6 Subnormality of unbounded composition operators over 1 one-circuit directed graphs: exotic examples 0 2 b Piotr Budzyn´ski, Zenon Jan Jabl(cid:32)on´ski, Il Bong Jung, and Jan Stochel e F Abstract. Arecentexampleofanon-hyponormalinjectivecompositionop- 2 eratorinanL2-spacegeneratingStieltjesmomentsequences,inventedbythree 2 of the present authors, was built over a non-locally finite directed tree. The main goal of this paper is to solve the problem of whether there exists such ] anoperatoroveralocallyfinitedirectedgraphand,intheaffirmativecase,to A findthesimplestpossiblegraphwiththeseproperties(simplicityreferstolo- F calvalency). Theproblemissolvedaffirmativelyforthelocallyfinitedirected h. graph G2,0, which consists of two branches and one loop. The only simpler directedgraphforwhichtheproblemremainsunsolvedconsistsofonebranch t a and one loop. The consistency condition, the only efficient tool for verifying m subnormalityofunboundedcompositionoperators,isintensivelystudiedinthe [ contextofG2,0,whichleadstoaconstructivemethodofsolvingtheproblem. ThemethoditselfispartlybasedontransformingtheKreinandtheFriedrichs 2 measures coming either from shifted Al-Salam-Carlitz q-polynomials or from v aquarticbirthanddeathprocess. 1 6 2 6 0 . 1 0 6 1 : v i X r a 2010 Mathematics Subject Classification. Primary47B33,47B20;Secondary47B37,44A60. Key words and phrases. Subnormal operator, operator generating Stieltjes moment se- quences, composition operator, consistency condition, graphs induced by self-maps, Hamburger andStieltjesmomentsequences. The research of the first author was supported by the Ministry of Science and Higher Edu- cation of the Republic of Poland. The research of the second and fourth authors was supported bytheNCN(NationalScienceCenter), decisionNo. DEC-2013/11/B/ST1/03613. Theresearch ofthethirdauthorwassupportedbyBasicScienceResearchProgramthroughtheNationalRe- search Foundation of Korea(NRF) funded by the Ministry of Science, ICT and future Planning (KRF-2015R1A2A2A01006072). 1 2 P.BUDZYN´SKI,Z.J.JABL(cid:32)ON´SKI,I.B.JUNG,ANDJ.STOCHEL Contents 1. Preliminaries 2 1.1. Introduction 2 1.2. Notation and terminology 6 2. Determinacy in moment problems 7 2.1. Basic concepts 7 2.2. Transforming moments via homotheties 9 2.3. The Al-Salam-Carlitz moment problem 13 2.4. Index of H-determinacy 16 2.5. The Carleman condition 17 3. Composition operators over one-circuit directed graphs 17 3.1. Criteria for hyponormality and subnormality 17 3.2. A class of directed graphs with one circuit 20 3.3. Injectivity problem 24 3.4. The Radon-Nikodym derivatives 28 4. Subnormality of C via the consistency condition (CC) 30 φη,κ 4.1. Characterizations of (CC) 30 4.2. Modelling subnormality via (CC) 32 4.3. Criteria for subnormality related to x 34 0 4.4. Extending to families satisfying (CC) 38 5. Examples of exotic non-hyponormal operators 42 5.1. Outline 42 5.2. General scheme 43 5.3. Three key lemmata 44 5.4. The gap between the conditions (i-d) and (i-d(cid:48)) 49 5.5. Exotic non-hyponormality 51 5.6. Addendum 55 Acknowledgements 58 References 58 1. Preliminaries 1.1. Introduction. The theory of bounded subnormal operators was initi- ated by Halmos (cf. [32]). The definition and the first characterization of their unboundedcounterpartsweregivenindependentlybyBishop(cf.[9])andFoia¸s(cf. [29]). The foundations of the theory of unbounded (i.e., not necessarily bounded) subnormal operators were developed by the fourth-named author and Szafraniec (cf.[62, 63, 64, 65]). Thestudyofthistopicturnedouttobehighlysuccessful. It led to a number of challenging problems and nontrivial results in various branches of mathematics including functional analysis and mathematical physics (see, e.g., [23, 24, 25] for the case of bounded operators and [50, 43, 26, 45, 46, 47, 44] forunboundedones). Thisareaofintereststillplaysavitalroleinoperatortheory. The first characterization of bounded subnormal operators was given by Hal- moshimself. ItwassuccessivelysimplifiedbyBram(cf.