. The main goal of thepresent paper is to provide an abstract set-up for the inverse theorems and togive them a systematic treatment. This set-up appears to be coherent with directtheorems obtained in [17] and it constitutes in a sense a final block of the theorydeveloped in [17]. As in the case of direct theorems treated in [17], known inversetheorems on rates for particular cases (e.g. for polynomial rates) can be includedin our framework.In [17] our direct ergodic theorems involved the ranges of complete Bernsteinfunctions of semigroup generators (as e.g. in (1.3)). In the present study of the in-verse theorems, it will be convenient to restrict our attention to the class of Stieltjesfunctions and to deal with their domains rather than the ranges of (reciprocal) oper-ator complete Bernstein functions. Such a setting enabled us to apply an (adapted)abstract characterization of the domains of operator complete Bernstein functionsdue to Hirsch [22]. (A similar result in a slightly more general setting was obtainedlater by R. Schilling, see e.g. Theorem 12 . , Remark 12 . , and Corollary 12.21in [30] and also [29].)The paper is based on ideas worked out in [16], [17], and [19]. However, itsfiner details are essentially different from the arguments used in those papers andit complements the results obtained in [16], [17], and [19]. To give a flavor ofinverse theorems proved by our technique we indicate a partial converse of thedirect theorem formulated above. It illustrates our approach of adding an ‘extrarate’ to the decay of the means in order to invert the direct statements.Assume that ran( A ) = X. If g is a Stieltjes function of the form g ( z ) = Z ∞ µ ( ds ) z + s , z > , where µ is a (non-negative) Radon measure on (0 , ∞ ) such that Z ∞ µ ( ds )1 + s < ∞ ,g (0+) = ∞ , and x ∈ X satisfies Z ∞ g (1 /t ) k C t ( A ) x k t dt < ∞ , then x ∈ dom( g ( A )), or, equivalently, x ∈ ran([1 /g ]( A )) . Note that there are close relations between inverse mean ergodic theorems forbounded C -semigroups ( T ( t )) t ≥ and bounded discrete semigroup ( T n ) n ≥ , andour approach, in fact, unifies continuous and discrete frameworks. It allows one tostudy the continuous and the discrete cases simultaneously and to obtain resultsparallel in spirit and proofs. However, because of space limitations, the functionalcalculus approach to inverse mean ergodic theorems in the discrete case will bepresented elsewhere.We also show that our statements are sharp and cannot in general be improved.In fact, it appears that they are are optimal even for a very simple multiplicationoperator on an L space. However, even in this simple case, there are nontrivialtechnical difficulties to overcome. Thus a substantial part of the paper is devotedto proving optimality of our results in various senses. ALEXANDER GOMILKO AND YURI TOMILOV
Our Appendix addresses important and related to inverse theorems problemwhich however stay a bit aside from the mainstream of the exposition and thusshifted to a separate part. We believe that it is of independent interest. There weprove that the means cannot be too small in an “integral” sense.1.1.
Some Notations and Definitions.
For a closed linear operator A on acomplex Banach space X we denote by dom( A ) , ran( A ), ker( A ), and ρ ( A ) the domain , the range , the kernel , and the resolvent set of A , respectively. The norm-closure of the range is written as ran( A ). The space of bounded linear operators on X is denoted by L ( X ). Finally, we set R + := [0 , ∞ ) and C + := { λ ∈ C : Re λ > } . Preliminaries
Functional calculus: Bernstein and Stieltjes functions.
In this subsec-tion we recall basic properties of operator Bernstein and Stieltjes functions andprove several auxiliary statements on functional calculi useful for the sequel. More-over we arrange the material in the way most suitable for our purposes. Thedeveloped machinery will be used intensively in the next sections.Let M( R + ) be a Banach algebra of bounded Radon measures on R + . Define the
Laplace transform of µ ∈ M( R + ) as( L µ )( z ) := Z R + e − sz µ (d s ) , z ∈ C + , and note that L µ extends to a continuous function on C + . Note that the spaceA ( C + ) := {L µ : µ ∈ M( R + ) } is a commutative Banach algebra with pointwise multiplication and with respectto the norm(2.1) kL µ k A := k µ k M( R + ) = | µ | ( R + ) , and the Laplace transform L : M( R + ) −→ A ( C + )is an isometric isomorphism.Let − A be the generator of a bounded C -semigroup ( T ( s )) s ≥ on a Banachspace X . Then the mapping g = L µ = Z R + e − s · µ (d s ) g ( A ) := Z R + T ( s ) µ (d s )(where the integral converges in the strong topology) is a continuous algebra ho-momorphism of A ( C + ) into L ( X ) . The homomorphism is called the
Hille-Phillips(HP-) functional calculus for A . Its basic properties can be found in [20, ChapterXV].The HP-calculus has an extension to a larger function class. This extension isconstructed as follows: if f : C + → C is holomorphic such that there exists afunction e ∈ A ( C + ) with ef ∈ A ( C + ) and the operator e ( A ) is injective, thenwe define f ( A ) := e ( A ) − ( ef )( A )with its natural domain dom( f ( A )) := { x ∈ X : ( ef )( A ) x ∈ ran( e ( A )) } . In thiscase f is called regularizable , and e is called a regularizer for f . Such a definition of f ( A ) does not depend on the particular regularizer e and f ( A ) is a closed operator HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 5 on X . Moreover, the set of all regularizable functions f is an algebra depending on A (see e.g. [18, p. 4-5] and [13, p. 246-249]), and the mapping f f ( A )from this algebra into the set of all closed operators on X is called the extendedHille–Phillips calculus for A . The next product rule of this calculus (see e.g. [18,Chapter 1]) will be crucial for the sequel: if f is regularizable and g ∈ A ( C + ) ,then (2.2) g ( A ) f ( A ) ⊆ f ( A ) g ( A ) = ( f g )( A ) , where we take the natural domain for a product of operators.This regularization approach can be applied to the study of operator Bernsteinfunctions. First recall that f ∈ C ∞ (0 , ∞ ) is called a complete monotone functionif f ( t ) ≥ − n d n f ( t ) dt n ≥ n ∈ N and t > . A function f ∈ C ∞ (0 , ∞ ) is called Bernstein function if its derivative is completelymonotone. By [30, Theorem 3.2], f is Bernstein if and only if there exist constants a, b ≥ µ on (0 , ∞ ) satisfying Z ∞ s s µ ( ds ) < ∞ and such that(2.3) f ( z ) = a + bz + Z ∞ (1 − e − sz ) µ ( ds ) , z > . The formula (2.3) is called the Levy-Khintchine representation of f. The triple( a, b, µ ) is uniquely determined by the corresponding Bernstein function f and iscalled the Levi-Khintchine triple.It was proved in [17, Lemma 2.5] that Bernstein functions belong to the extendedHP-functional calculus and every Bernstein function is regularizable by any of thefunctions e λ ( z ) = ( λ + z ) − , Re λ > . This led to the following the operator Levy-Khintchine representation for a Bernstein function f of A (cf. (2.3)) essentially dueto Phillips [27]. Theorem 2.1.