[10]),Embry(cf.[28])and Lambert (see [48]; see also [63, Theorem 7] where the assumption of injectivity was removed). The Lambert characterization states that a bounded Hilbert space operator is subnormal if and only if it generates Stieltjes moment sequences (see SUBNORMALITY OF COMPOSITION OPERATORS: EXOTIC EXAMPLES 3 Section 1.2 for definitions). It turns out that this characterization also works for unbounded operators which have sufficiently many analytic vectors (cf. [63, Theo- rem 7]). However, it is no longer true for arbitrary unbounded operators (cf. [13, Section3.2]). RecallthatsubnormaloperatorswithdensesetofC∞-vectorsalways generate Stieltjes moment sequences (see [15, Proposition 3.2.1]). It is also worth pointing out that subnormal composition operators in L2-spaces, as opposed to abstractsubnormaloperators,arealwaysinjective(see[15,Corollary6.3]). Hence, there arises the question whether or not composition operators in L2-spaces gener- ating Stieltjes moment sequences are injective (see Problem 3.3.6). In a recent paper [16], we have developed a completely new, even in the boundedcase, approachtostudyingsubnormalityofcompositionoperators(inL2- spacesoverσ-finitemeasurespaces)whichinvolvesmeasurablefamiliesofprobabil- itymeasuressatisfyingtheso-calledconsistencycondition. Thisapproachprovides a criterion (read: sufficient condition) for subnormality of composition operators, which does not refer to the density of domains of powers. The corresponding tech- niqueforweightedshiftsondirectedtreesworkedoutin[12](seealso[18])enabled ustoconstructanunexpectedexampleofasubnormalcompositionoperatorwhose square has trivial domain (cf. [17]). As shown in [42, Example 4.2.1], there are unbounded injective operators gen- erating Stieltjes moment sequences which are not even hyponormal, and thus not subnormal. In fact, it was proved there that if T is a leafless directed tree which has exactly one branching vertex and if the branching vertex itself has infinite va- lency1, then there exists a non-hyponormal (injective) weighted shift on T with nonzero weights generating Stieltjes moment sequences. Up to isomorphism, there is only one rootless directed tree of this kind, denoted in [41, p. 67] by T . A ∞,∞ weightedshiftonT withnonzeroweightsisunitarilyequivalenttoaninjective ∞,∞ compositionoperatorinanL2-spaceoveradiscretemeasurespace(see[42,Lemma 4.3.1] and [41, Theorem 3.2.1]). Since the directed graph induced by the symbol of such a composition operator coincides with T (see Section 3.2 for the defi- ∞,∞ nition), it is not locally finite. This raises the question as to whether there exists a non-hyponormal injective composition operator over2 a locally finite connected directed graph generating Stieltjes moment sequences, and, if this is the case, how simple such a directed graph can be, where simplicity is understood with respect to local valency (cf. Remark 3.2.2). The present paper addresses both of these questions. Taking into account the simplicity leads to considering directed graphs induced by self-maps whose vertices, all but one, say ω, have valency one, and the valency of ω is greater than or equal to 1. Such directed graphs are described in Theorem 3.2.1 and Remark 3.2.2. In view of part 1) of Remark 3.2.2, the situation in which the valency of ω is equal to one is excluded by an unbounded variant of Herrero’scharacterization3ofsubnormalinjectivebilateralweightedshifts(see[63, Theorem 5]; see also [14, Theorem 3.2] for a recent approach). If the valency of ω is strictly greater than one, then, by Theorem 3.2.1, we have two cases. The first, which is described in Theorem 3.2.1(ii-b), reduces to the directed tree T , ∞,∞ the case studied in [42]. Unfortunately, the method invented in [42] does not give 1Thevalencyofavertexv isunderstoodasthenumberofoutgoingedgesatv. 