Let − A generate a bounded C -semigroup ( T ( s )) s ≥ on X , andlet f ∼ ( a, b, µ ) be a Bernstein function. Then f ( A ) is defined in the extendedHP-calculus. Moreover, dom( A ) ⊆ dom( f ( A )) and (2.4) f ( A ) x = ax + bAx + Z ∞ ( I − T ( s )) x µ ( ds ) for each x ∈ dom( A ) , and dom( A ) is a core for f ( A ) . If a > , then ran( f ( A )) = X and f ( A ) is invertible. For the detailed theory of operator Bernstein functions we refer to [30].The class of Bernstein functions is quite large and to ensure good algebraicand function-theoretic properties of Bersntein functions it is convenient and alsosufficient for many purposes to consider its subclass consisting of complete Bernsteinfunctions. A Bernstein function is called complete if its representing measure in theLevy-Khintchine formula (2.3) has a completely monotone density with respect toLebesgue measure, see [30, Definition 6.1].
ALEXANDER GOMILKO AND YURI TOMILOV
To discuss other representations of complete Bernstein functions, more suitablefor the goals of this paper, we will also need yet another related class of functions.A function g : (0 , ∞ ) → R + is called Stieltjes if it can be written as(2.5) g ( z ) = a + bz + Z ∞ µ ( ds ) z + s , z > , where a, b ≥ µ is a positive Radon measure on (0 , ∞ ) satisfying(2.6) Z ∞ µ ( ds )1 + s < ∞ . Since the representation (2.5) is unique, the measure µ is called a Stieltjes mea-sure for g and (2.5) is called the Stieltjes representation for g , see e.g. [30, Chapter2]. We will then write g ∼ ( a, b, µ ) . Note that(2.7) a = g ( ∞ ) , b = lim z → zg ( z ) . The following result (see [30, Theorem 6.2 and Corollary 7.4]) shows, in partic-ular, a reciprocal duality between complete Bernstein and Stieltjes functions, andit will be crucial for the sequel.
Theorem 2.2.
A non-zero function g is a Stieltjes function if and only zg ( z ) , z > , is a complete Bernstein function, if and only if /g is a complete Bernstein function.Remark . Thus every complete Bernstein function f admits a unique represen-tation(2.8) f ( z ) = a + bz + Z ∞ zz + s µ ( ds ) , z > , where a, b ≥ µ is a positive Radon measure on (0 , ∞ ) satisfying (2.6), andwe can speak of the Stieltjes representation ( a, b, µ ) of f, and write f ∼ ( a, b, µ ) . However, there are also other representations for complete Bernstein functionsin the literature. For example, we note the representation(2.9) f ( z ) = a + bz + Z ∞ zν ( ds )1 + zs , Z ∞ ν ( ds )1 + s < ∞ , used in particular in [21] and [22]. The representations (2.8) are (2.9) are equivalentin the sense that one of them is transformed by the change of variable s = 1 /t intoanother so that the measures µ and ν satisfy the same integrability condition (2.6).We will be interested in Stieltjes functions g with the Stieltjes representation ofthe form (0 , , µ ) , and satisfying g (0+) = ∞ . Before going further, we give severalelementary examples of such functions important for the sequel.
Example 2.4. a) The functions g γ ( z ) := z − γ , γ ∈ (0 , , are Stieltjes andlim s → g ( s ) = ∞ , g γ ( z ) = sin πγπ Z ∞ ds ( z + s ) s γ , z > . Accordingly, f γ ( z ) = zg γ ( z ) = z − γ , γ ∈ (0 , , are complete Bernstein functions.b) By [17, Example 2.9] the function g ( z ) := log zz − Z ∞ ds ( z + s )(1 + s ) , z > , is Stieltjes with g (0+) = ∞ , and so is the function g ( z ) := z − z log z by Theorem 2.2. HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 7
Let us show now that complete Bernstein and Stieltjes functions of the generator − A can be expressed in resolvent terms in accordance with the formulas (2.5) and(2.8). To this aim, we will need the notion of a sectorial operator. Recall that alinear operator V on X is called sectorial if ( −∞ , ⊂ ρ ( V ) and there exists c > s k ( s + V ) − k ≤ c, s > . Theorem 2.5.
Let − A be the generator of a bounded C -semigroup on X .(i) If f ∼ (0 , , µ ) is a complete Bernstein function, then (2.10) f ( A ) x = Z ∞ A ( s + A ) − x µ ( ds ) for every x ∈ dom( A ) . Moreover, dom( A ) is a core for f ( A ) . (ii) If g ∼ (0 , , µ ) is a Stieltjes function and A has dense range, then g belongsto the extended HP-calculus and (2.11) g ( A ) x = Z ∞ ( s + A ) − x µ ( ds ) for every x ∈ ran( A ) . Moreover, ran( A ) is a core for g ( A ) . Proof.