2AcompositionoperatorC isoveradirectedgraphG ifG isinducedbythesymbolofC. 3TheHerreroresult(see[36];seealso[27])isabilateralanalogueoftheBerger-Gellar-Wallen characterizationofboundedsubnormalinjectiveunilateralweightedshifts(cf.[33, 31]). 4 P.BUDZYN´SKI,Z.J.JABL(cid:32)ON´SKI,I.B.JUNG,ANDJ.STOCHEL any hope of answering our questions. In the second case, which is described in Theorem 3.2.1(ii-a), the directed graph under consideration has exactly one circuit of length κ+1 starting at ω and η branches of infinite length attached to ω, where η ∈ {1,2,3,...}∪{∞} and κ ∈ {0,1,2,...} (see Figure 2 with ω = x ); denote it κ by G . The culminating result of the present paper, Theorem 5.5.2, shows that η,κ thereexistsanon-hyponormalinjectivecompositionoperatoroverthelocallyfinite directed graph G generating Stieltjes moment sequences. This answers our first 2,0 question in the affirmative. Regarding simplicity, the only simpler directed graph which potentially may admit a composition operator with the above-mentioned properties is G (the subnormality over G was studied in [16, Section 3.4]). 1,0 1,0 However, so far this particular case remains unsolved because composition opera- tors over G obtained by our method are automatically subnormal (cf. Theorem 1,0 5.4.2(iv)). A large part of the present paper is devoted to the study of subnormality of compositionoperatorsoverthedirectedgraphG . Theyallhavethesamesymbol η,κ φ whereas masses attached to vertices that define the underlying L2-space are η,κ subjecttochanges. Sincetheonlyknowncriterion4forsubnormalityofunbounded composition operators relies on the consistency condition (CC) (cf. [16, Theorem 9]), we first characterize families of Borel probability measures (on the positive half-line) indexed by the vertices of G which satisfy (CC) (cf. Theorem 4.1.1). η,κ This enables us to model all such families via collections of measures indexed by the set {x : i ∈ J } which satisfy some natural conditions (cf. Procedure 4.2.1), i,1 η where {x : i ∈ J } are ends of edges outgoing from ω = x not lying on the i,1 η κ circuit (see Figure 2) and J is the set of all positive integers less than or equal η to η. The end x of the edge that outgoes from x and lies on the circuit also 0 κ plays an important role in our considerations. Namely, assuming both that the Radon-Nikodym derivatives {h }∞ (see Section 3.1) calculated at x and x , φn n=0 0 i,1 i ∈ J , form Stieltjes moment sequences and that appropriate sequences coming η from {h (x )}∞ are S-determinate, we show that the corresponding composi- φn 0 n=0 tion operator over G is subnormal (cf. Theorem 4.3.3). The case of G does η,κ 1,κ not require any determinacy assumption and may be written purely in terms of the Hankel matrices [h (x )]∞ and [h (x )]∞ (cf. Proposition 4.3.4). φi+j 0 i,j=0 φi+j+1 0 i,j=0 The proofs of Theorem 4.3.3 and Proposition 4.3.4 rely on constructing families of measures satisfying (CC). These two results are in the spirit of Lambert’s char- acterization of subnormality of bounded composition operators (cf. [49]) which is no longer true for unbounded operators (cf. [42, Theorem 4.3.3] and [15, Section 11]). The case of bounded composition operators over G , which is also covered η,κ by Lambert’s criterion, follows easily from Theorem 4.3.3 (cf. Proposition 4.3.6). The optimality of the assumptions of Propositions 4.3.4 and 4.3.6 is illustrated by Examples 4.3.5 and 4.3.7. Itfollowsfrom[16,Theorems9and17](seealsoTheorem3.1.3)thatunderthe assumptionthath takesfinitevaluesforallpositiveintegersn,anyfamilyofBorel φn probability measures satisfying (CC) consists of representing measures of Stieltjes momentsequences{h (x)}∞ ,wherexvariesovertheverticesofG . InSection φn n=0 η,κ 4.4 we discuss the question of extending a given family {P(x ,·)} of Borel i,1 i∈Jη probability measures to a wider one (indexed by G ) satisfying the consistency η,κ condition (CC). According to Theorem 4.4.1, such extension exists if and only if 4Generalcriteriain[9, 29, 67, 68]seemhardlytobeapplicabletocompositionoperators. SUBNORMALITY OF COMPOSITION OPERATORS: EXOTIC EXAMPLES 5 for every i ∈ J , {h (x )}∞ is a Stieltjes moment sequence represented by a η φn i,1 n=0 measureP(x ,·)satisfyingtheconditions(i-b),(i-c)and(i-d)ofthistheorem. The i,1 condition (i-b) refers to moments of the measures P(x ,·), i∈J . The remaining i,1 η twoareofdifferentnature,namely(i-c)isasystemofκequations(thecaseofκ=0 is not excluded), while (i-d) is a single inequality. In Theorem 4.4.2 we introduce the condition (i-d(cid:48)) which is a weaker version of (i-d). This turns out to be the key ideathatleadstoconstructingexoticexamples. AssumingtheS-determinacyofthe sequence {h (x )+c}∞ for any c ∈ (0,∞), it is proved in Theorem 4.4.2 that φn κ n=0 the conditions (i-d) and (i-d(cid:48)) are equivalent (provided the remaining ones (i-a), (i-b) and (i-c) are satisfied). However, this is no longer true if the S-determinacy assumptionisdropped. WeshowthisbyusingProcedure5.2.1thatheavilydepends on the existence of a pair of N-extremal measures satisfying some constraints (cf. Lemma 5.3.1). The task of finding such a pair is challenging. It is realized by transformingviaspecialhomothetiestheKreinandtheFriedrichsmeasures(which are particular instances of N-extremal measures). The crucial properties of these transformations are described in Lemma 5.3.2. The proof of the existence of the gap between (i-d) and (i-d(cid:48)) is brought to completion in Theorems 5.4.1 (the case of η (cid:62)2) and 5.4.2 (the case of η =1). Adapting the above technique, we show in Theorem 5.5.2 that for any integer η (cid:62) 2, there exists a non-hyponormal injective composition operator over G which generates Stieltjes moment sequences. The η,0 case of η = ∞ is treated in Theorem 5.5.1. The parallel question of determinacy of moment sequences {h (x)}∞ , x ∈ {x }∪{x : i ∈ J , j ∈ N}, is studied in φn n=0 κ i,j η Section 5 by using the index of H-determinacy introduced by Berg and Duran in [5]. As noted above, the proofs of the main results of the present paper (Theorems 5.4.1, 5.4.2, 5.5.1 and 5.5.2) essentially depend on subtle properties of N-extremal measures. The question of determinacy of moment sequences is of considerable importanceinourstudyaswell. Therefore,forthesakeofcompleteness,wecollect in Section 2 basic concepts of the classical theory of moments and include some new results in this field. Using [5, Theorem 3.6], we show that a measure which comes from an N-extremal measure by removing an infinite number of its atoms has infinite index of H-determinacy (cf. Theorem 2.4.1). The Carleman condi- tion, which always guarantees the H-determinacy of Stieltjes moment sequences, is investigated in Section 2.5. The process of transforming moment sequences and their representing measures, including N-extremal ones, via homotheties is described in Section 2.2 (the particular case of transformations induced by trans- lations has already been studied via different approaches in [52, 57]). Particular attention is paid to transforming the Krein and the Friedrichs measures (cf. Theo- rem 2.2.3). As a consequence, a new way of parametrizing N-extremal measures of H-indeterminateStieltjesmomentsequencesisinvented(cf.Theorem2.2.5)andan unexpected trichotomy property of N-extremal measures of H-indeterminate Ham- burger moment sequences is proven (cf. Theorem 2.2.6). The N-extremal measures used in the proofs of Theorems 5.4.1, 5.4.2 and 5.5.1 are derived from