The proof of (2.10) relies on a direct transforming (2.4) to the form (2.10)by means of the definition of a complete Bernstein function and can be found in[30, p. 149]. The fact that dom( A ) is a core for f ( A ) follows from Theorem 2.1.Thus (i) is a straightforward consequence of Theorem 2.1.To prove (ii) note that by Theorem 2.2 if g is a Stieltjes function then g ( z ) = q ( z ) /z for some complete Bernstein function q. Since by [17, Lemma 2.5] q is regular-izable by 1 / ( z + 1) and A is injective in view of (1.2), the function g is regularizableby z/ ( z + 1) and belongs to the extended HP-calculus. Moreover, if x ∈ ran( A ) , then using the product rule for the extended HP-calculus we obtain g ( A ) x = (cid:20) z + 1 z · zg · z + 1 (cid:21) ( A ) x = ( A + I ) A − Z ∞ A ( s + A ) − ( A + I ) − x µ ( ds )= Z ∞ ( s + A ) − x µ ( ds ) . It remains to prove that ran( A ) is a core for g ( A ) . Observe that by e.g. [18,Proposition 2.2.1,b] the operator A − is sectorial with dense domain ran( A ) . Henceif e t ( A ) = t ( t + A − ) − , t > , then e t ( A ) x → x for every x ∈ X as t → ∞ by[18, Proposition 2.2.1, c]. Since e t ∈ A ( C + ) for each t >
0, the product rule(2.2) implies that if x ∈ dom( g ( A )) and g ( A ) x = y then g ( A ) e t ( A ) x = e t ( A ) y. Asran( e t ( A )) = dom( A − ) = ran( A ) , the statement follows. (cid:3) Let g be a Stieltjes function with the Stieltjes representation (0 , , µ ) , i.e.(2.12) g ( z ) = Z ∞ µ ( ds ) z + s , z > , Z ∞ µ ( ds )1 + s < ∞ . If a complete Bernstein function h is given by(2.13) h ( z ) := g (1 /z ) = Z ∞ z µ ( ds )1 + zs . ALEXANDER GOMILKO AND YURI TOMILOV and a linear operator V is sectorial then the operator h ( V ) can be defined as theclosure of a (closable) linear operator h ( V ) given by the formula(2.14) h ( V ) x = Z ∞ V (1 + sV ) − x µ ( ds ) , x ∈ dom( V ) . This definition is due to Hirsch and it was introduced and thoroughly studied inhis paper [21]. Thus h ( V ) is a closed linear operator and dom( h ( V )) ⊃ dom( V ) . If A − is a sectorial operator with dense domain ran( A ) , then setting V = A − in (2.14) we obtain h ( A − ) x = Z ∞ ( A + s ) − x µ ( ds ) = g ( A ) x, x ∈ ran( A ) . (2.15)Hence the operators h ( A − ) and g ( A ) coincide on their core dom( A − ) = ran( A )and therefore coincide. In other words, g ( A ) defined in the extended HP-calculuscoincides with h ( A − ) defined by means of (2.14).Hirsch proved in [21] a number of properties of complete Bernstein functions ofsectorial operators. We will need two of them which we state as a lemma. For theirproofs see [21, Theorem 1] and [21, Theorem 3]. Lemma 2.6.
Let f and q be complete Bernstein functions and let A be a sectorialoperator with dense range. Then(i) f ( A ) is a sectorial operator with dense range;(ii) ( f ◦ q )( A ) = f ( q ( A )) . The property (2.15) will allow us to link several results from [21] and [22] to oursetting of Stieltjes functions of semigroup generators.
Lemma 2.7.
Let − A be the generator of a bounded C -semigroup on X, with denserange. If f is a complete Bernstein function and g is a Stieltjes function, then theircomposition f ◦ g belongs to the extended HP-calculus, and f ( g ( A )) = ( f ◦ g )( A ) . As a consequence, (2.16) dom(( f ◦ g )( A )) ⊃ dom( g ( A )) . Proof.
By assumption, A − is sectorial with dense domain. Note that if g is aStieltjes function and a complete Bernstein function h is given by (2.13), then g ( A ) = h ( A − ) , where h ( A − ) is defined by means of (2.14). By Lemma 2.6, (i) theoperator h ( A − ) is sectorial with dense range, so g ( A ) is the same.Observe also that the composition f ◦ g is Stieltjes, see e.g [30, Theorem 7.5].Then using Lemma 2.6, (ii) and (2.15) we conclude that f ( g ( A )) = f ( h ( A − )) = ( f ◦ h )( A − ) = [( f ◦ h )(1 /z )]( A ) = ( f ◦ g )( A ) . This implies, in particular, by Theorem 2.1 that dom(( f ◦ g )( A )) = dom f ( g ( A )) ⊃ dom( g ( A )) . (cid:3) The next statement describing domains of operator Stieltjes functions in resol-vent terms is basic for the paper. It is in fact a reformulation of [22, Theorem 2]based on (2.15).
HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 9
Theorem 2.8 (Hirsch Criterion) . If − A is the generator of a bounded C -semigroupon X such that ran( A ) = X and g is a Stieltjes function given by (2.12) , then x ∈ dom( g ( A )) ⇐⇒ ∃ weak − lim δ → Z δ ( A + s ) − x µ ( ds ) ⇐⇒ ∃ strong − lim δ → Z δ ( A + s ) − x µ ( ds ) . Rates in mean ergodic theorem
For the whole of this section, we make the following assumptions: − A is the generator of a C − semigroup ( T ( t )) t ≥ ,M := sup t ≥ k T ( t ) k < ∞ , and ran( A ) = X, (recall that in this case, by (1.2), ker( A ) = { } ) and g is a Stieltjes function , g ∼ (0 , , µ ) , g (0+) := ∞ . Let us comment on the above assumptions on g . For technical reasons, it willbe more convenient for us to consider Stieltjes functions as above than those of thegeneral form (2.5), (2.6). To see that we do not loose generality indeed, note that wemay assume a = 0 (that is lim s →∞ g ( s ) = 0) in view of dom( g ( A )+ a ) = dom( g ( A )).If b = 0 then by passing to the reciprocal complete Bernstein function 1 /g and using[30, Corollary 12.7] we infer that dom( g ( A )) = ran( A ) , so that the Ces´aro meansC t ( A ) x for x ∈ dom( g ( A )) decay at the extremal rate:(3.1) k C t ( A ) x k = O (cid:0) t − (cid:1) , t → ∞ , x ∈ dom( g ( A )) . The inverse theorems given below become void in this case, see Remark 5.3 inAppendix. Finally if g (0+) < ∞ then our direct theorem on rates from [17] (seealso Theorem 3.2 below) does not yield any rate of decay of C t ( A ) restricted todom( g ( A )) since 1 /g (1 /t ) , t → ∞ in this case.We start with an elementary inequality which will nevertheless be essential inthe proof of direct theorems for rates in both discrete and continuous cases. Lemma 3.1.
Let f ∼ (0 , , µ ) be a complete Bernstein function. Then (3.2) 1 t Z ∞ − e − st s µ ( ds ) ≤ f ( t − ) , t > . Proof.
Since 1 − e − x x ≤
21 + x , x > , we have 1 − e − st ts ≤
21 + ts = 2 t − t − + s , s, t > . Hence 1 t Z ∞ − e − st s µ ( ds ) ≤ Z ∞ t − µ ( ds ) t − + s = 2 f ( t − ) , t > . (cid:3) First we derive a convergence rate for C t ( A ) x for x ∈ dom( g ( A )) , or equivalentlyfor x ∈ ran([1 /g ]( A )) with 1 /g being a complete Bernstein function. The follow-ing theorem is a partial case of [17, Theorem 3.4 and Proposition 4.2] where theconvergence rate was obtained in terms of the limit behavior of 1 /g at zero for thewhole class of Bernstein functions. However, in the particular situation of completeBernstein functions we give an argument which is simpler and more transparentthan that from [17]. Moreover, it illustrates nicely our functional calculus approachand makes the presentation self-contained. Theorem 3.2. If x ∈ dom( g ( A )) then (3.3) k C t ( A ) x k ≤ M k g ( A ) x k g ( t − ) , t > . Proof.
Remark first that if g a Stieltjes function, then f = 1 /g is a completeBernstein function and by [18, Theorem 1.2.2, d)] one has(3.4) ( f ( A )) − = (1 /f )( A ) = g ( A ) , hence dom( g ( A )) = ran( f ( A )) . Let y ∈ dom( A ) ⊂ dom( f ( A )) and t > . Then from (2.10) it follows that t C t ( A ) f ( A ) y = Z ∞ Z t T ( τ ) A ( A + s ) − y dτ µ ( ds )= Z ∞ [1 − T ( t )]( A + s ) − y µ ( ds ) . Since[ I − T ( t )]( A + s ) − y = Z ∞ t (cid:16) e − s ( τ − t ) − e − sτ (cid:17) T ( τ ) y dτ + Z t e − sτ T ( τ ) y dτ, we infer that k [1 − T ( t ))( A + s ) − y k ≤ M k y k (cid:26)Z ∞ t (cid:16) e − s ( τ − t ) − e − sτ (cid:17) dτ + Z t e − sτ dτ (cid:27) = 2 M k y k · − e − st s . Thus using Lemma 3.1 we have k C t ( A ) f ( A ) y k ≤ M k y k Z ∞ − e − st ts µ ( ds ) ≤ M f ( t − ) k y k , y ∈ dom( A ) . Since dom( A ) is a core for f ( A ) , by passing to closures in the last inequality, wefinally obtain(3.5) k C t ( A ) f ( A ) y k ≤ M f ( t − ) k y k , y ∈ dom( f ( A )) . Then (3.4) and (3.5) imply (3.3). (cid:3)
Remark . Note that Theorem 3.2 can be formulated in terms of complete Bern-stein functions as in (3.5). It is this form of (3.3) that we have obtained in [17,Theorem 3.4 and Proposition 4.2].
HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 11
The following result, Theorem 3.4, is our main inverse theorem for rates in thecontinuous setting. At first glance, its assumptions differ from the conclusions of(direct) Theorem 3.2. We show however that the result yields several statementswhich are are “almost” converse of Theorem 3.2. The word “almost” is crucial: weprove that the result is optimal and thus there is an unavoidable, in general, gapbetween our direct and inverse mean ergodic theorems with rates.
Theorem 3.4. If x ∈ X is such that (3.6) Z ∞ | g ′ (1 /t ) |k C t ( A ) x k t dt < ∞ , then x ∈ dom( g ( A )) . Proof.
Note that for any s > x ∈ X ( A + s ) − x = Z ∞ e − st T ( t ) x dt = Z ∞ e − st (cid:18)Z t T ( τ ) x dτ (cid:19) ′ dt = Z ∞ ste − st C t ( A ) x dt. Therefore k ( A + s ) − x k ≤ M k x k + Z ∞ ste − st k C t ( A ) x k dt and Z k ( A + s ) − x k µ ( ds ) ≤ M k x k Z µ ( ds )+ Z ∞ (cid:18)Z ste − st µ ( ds ) (cid:19) k C t ( A ) x k dt. To estimate the inner integral observe that for every τ ≥ τ e − τ ≤ τ ) , since 4 e τ − τ (1 + τ ) > X i =0 t i i ! ! − τ − τ − τ > τ τ − > . Now using (3.7) with τ = ts we have Z ∞ tse − ts µ ( ds ) ≤ Z ∞ µ ( ds )(1 + ts ) = 4 t Z ∞ µ ( ds )(1 /t + s ) = 4 | g ′ (1 /t ) | t . Thus Z k ( A + s ) − x k µ ( ds ) ≤ M k x k Z µ ( ds )(3.8) + 4 Z ∞ | g ′ (1 /t ) | t k C t ( A ) x k dt < ∞ , and x ∈ dom( g ( A )) by Hirsch’s Theorem 2.8. (cid:3) The next direct corollary of Theorem 3.4 is formulated in terms of a norm esti-mate for C t ( A ) thus removing assumptions on the derivative of g. Corollary 3.5. If x ∈ X and a measurable function ǫ : ( g (1) , ∞ ) (0 , ∞ ) satisfy (3.9) k C t ( A ) x k = O (cid:18) g (1 /t ) ǫ ( g (1 /t )) (cid:19) , t → ∞ , Z ∞ g (1) dττ ǫ ( τ ) < ∞ , then x ∈ dom( g ( A )) . Proof.
If (3.9) holds, then there exists c > Z ∞ | g ′ (1 /t ) |k C t ( A ) x k t dt ≤ c Z ∞ dg (1 /t ) g (1 /t ) ǫ ( g (1 /t )) = c Z ∞ g (1) dττ ǫ ( τ ) < ∞ , and Theorem 3.4 implies x ∈ dom( g ( A )). (cid:3) Now we derive a corollary of Theorem 3.4 which is almost converse to Theorem3.2. It is however strictly weaker than Theorem 3.4 and at the same time it cannotessentially be improved as we will show in Section 4.
Corollary 3.6. If x ∈ X is such that (3.10) Z ∞ g (1 /t ) k C t ( A ) x k t dt < ∞ , then x ∈ dom( g ( A )) .Proof. It suffices to observe that(3.11) | g ′ ( τ ) | = Z ∞ µ ( ds )( τ + s ) ≤ τ Z ∞ µ ( ds ) τ + s = g ( τ ) τ , τ > . (In fact, a more general estimate is given in [23, Lemma 3.9.34].) The claim followsnow from Theorem 3.4. (cid:3) Remark . Note that Corollary 3.6 can be formulated in terms of the norm esti-mates for C t ( A ) x rather than an integral condition on k C t ( A ) x k . Indeed, (3.10) isequivalent to the condition(3.12) k C t ( A ) x k = O (cid:18) ǫ ( t ) g (1 /t ) (cid:19) , t → ∞ , where ǫ : (1 , ∞ ) → (0 , ∞ ) is a measurable function satisfying Z ∞ dττ ǫ ( τ ) < ∞ . Observe that if g ( z ) = z − α , α ∈ (0 , , then (3.9) and (3.12) are, in a sense,equivalent. Indeed, if (3.9) holds then setting ˜ ǫ ( τ ) = ǫ ( τ α ) we have Z ∞ dττ ˜ ǫ ( τ ) = 1 α Z ∞ dττ ǫ ( τ ) , and (3.12) is satisfied with ǫ replaced by ˜ ǫ. Conversely, if (3.12) holds then setting˜ ǫ ( τ ) = ǫ ( τ /α ) we infer that (3.12) is true with ǫ replaced by ˜ ǫ. Using Remark 3.7 we state now the following straightforward consequence ofCorollary 3.6.
Corollary 3.8.