the Krein and the Friedrichs measures of an S-indeterminate Stieltjes moment sequence, first by scaling them and then by transforming them via carefully chosen homotheties (cf. Lemma 5.3.2). As for the proof of Theorem 5.5.2, the above method requires the usage of the Krein and the Friedrichs measures coming from shifted Al-Salam- Carlitz q-polynomials (or, alternatively, from a quartic birth and death process, 6 P.BUDZYN´SKI,Z.J.JABL(cid:32)ON´SKI,I.B.JUNG,ANDJ.STOCHEL see Remark 5.5.3). The existence, determinacy and explicit form of orthogonaliz- ing measures for Al-Salam-Carlitz q-polynomials {V(a)(x;q)}∞ are discussed in n n=0 Section 2.3. It is worth mentioning that explicit examples of N-extremal measures such as those used in the present paper are to the best of our knowledge very rare (see, e.g., [39, 38]). The necessary facts concerning composition operators in L2-spaces over dis- crete measure spaces, including criteria for their hyponormality and subnormality, are recapitulated in Section 3.1. A variety of relations between Radon-Nikodym derivatives {h }∞ calculated in different vertices of the directed graph G are φn n=0 η,κ established in Section 3.4. 1.2. Notationandterminology. DenotebyC,R,R ,Z,Z andNthesets + + ofcomplexnumbers,realnumbers,nonnegativerealnumbers,integers,nonnegative integers and positive integers, respectively. Set R =R ∪{∞} and N =N\{1}. + + 2 Given k ∈Z ∪{∞}, we write J ={i∈N: i(cid:54)k} (clearly J =∅). The identity + k 0 map on a set X is denoted by id . We write card(X) for the cardinality of a set X (cid:70) X and χ for the characteristic function of a subset ∆ of X. The symbol “ ” ∆ denotes the disjoint union of sets. A mapping from X to X is called a self-map of X. The σ-algebra of all Borel subsets of a topological space X is denoted by B(X). All measures considered in this paper are positive. Since any finite Borel measure ν on R is automatically regular (cf. [55, Theorem 2.18]), we can consider its closed support; we denote it by supp(ν). Given t∈R, we write δ for the Borel t probability measure on R such that supp(δ ) = {t}. In this paper we will use the t notation (cid:82)b and (cid:82)∞ in place of (cid:82) and (cid:82) respectively (a,b ∈ R). We also a a [a,b] [a,∞) use the convention that 00 = 1. The ring of all complex polynomials in one real variable t (which in the context of L2-spaces are regarded as equivalence classes) is denoted by C[t]. Let A be an operator in a complex Hilbert space H (all operators considered in this paper are linear). Denote by D(A) the domain of A. If A is closable, then the closure of A is denoted by A¯. Set D∞(A) = (cid:84)∞ D(An) with A0 = I, where n=0 I is the identity operator on H. We say that A is positive if (cid:104)Af,f(cid:105) (cid:62) 0 for all f ∈ D(A). A is said to be normal if it is densely defined, D(A) = D(A∗) and (cid:107)A∗f(cid:107)=(cid:107)Af(cid:107) for all f ∈D(A) (or, equivalently, if and only if A is closed, densely defined and A∗A = AA∗, see [69, Proposition, page 125]). A is called subnormal if A is densely defined and there exists a normal operator N in a complex Hilbert spaceKwithH⊆K(isometricembedding)suchthatD(A)⊆D(N)andAf =Nf forallf ∈D(A). Aissaidtobehyponormalifitisdenselydefined,D(A)⊆D(A∗) and (cid:107)A∗f(cid:107) (cid:54) (cid:107)Af(cid:107) for all f ∈ D(A). Following [42], we say that A generates Stieltjes moment sequences if D∞(A) is dense in H and {(cid:107)Anf(cid:107)2}∞ is a Stieltjes n=0 momentsequenceforeveryf ∈D∞(A)(seeSection2.1belowforthedefinitionand basic properties of Stieltjes moment sequences). It is known that if A is subnormal and D∞(A) is dense in H, then A generates Stieltjes moment sequences (cf. [13, Proposition 3.2.1]). However, the reverse implication is not true in general (see [59]; see also [42]). In what follows B(H) stands for the