If for x ∈ X there exists α ∈ (0 , such that (3.13) k C t ( A ) x k = O (cid:18) g (1 /t ) log α (2 + g (1 /t )) (cid:19) , t → ∞ , then x ∈ dom( g ( A )) . HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 13
Specifying (3.13) for a power function we obtain the domain/range condition forfractional powers of A . Corollary 3.9.
If for x ∈ X there exist α, β ∈ (0 , such that k C t ( A ) x k = O (cid:18) t β log α t (cid:19) , t → ∞ , then x ∈ dom( A − β ) = ran( A β ) . The next result proposes a different ideology for proving the inverse mean ergodictheorems with rates. To be able to place an element x into dom( g ( A )) in the resultsabove we had to add an “extra rate” to the rate r ( t ) = ( g (1 /t )) − of the decay ofC t ( A ) x obtained in Theorem 3.2. Now, instead of adding an “extra rate”, we add an“extra domain” to dom( g ( A )). In this way, we will show that x belongs to a slightlylarger space than dom( g ( A )) under the assumption k C t ( A ) x k = O(( g (1 /t )) − ).(Recall that by Theorem 3.2 k C t ( A ) x k = O(( g (1 /t )) − ) would follow from merely x ∈ dom( g ( A )) . ) To this aim, recall first (from Section 2) that for any completeBernstein function f and Stieltjes function g the function f ◦ g is a Stieltjes. More-over, by (2.16), we have dom(( f ◦ g )( A )) ⊃ dom( g ( A )) , and the inclusion is in general strict. While the assumption k C t ( A ) x k = O(( g (1 /t )) − )does not imply x ∈ dom( g ( A )) we prove that it does suffices to guarantee x ∈ dom(( q ◦ g )( A )) for a large class of Bernstein functions q. Theorem 3.10.
Suppose q is a complete Bernstein function such that (3.14) Z ∞ q ( τ ) τ dτ < ∞ , lim s → q ( s ) = 0 . If x ∈ X satisfies (3.15) k C t ( A ) x k = O (cid:18) g (1 /t ) (cid:19) , t → ∞ , then x ∈ dom(( q ◦ g )( A )) . In particular, if (3.15) holds, then for any α ∈ (0 , onehas x ∈ dom([ g α ]( A )) . Proof.
Since lim s →∞ g ( s ) = 0 , from our assumption on q it follows that(3.16) lim s →∞ q ( g ( s )) = 0 . Furthermore q ( t ) /t is a Stieltjes function, hence it decreases on (0 , + ∞ ) and lim t →∞ q ( t ) /t exists and finite. By (3.14), we have lim t →∞ q ( t ) /t = 0 . Moreover, g (1 /t ) = t Z ∞ µ ( ds )1 + ts ≤ t Z ∞ µ ( ds )1 + s , t ≥ . Thus since q is increasing we obtain(3.17) q ( g (1 /t )) ≤ q ( d ( µ ) t ) , t ≥ , d ( µ ) := Z ∞ µ ( ds )1 + s . Therefore,(3.18) lim s → sq ( g ( s )) ≤ lim t →∞ q ( d ( µ ) t ) t = 0 . Using (3.16) and (3.18) we infer that the Stieltjes function q ◦ g has the represen-tation (0 , , ν ) . Next we apply Theorem 2.8 to x and ( q ◦ g )( A ) . Using the hypothesis on thedecay of k C t ( A ) x k and the estimate (3.8) from the proof of Theorem 3.4, we obtain Z k ( A + s ) − x k ν ( ds ) ≤ M k x k Z ν ( ds )+ 4 c Z ∞ g (1 /t ) ( q ( g (1 /t )) ′ dt, for some c > . Furthermore Z ∞ q (1 /t ) ( q ( g (1 /t )) ′ dt = Z ∞ q ′ ( g (1 /t )) g (1 /t ) dg (1 /t ) = Z ∞ τ q ′ ( τ ) τ dτ = − q ( g (1)) g (1) + Z ∞ τ q ( τ ) τ dτ ≤ Z ∞ g (1) q ( τ ) τ dτ. Thus finally Z k ( A + s ) − x k ν ( ds ) ≤ M k x k Z ν ( ds ) + 4 c Z ∞ g (1) q ( τ ) τ dτ < ∞ , and then x ∈ dom (( q ◦ g )( A )) . The last statement follows from the fact that z α , α ∈ (0 , , is a complete Bernstein function satisfying (3.14). (cid:3) Note that if g ∼ (0 , , µ ) is a Stieltjes function then(3.19) g ( t ) = Z ∞ e − ts m ( s ) ds, m ( s ) = Z ∞ e − sτ µ ( dτ ) , t, s > , where m is integrable in the neighborhood of zero and lim t →∞ m ( t ) = 0 . Using (3.19) and our direct mean ergodic theorems with rates, we prove below acharacterization of dom ( g ( A )) in terms of C t ( A ) which complements [22, Corollaire,p. 214-215]. It involves certain means of C t ( A ) thus avoiding a need of adding“extra rate” or “extra range”. It is however less explicit than theorems above. Proposition 3.11.
An element x belongs to dom( g ( A )) if and only if(i) lim t →∞ tm ( t )C t ( A ) x = 0; (ii) lim t →∞ Z t sm ′ ( s )C s ( A ) x ds exists.Proof. By [22, Corollaire, p. 214-215] x ∈ dom( g ( A )) if and only if(3.20) lim t →∞ Z t m ( s ) T ( s ) x ds exists . Since for any x ∈ X and t > Z t m ( s ) T ( s ) x ds = tm ( t )C t ( A ) x − Z t sm ′ ( s )C s ( A ) x ds, it suffices to show that (3.20) implies (i) and (ii). Using (3.19) and e − s ≤ / (1 + s ), s ≥ , we infer that(3.22) sm ( s ) ≤ g (1 /s ) , s > . HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 15
Since by assumption ran( A ) is dense, [16, Theorem 3.4 and Proposition 4.2] implythat g (1 /t )C t ( A ) x → , t → ∞ , and then (3.22) implies (i) . If (i) holds, then from(3.21) it follows that (ii) holds as well. (cid:3) If g ( z ) = z − α , α ∈ (0 , Theorem 3.12. If α ∈ (0 , then x ∈ dom( A − α ) if and only if (3.23) lim t →∞ Z t s α − C s ( A ) x ds exists . Proof.
Since z − α = 1Γ( α ) Z ∞ s α − e − sz ds, z > , we have m ( s ) = α ) s α − , s >
0, and by Proposition 3.11 it suffices to prove that(3.23) implies(3.24) t α C t ( A ) x → , t → ∞ . To this aim note that if (3.23) holds, then Z tt s α − C s ( A ) x ds = Z tt s α − ds Z t T ( τ ) x dτ + T ( t ) Z t ( r + t ) α − Z r T ( τ ) x dτ dr = (1 − α − )1 − α t α C t ( A ) x + T ( t ) Z t ( r + t ) α − r C r ( A ) xdr, where the last sum goes to zero as t → ∞ . Thus to prove (3.24) it suffices to showthat lim t →∞ Z t ( r + t ) α − r C r ( A ) x dr = 0 . Setting G t ( r ) := r − α ( r + t ) − α , ( t > R ( r ) := Z ∞ r s α − C s ( A ) x ds, r ≥ , where the second function is well-defined, continuously differentiable and boundedon [0 , ∞ ) by our assumption, write(3.25) Z t ( r + t ) α − r C r ( A ) x dr = − G t ( t ) R ( t ) + Z t G ′ t ( r ) R ( r ) dr. We prove that both terms on the right hand side of (3.25) converge to zero as t → ∞ , and thus obtain the statement.First note that for all t > r > < G t ( r ) ≤ , r >
0; lim t →∞ G t ( r ) = 0 . Hence, by our assumption, lim t →∞ k G t ( t ) R ( t ) k = 0 . To prove the convergence tozero of the other term note that G ′ t ( r ) = (2 − α ) (cid:18) rr + t (cid:19) − α t ( r + t )
26 ALEXANDER GOMILKO AND YURI TOMILOV is positive on (0 , ∞ ) for each t > . Let ǫ > b = b ( ǫ ) such that k R ( t ) k ≤ ǫ if t ≥ b. Now using (3.26) we have for largeenough t (cid:13)(cid:13)(cid:13)(cid:13)Z t G ′ t ( r ) R ( r ) dr (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z b G ′ t ( r ) k R ( r ) k dr + Z Tb G ′ t ( r ) dr sup r ≥ b k R ( r ) k≤ sup r ≥ b k R ( r ) k G T ( b ) + ǫG t ( t ) < ǫ, and the statement follows. (cid:3) Optimality of domain conditions
In this section we give a number of results showing that the inverse theoremson rates proved in the previous sections are optimal. The results will illustrate, inparticular, that the implications of the form k C t ( A ) x k = O(1 /g (1 /t )) , t → ∞ ⇒ x ∈ dom( g ( A )) , are far from being true, in general, and direct theorems on rates obtained in [17]cannot be inverted. Thus to get positive statements one has to add either “extrarate” or “extra range” assumptions as it was done above.We start with introducing basic objects for constructing our examples. Let L := L (1 , ∞ ), and define a bounded operator A on L by(4.1) ( Au )( s ) := u ( s ) s , u ∈ L . Note that the − A generates a contraction C -semigroup ( T ( t )) t ≥ given by(4.2) ( T ( t ) u )( s ) := e − t/s u ( s ) , t ≥ . Thus we have in particular(C t ( A ) u )( s ) = 1 t Z t e − τ/s u ( s ) dτ = s (1 − e − t/s ) t u ( s ) , and(4.3) t k C t ( A ) u k L = k w t u k L , w t ( s ) := s (1 − e − t/s ) u ∈ L . A direct application of functional calculi rules reveals that for any Stieltjes function g the operator g ( A ) is of the formdom( g ( A )) = (cid:26) u ∈ L : Z ∞ g (1 /s ) | u ( s ) | ds < ∞ (cid:27) , ( g ( A ) u )( s ) = g (1 /s ) u ( s ) , for a.e. s ≥ . It will be convenient to introduce the following family of norms on L :(4.4) N t ( u ) := 1 t Z t s | u ( s ) | ds + Z ∞ t | u ( s ) | ds, u ∈ L , t ≥ . The norms are equivalent to the original norm on L :(4.5) t − k u k L ≤ N t ( u ) ≤ k u k L , u ∈ L . Moreover, using the inequalities(1 − e − ) τ ≤ − e − τ ≤ τ, τ ∈ (0 , , − e − ≤ − e − τ ≤ , τ ≥ , HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 17 and (4.3), we obtain(4.6) (1 − e − ) N t ( u ) ≤ k C t ( A ) u k L ≤ N t ( u ) , t > , u ∈ L , Let in this section g be a Stieltjes function such that g ∼ (0 , , µ ) and g (0+) = ∞ . First we show that Theorem 3.4 is sharp in the sense that for A defined by(4.1) and for a large class of g the properties (3.6) and x ∈ dom( g ( A )) are in factequivalent. Theorem 4.1.
The condition (4.7) Z ∞ | g ′ (1 /t ) | k C t ( A ) u k L t dt < ∞ holds for all u ∈ dom( g ( A )) if and only if g is integrable on (0 , and there exists c > such that (4.8) 1 t Z t g ( τ ) dτ ≤ cg ( t ) , t ∈ (0 , . Proof.
Assume that (4.7) holds for all u ∈ dom( g ( A )) . Since the operator g ( A ) isclosed, (cid:16) dom( g ( A )) , k·k dom( g ( A )) (cid:17) is a Banach space with the graph norm k u k dom( g ( A )) := k g ( A ) u k L + k u k L = Z ∞ ( g (1 /s ) + 1) | u ( s ) | ds. Consider a linear operator G : dom( g ( A )) L ((1 , ∞ ); L ) ,Gu := | g ′ (1 /t ) | C t ( A ) ut , u ∈ dom( g ( A )) . From our assumption it follows that G is well defined. By a standard argument(after passing to appropriate a.e. convergent subsequences) G is closed. Therefore,by the closed graph theorem and (4.6), it follows that(4.9) Z ∞ | g ′ (1 /t ) | N t ( u ) t dt ≤ c k u k dom( g ( A )) , for some constant c > u ∈ dom( g ( A )).Using (4.4) and Fubini’s theorem we can write down the right hand side of (4.9)as follows: Z ∞ | g ′ (1 /t ) | N t ( u ) t dt = Z ∞ | g ′ (1 /t ) | t (cid:26) t Z t s | u ( s ) | ds + Z ∞ t | u ( s ) | ds (cid:27) dt = Z ∞ | u ( s ) | W g ( s ) ds, where W g ( s ) := − s Z ∞ s g ′ (1 /t ) t dt − Z s g ′ (1 /t ) t dt. Integrating by parts and taking into account that lim τ → τ g ( τ ) = 0 (by thebounded convergence theorem) we infer that g ∈ L (0 ,
1) and for every s ≥ W s ( g ) = − s Z /s g ′ ( τ ) τ dτ − Z /s g ′ ( τ ) dτ = s Z /s g ( τ ) dτ − g (1) . Thus (4.9) is satisfied for all u ∈ dom( g ( A )) if and only if(4.11) Z ∞ W g ( s ) | u ( s ) | ds ≤ c Z ∞ ( g (1 /s ) + 1) | u ( s ) | ds, u ∈ dom( g ( A )) , where W s ( g ) is given by (4.10). In turn, (4.9) is equivalent to(4.12) W g ( s ) ≤ c ( g (1 /s ) + 1) , s ≥ . Indeed, it suffices to note that dom( g ( A )) contains all integrable functions withcompact support. Writing down (4.11) for u = 1 ( a,b ) , 1 ≤ a < b < ∞ , we obtainthat Z ba W g ( s ) ds ≤ c Z ba ( g (1 /s ) + 1) ds. Hence F ( s ) := Z s [ c ( g (1 /t ) + 1) − W g ( t )] dt, s ≥ , is an increasing function, and then F ′ ( s ) = c ( g (1 /s ) + 1) − W g ( s ) ≥ s ≥ s Z /s g ( τ ) dτ − g (1) ≤ c ( g (1 /s ) + 1) , s ≥ , that is 1 t Z t g ( τ ) dτ ≤ cg ( t ) + c + g (1) , t ∈ (0 , . Since the function g is decreasing on (0 , ∞ ) , the last inequality is equivalent to (4.8)(with in general a new constant c > (cid:3) Remark . A natural question is what are the functions g satisfying (4.8). Toshow that the class of such functions is quite large we note that if(4.13) τ α g ( τ ) is increasing on (0 , α ∈ (0 , g could be a power function. Indeed,if (4.13) holds, then(4.14) Z t g ( τ ) dτ ≤ t α g ( t ) Z t dττ α = t − α g ( t ) , t ∈ (0 , . On the other hand there are Stieltjes functions g ∼ (0 , , µ ) with g (0+) = ∞ forwhich (4.8) is not true. For instance, if g ( z ) := z − z log z then g L (0 , . Now we prove that Corollary 3.6 is optimal.