C∗-algebra of all bounded operators in H whose domains are equal to H. SUBNORMALITY OF COMPOSITION OPERATORS: EXOTIC EXAMPLES 7 2. Determinacy in moment problems 2.1. Basic concepts. DenotebyM thesetofallBorelmeasuresν onRsuch that (cid:82) |t|ndν(t) < ∞ for all n ∈ Z . Set M+ = {ν ∈ M: supp(ν) ⊆ R }. A R + + sequence γ = {γ }∞ ⊆ R is said to be a Hamburger (resp. Stieltjes) moment n n=0 sequence if there exists ν ∈M (resp. ν ∈M+) such that (cid:90) γ = tndν(t), n∈Z ; n + R thesetofallsuchmeasures,calledH-representing(resp.S-representing)measuresof γ, is denoted by M(γ) (resp. M+(γ)). A Hamburger (resp. Stieltjes) moment se- quenceγ issaidtobeH-determinate(resp.S-determinate)ifcard(M(γ))=1(resp. card(M+(γ)) = 1); otherwise, we call it H-indeterminate (resp. S-indeterminate). Wesaythatameasureν ∈M (resp.ν ∈M+)isH-determinate(resp.S-determinate) if the sequence {(cid:82) tndν(t)}∞ is H-determinate (resp. S-determinate). Simi- R n=0 larly, we define H-indeterminacy and S-indeterminacy of measures. Clearly, an S- indeterminate Stieltjes moment sequence is H-indeterminate. It is well-known that a Hamburger moment sequence which has a compactly supported H-representing measure is H-determinate (cf. [30]). Note that H-determinacy and S-determinacy coincide for Stieltjes moment sequences having S-representing measures vanishing on {0} (see [21, Corollary, p. 481]; see also [42, Lemma 2.2.5]). Let γ = {γ }∞ be a Hamburger moment sequence. A measure ν ∈ M(γ) n n=0 is called an N-extremal measure of γ if γ is H-indeterminate and C[t] is dense in L2(ν). Wesaythatν ∈M isanN-extremalmeasureifν isanN-extremalmeasure of the Hamburger moment sequence {(cid:82) tndν(t)}∞ . Denote by M (γ) the set of R n=0 e all N-extremal measures of γ and put M+(γ)=M (γ)∩M+. e e Note that if γ ={γ }∞ isan H-indeterminateHamburgermoment sequence, n n=0 thencard(M (γ))=c(cf.[57,Theorem4andRemark,p.96]). Moreover,wehave: e Lemma 2.1.1 ([57, Theorems 5 and 4.11]). If γ ={γ }∞ is an H-indetermi- n n=0 nate Hamburger moment sequence, then R = (cid:70) supp(ν), and the closed ν∈Me(γ) support of any ν ∈M (γ) is countably infinite with no accumulation point in R. e Now we state the M. Riesz characterizations of H-determinacy and N-extrem- ality (see [54, p. 223] or [30, Theorem, p. 58]) and the Berg-Thill characterization of S-determinacy (see [6, Theorem 3.8] or [7, Proposition 1.3]). Lemma 2.1.2. (i) A measure ν ∈ M is H-determinate (resp. N-extremal) if and only if C[t] is dense in L2((1+t2)dν(t)) (resp. C[t] is dense in L2(ν) and not dense in L2((1+t2)dν(t))). (ii) A measure ν ∈M+ is S-determinate if and only if C[t] is dense in both L2((1+t)dν(t)) and L2(t(1+t)dν(t)). The above enables us to formulate a comparison test for determinacy. Proposition 2.1.3 (Comparison test). Let ρ and ν be Borel measures on R (resp.R )suchthatν ∈M (resp.ν ∈M+)andρ(σ)(cid:54)Mν(σ)foreveryσ ∈B(R) + (resp.σ ∈B(R ))andforsomeM ∈R . Thenρ∈M (resp.ρ∈M+). Moreover, + + ρ is H-determinate (resp. S-determinate) whenever ν is. Proof. We deal only with the case of H-determinacy; the other case can be treated similarly. The standard measure-theoretic argument implies that ρ ∈ M. Since ρ (cid:54) Mν, we deduce from [55, Theorem 3.13] that L2((1+t2)dν(t)) (cid:51) f (cid:55)→ 8 P.BUDZYN´SKI,Z.J.JABL(cid:32)ON´SKI,I.B.JUNG,ANDJ.STOCHEL f ∈L2((1+t2)dρ(t)) is a well-defined bounded operator with dense range. Hence, applying Lemma 2.1.2 completes the proof. (cid:3) Corollary2.1.4. Supposethat{γ }∞ and{γ }∞ areHamburger(resp. 1,n n=0 2,n n=0 Stieltjes) moment sequences such that {γ + γ }∞ is H-determinate (resp. 1,n 2,n n=0 S-determinate). Then both {γ }∞ and {γ }∞ are H-determinate (resp. S- 1,n n=0 2,n n=0 determinate). Remark 2.1.5. It follows from Corollary 2.1.4 that if {γ }∞ is a Stieltjes n n=0 momentsequencesuchthat{γ +c}∞ isS-determinateforsomec∈(0,∞), then