Theorem 4.3.
The condition (4.15) Z ∞ g (1 /t ) k C t ( A ) u k L t dt < ∞ holds for all u ∈ dom( g ( A )) if and only if g is integrable on (0 , and there exists c > such that (4.16) 1 t Z t g ( τ ) dτ + Z t g ( τ ) τ dτ ≤ cg ( t ) , t ∈ (0 , . HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 19
Proof.
The proof is similar to the proof of Theorem 4.1 and will only be sketched.Note that Z ∞ g (1 /t ) N t ( u ) t dt = Z ∞ g (1 /t ) | t (cid:26) t Z t s | u ( s ) | ds + Z ∞ t | u ( s ) | ds (cid:27) dt = Z ∞ | u ( s ) | V s ( g ) ds, where V s ( g ) := s Z /s g ( τ ) dτ + Z /s g ( τ ) τ dτ. Then by the argument analogous to that from the proof of Theorem 4.1, (4.15) isequivalent to the inequality(4.17) V s ( g ) ≤ c ( g (1 /s ) + 1) , s ≥ , for some constant c > , which in turn is equivalent to (4.16). (cid:3) Remark . Observe that if there exist α, β ∈ (0 ,
1) such that 0 < β < α < τ α g ( τ ) is increasing on (0 , , τ β g ( τ ) is decreasing on (0 , , then (4.16) holds. Indeed, in view of (4.14) it suffices to note that Z t g ( τ ) τ dτ ≤ t β g ( t ) Z t dττ β ≤ g ( t ) β , t ∈ (0 , . Example 4.5.
Observe that the Stieltjes function g ( z ) = ( z − − log z satisfies(4.8) but it does not satisfy (4.16). Indeed, if t ∈ (0 , /
2) then we have1 t Z t log τ dττ − ≤ − t Z t log τ dτ ≤ tt − g ( t ) . On the other hand, Z t log τ dτ ( τ − τ ≥ − Z t log τ dττ = log t , and (4.16) is violated.Thus if g satisfies the conditions of Theorem 4.3 and u dom( g ( A )) thenthe integral in (4.15) diverges. In this case (4.15) can hardly be written in theform of sup-norm estimates. To circumvent this drawback, we use the notion ofslowly varying function. Recall that (see [31, Chapter 1]) a measurable function ǫ : ( a, ∞ ) (0 , ∞ ), a ≥
0, is called slowly varying (at infinity) if for all λ > t → + ∞ ǫ ( λt ) /ǫ ( t ) = 1 . We proceed with a result which in a sense complementsTheorem 4.3.
Theorem 4.6.
Assume that (4.13) holds. Let ǫ be a slowly varying on ( g (1) , ∞ ) function. Then the function y ( s ) := − g ′ (1 /s ) s g (1 /s ) ǫ ( g (1 /s )) , s > , is positive, belongs to L , and satisfies (4.19) N t ( y ) = O (cid:18) g (1 /t ) ǫ ( g (1 /t )) (cid:19) , t → ∞ . Moreover, y ∈ dom( g ( A )) if and only if (4.20) Z ∞ g (1) dττ ǫ ( τ ) < ∞ . Proof.
Since g (0+) = ∞ , by the change of variable τ = g (1 /s ) , τ ∈ ( g (1 /t ) , ∞ ), weobtain for any t ≥ Z ∞ t y ( s ) ds = − Z ∞ t g ′ (1 /s ) dss g (1 /s ) ǫ ( g (1 /s )) = Z ∞ g (1 /t ) dττ ǫ ( τ )(4.21) ≤ ǫ ( g (1 /t ) Z ∞ g (1 /t ) dττ = 1 g (1 /t ) ǫ ( g (1 /t )) , hence y ∈ L . Similarly, Z ∞ g (1 /s ) y ( s ) ds = − Z ∞ g ′ (1 /s ) dss g (1 /s ) ǫ ( g (1 /s )) = Z ∞ g (1) dττ ǫ ( τ ) , and, in view of the description of dom( g ( A )) in (4.4), the statement follows.It remains to prove that y satisfies (4.19). Observe that by means of (4.4) and(4.21) one can rewrite (4.19) as(4.22) 1 t Z t sy ( s ) ds ≤ cg (1 /t ) q ( g (1 /t )) , t ≥ , for some constant c > . By [31, Section 1.5], for any δ > t − δ ǫ ( t )is equivalent as t → ∞ to a positive function decreasing on ( g (1) , ∞ ), and thefunction t δ ǫ ( t ) is equivalent as t → ∞ to a positive function increasing on ( g (1) , ∞ ) . Therefore, since g (0+) = ∞ and g is decreasing, for any δ > , g − δ ( τ ) ǫ ( g ( τ ))is equivalent to a function increasing on (1 , ∞ ) . Choose now positive δ such that β := (1 + δ ) α ∈ (0 , , where α is defined in (4.13). Then τ β g ( τ ) ǫ ( g ( τ )) = ( τ α g ( τ )) δ g − δ ( τ ) ǫ ( g ( τ ))is equivalent to a function increasing on (0 , . In other words, the function g (1 /s ) ǫ ( g (1 /s ))is equivalent to a measurable function ψ such that s β /ψ ( s ) is increasing on (1 , ∞ ) . Hencesince τ | g ′ ( τ ) | ≤ g ( τ ) , τ > , by (3.11), we obtain for every t ≥ t Z t sy ( s ) ds = 1 t Z t | g ′ (1 /s ) | dssg (1 /s ) ǫ ( g (1 /s )) ≤ t Z t dsg (1 /s ) ǫ ( g (1 /s )) ≤ ct Z t s β dss β ψ ( s ) ≤ ct t β ψ ( t ) Z t dss β ≤ C t β tg (1 /t ) ǫ ( g (1 /t )) Z t dss β ≤ C − β ) g (1 /t ) ǫ ( g (1 /t )) , where c, C are positive constants, thus the proof is complete. (cid:3) Theorem 4.6 and (4.6) imply the following statement (cf. Corollary 3.5).