n n=0 {γ }∞ is S-determinate. This may suggest that if {γ }∞ is an S-determinate n n=0 n n=0 Stieltjes moment sequence, then so is {γ + c}∞ for some c ∈ (0,∞). How- n n=0 ever, in general, this is not true. In fact, one can show more; namely there exists an H-determinate Stieltjes moment sequence {γ }∞ such that {γ + c}∞ is n n=0 n n=0 S-indeterminate for all c ∈ (0,∞). Indeed, as noticed by C. Berg (private com- munication), if ν is an N-extremal measure of an S-indeterminate Stieltjes moment sequence such that infsupp(ν)=1 (e.g., the orthogonalizing measure β(a;q) for the Al-Salam-Carlitz q-polynomials {V(a)(x;q)}∞ with 0 < q < a (cid:54) 1 is N-extremal n n=0 and its closed support is equal to {q−n}∞ , see Section 2.3), then the measure n=0 µ := ν −ν({1})δ ∈ M+ is H-determinate and for every c ∈ (0,∞), the measure 1 µ+cδ is N-extremal (see [5, Theorem 3.6 and Lemma 3.7]) and consequently, 1 since infsupp(µ+cδ )>0, it is S-indeterminate (see [21, Corollary, p. 481] or [42, 1 Lemma 2.2.5]). The following lemma will be used in Section 4.3. Lemma 2.1.6. If {γ }∞ ⊆R , then the following conditions are equivalent: n n=0 + (i) {γ }∞ is a Stieltjes moment sequence which has an S-representing mea- n n=0 sure vanishing on [0,1), (ii) 0 (cid:54) (cid:80)n γ λ λ¯ (cid:54) (cid:80)n γ λ λ¯ for all finite sequences {λ }n i,j=0 i+j i j i,j=0 i+j+1 i j i i=0 of complex numbers, (iii) {γ }∞ isaStieltjesmomentsequenceand(cid:80)n (γ −γ )λ λ¯ (cid:62)0 n n=0 i,j=0 i+j+1 i+j i j for all finite sequences {λ }n of complex numbers. i i=0 Proof. (i)⇒(ii) Obvious. (ii)⇔(iii) Apply the Stieltjes theorem (cf. [4, Theorem 6.2.5]). (ii)⇒(i) Let Λ: C[t] → C be a linear functional such that Λ(tn) = γ for all n n∈Z . Takep∈C[t]whichisnonnegativeon[1,∞). Sincep(t+1)isnonnegative + on[0,∞),thereexistq ,q ∈C[t]suchthatp=(t−1)|q |2+|q |2 (cf.[53,Problem 1 2 1 2 45, p. 78]). Hence Λ(p) (cid:62) 0. Applying the Riesz-Haviland theorem (cf. [35]) completes the proof. (cid:3) The question of when M+(γ) is nonempty has the following answer. e Lemma 2.1.7. Suppose γ is an H-indeterminate Stieltjes moment sequence. Then M+(γ)(cid:54)=∅. Moreover, γ is S-determinate if and only if card(M+(γ))=1. e e Proof. Let A be a symmetric operator in a complex Hilbert space H and e ∈ D∞(A) be such that D(A) is the linear span of {Ane: n ∈ Z }, and γ = + n (cid:104)Ane,e(cid:105) for all n ∈ Z (cf. [57, (1.10)]). By assumption, A is a positive operator + with deficiency indices (1,1) (cf. [57, Theorem 2 and Corollary 2.9]). Hence, the Friedrichs extension S of A differs from A¯ (cf. [69, Theorem 5.38]). As a con- sequence, (cid:104)E(·)e,e(cid:105) ∈ M+(γ), where E is the spectral measure of S (cf. [42, p. e SUBNORMALITY OF COMPOSITION OPERATORS: EXOTIC EXAMPLES 9 3951]). Thisalsoprovesnecessityinthe“moreover”part. Thesufficiencyisadirect consequence of [57, Theorem 4]. (cid:3) Recall that if γ = {γ }∞ is an S-indeterminate Stieltjes moment sequence, n n=0 then card(M+(γ))=c and there exist distinct measures α,β ∈M+(γ) (uniquely e e determined by (2.1.1)) such that for every ρ∈M+(γ)\{α,β}, e 0=infsupp(α)<infsupp(ρ)<infsupp(β); (2.1.1) α and β are called the Krein and the Friedrichs measures of γ, respectively. These twoparticularN-extremalmeasurescomefromtheKreinandtheFriedrichsexten- sions of a positive operator attached to γ. The reader is referred to [52] for the case of Friedrichs extensions and to [57] for a complete and up-to-date operator approach to moment problems (see also [42, Section 2]). 