Corollary 4.7.
Assume that the functions g and ǫ satisfy the conditions of Theorem4.6. If Z ∞ g (1) dττ ǫ ( τ ) = ∞ , HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 21 then there exists y ∈ L such that (4.23) k C t ( A ) y k L = O (cid:18) g (1 /t ) ǫ ( g (1 /t )) (cid:19) , t → ∞ , but y dom( g ( A )) .Remark . By [17, Theorem 4.6] if ǫ : (0 , ∞ ) → (0 , ∞ ) is an increasing functionsuch that lim t →∞ ǫ ( t ) = ∞ , then there exists x ∈ dom( g ( A )) such thatsup t ≥ g (1 /t ) ǫ ( g (1 /t )) k C t ( A ) x k L = ∞ . Thus conditions like (4.23) cannot hold for all elements from the correspondingdomain.
Example 4.9.
Observe that g ( z ) = z − γ , γ ∈ (0 , , is a Stieltjes function satisfying(4.13) for α ∈ ( γ, . Therefore, g satisfies also (4.8). (The latter fact can also bechecked directly.) Since(4.24) ǫ ( s ) := log( s + 2) log(log( s + 3)) , s ≥ , is slowly varying on (0 , ∞ ) , the functions g and ǫ satisfy the conditions of Theorem4.1. Hence by Corollary 4.7 there exists y ∈ L , y dom( A − γ ) , such that k C t ( A ) y k L = O (cid:18) t γ log( t ) log(log t ) (cid:19) , t → ∞ . Example 4.10.
Note that the Stieltjes function g ( z ) = log(1 + z − ) and thefunction ǫ defined by (4.24) satisfy the conditions of Theorem 4.6 (since g satisfies(4.13) for any α ∈ (0 , y ∈ L , y dom(log( I + A − )) , such that k C t ( A ) y k L = O (cid:18) t [log(log t ) log(log(log t ))] (cid:19) , t → ∞ . Appendix
Recall that if ( T ( t )) t ≥ is a bounded C -semigroup on X then for each x ∈ X \{ } the Ces´aro means C t ( A ) x cannot decay faster than 1 /t as t → ∞ . The propositionbelow shows that it is not possible to ‘improve’ this extremal rate of decay of C t ( A ) x by requiring the smallness of C t ( A ) x in an integral sense. Proposition 5.1.
Let ( T ( t )) t ≥ be a bounded C -semigroup on a Banach space X with generator − A . If for x ∈ X there exists { t k : k ≥ } ⊂ (0 , ∞ ) , t k → ∞ , k → ∞ , such that (5.1) weak − lim k →∞ t k Z t k s C s ( A ) x ds = 0 , then x = 0 . In particular, if weak − lim t →∞ t C t ( A ) x = 0 , then x = 0 . Proof.
Since tA ( I + A ) − C t ( A ) x = ( I − T ( t ))( I + A ) − x, t > , we have[ A ( I + A ) − ] t Z t s C s ( A ) x ds = A ( I + A ) − x − ( I − T ( t )) t ( I + A ) − x. As the operator A ( I + A ) − is bounded, the latter equality and (5.1) imply that A ( I + A ) − x = 0 and then x ∈ ker A . But if x ∈ ker A then1 t Z t s C s ( A ) x ds = 1 t Z t s ds x = t x, and, using (5.1) once again, we conclude that x = 0. (cid:3) Theorem 5.2.
Let ( T ( t )) t ≥ be a bounded C -semigroup on a Banach space X with generator − A . Let ϕ be a positive, increasing on [1 , ∞ ) function such that (5.2) Z ∞ dttϕ ( t ) = ∞ . If x ∈ X satisfies (5.3) Z ∞ k C t ( A ) x k ϕ ( t ) dt < ∞ , then x = 0 . In particular, if Z ∞ k C t ( A ) x k log(1 + t ) dt < ∞ , then x = 0 . Proof.
Define Θ( t ) := Z t s k C s ( A ) x k ds, t ≥ . If s >
1, then Z s k C t ( A ) x k ϕ ( t ) dt = Z s tϕ ( t ) d Θ( t )= Θ( s ) sϕ ( s ) − Θ(1) ϕ (1) − Z s Θ( t ) d (cid:18) tϕ ( t ) (cid:19) = Θ( s ) sϕ ( s ) − Θ(1) ϕ (1) + Z s Θ( t ) t ϕ ( t ) dt − Z s Θ( t ) t d (cid:18) ϕ ( t ) (cid:19) ≥ − Θ(1) ϕ (1) + Z s Θ( t ) t ϕ ( t ) dt. Hence by (5.3) it follows that(5.4) Z ∞ Θ( t ) dtt ϕ ( t ) < ∞ . Therefore by (5.2) and (5.4) we infer that there exists t k → ∞ , k → ∞ , suchthat lim k →∞ Θ( t k ) /t k = 0 . Therefore (5.1) holds and by Proposition 5.1 we have x = 0. (cid:3) HAT DOES A RATE IN A MEAN ERGODIC THEOREM IMPLY ? 23
Remark . Note that if g as in Section 3 and g ∼ (0 , b, µ ) , b > , then g ′ (1 /t ) t − and g (1 /t ) /t are separated from zero on (0 , ∞ ) so that the conditions (3.6) and(3.10) reduce to (5.3) with ϕ ( t ) ≡ x = 0 . Acknowledgements
The authors are grateful to M. Haase and M. Lin for useful remarks and fruitfuldiscussions. They would also like to thank M. Lin for sending them the unpublishedmanuscript [5].
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Results in Math. (1998), 381–394. Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul.Chopina 12/18, 87-100 Toru´n, Poland
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