2.2. Transforming moments via homotheties. In this section we investi- gate transformations acting on real sequences induced by homotheties of R. Such transformations are shown to preserve many properties of Hamburger and Stieltjes moment sequences. The particular case of transformations induced by translations has been considered in [52, Section 3] and [57, p. 96] (with different approaches). Fix ϑ∈(0,∞) and a∈R. Let us define the self-map ψ of R by ϑ,a ψ (t)=ϑ(t+a), t∈R. (2.2.1) ϑ,a Note that ψ is a strictly increasing homeomorphism of R onto itself such that ϑ,a ψ1,0 =idR, ψϑ˜,a˜◦ψϑ,a =ψϑ˜ϑ,a˜+a, ψϑ−,1a =ψ1,−aϑ (2.2.2) ϑ ϑ for all ϑ˜∈(0,∞) and a˜∈R. Next, we define the linear self-map Tϑ,a of RZ+ by n (cid:18) (cid:19) (cid:88) n (T γ) = an−jϑnγ , n∈Z , γ ={γ }∞ ⊆R. (2.2.3) ϑ,a n j j + n n=0 j=0 The proof of Lemma 2.2.1 below, being elementary, is omitted. Lemma 2.2.1. The following hold for all ϑ,ϑ˜∈(0,∞) and a,a˜∈R: T is a bijection of RZ+ onto itself, ϑ,a T =id , T T =T , T−1 =T . (2.2.4) 1,0 RZ+ ϑ˜,a˜ ϑ,a ϑ˜ϑ,a˜+a ϑ,a 1,−aϑ ϑ ϑ In view of (2.2.2) and Lemma 2.2.1, the correspondence ψ (cid:55)→ T defines ϑ,a ϑ,a a faithful representation of the group of all strictly increasing homotheties of R. Moreover, by (2.2.3) and (2.2.4), we have, for all ϑ∈(0,∞) and a∈R, n (cid:18) (cid:19) (cid:88) n (T−1γ) = (−a)n−jϑ−jγ , n∈Z , γ ={γ }∞ ⊆R. ϑ,a n j j + n n=0 j=0 In Lemma 2.2.2 and Theorem 2.2.3 below we state properties of T which ϑ,a are relevant for further considerations. If ν is a Borel measure on R and ϕ is a homeomorphism of R onto itself, then ν◦ϕ is the Borel measure on R given by ν◦ϕ(σ)=ν(ϕ(σ)), σ ∈B(R). (2.2.5) Lemma 2.2.2. Let ϑ∈(0,∞) and a∈R. Then (i) T is a self-bijection on the set of all Hamburger moment sequences, ϑ,a 10 P.BUDZYN´SKI,Z.J.JABL(cid:32)ON´SKI,I.B.JUNG,ANDJ.STOCHEL (ii) if γ is a Hamburger moment sequence, then the mapping M(γ)(cid:51)ν (cid:55)→ν◦ψ−1 ∈M(T γ) (2.2.6) ϑ,a ϑ,a is a well-defined bijection with the inverse given by M(T γ)(cid:51)ν (cid:55)→ν◦ψ ∈M(γ); ϑ,a ϑ,a in particular, γ is H-determinate if and only if T γ is H-determinate, ϑ,a (iii) if γ is an H-indeterminate Hamburger moment sequence, then so is T γ ϑ,a and the mapping defined by (2.2.6) maps M (γ) onto M (T γ), e e ϑ,a (iv) if γ is a nonzero Hamburger moment sequence and ν ∈M(γ), then supp(ν◦ψ−1)=ψ (supp(ν)), (2.2.7) ϑ,a ϑ,a infsupp(ν◦ψ−1)=ψ (infsupp(ν)), (2.2.8) ϑ,a ϑ,a with convention that ψ (−∞)=−∞, ϑ,a (v) if a (cid:62) 0, γ is an H-indeterminate Stieltjes moment sequence and ν ∈ M+(γ), then T γ is an H-indeterminate Stieltjes moment sequence and e ϑ,a ν◦ψ−1 ∈M+(T γ). ϑ,a e ϑ,a Proof. (i)&(ii) If γ is a Hamburger moment sequence and ν ∈ M(γ), then, by the measure transport theorem (cf. [3, Theorem 1.6.12]), we have (cid:90) (cid:90) (T γ) = (cid:0)ψ (t)(cid:1)ndν(t)= tndν◦ψ−1(t), n∈Z , ϑ,a n ϑ,a ϑ,a + R R whichmeansthatT γ isaHamburgermomentsequenceandν◦ψ−1 ∈M(T γ). ϑ,a ϑ,a ϑ,a The above combined with (2.2.2) and (2.2.4) completes the proof of (i) and (ii). (iii) Let γ be a Hamburger moment sequence and ν ∈M(γ). Since sup (1+ϕ(t)2)/(1+t2)<∞, ϕ∈(cid:8)ψ ,ψ−1(cid:9), t∈R ϑ,a ϑ,a we deduce from the measure transport theorem that the mapping W: L2((1+t2)dν◦ψ−1(t))(cid:51)f →f ◦ψ ∈L2((1+t2)dν(t)) ϑ,a ϑ,a is a well-defined linear homeomorphism (with the inverse g → g◦ψ−1) such that ϑ,a W(C[t]) = C[t]. Similarly, V : L2(ν ◦ψ−1) (cid:51) f → f ◦ψ ∈ L2(ν) is a unitary ϑ,a ϑ,a isomorphism such that V(C[t])=C[t]. This, (ii) and Lemma 2.1.2(i) yield (iii). (iv) The equality (2.2.7) is a direct consequence of the definition of the closed support of a measure (see also [61, Lemma 3.2] for a more general result). Clearly, (2.2.7) implies (2.2.8). (v) Apply (iii) and (iv). (cid:3) Theorem 2.2.3. Let ϑ ∈ (0,∞) and a ∈ R. Suppose γ = {γ }∞ is an n n=0 S-indeterminate Stieltjes moment sequence and β is its Friedrichs measure. Then T γ is an H-indeterminate Hamburger moment sequence and the following holds: ϑ,a (i) if c>0, then T γ is an S-indeterminate Stieltjes moment sequence and ϑ,a β◦ψ−1 is the Friedrichs measure of T γ, ϑ,a ϑ,a (ii) if c=0, then T γ is an S-determinate Stieltjes moment sequence, ϑ,a (iii) if c<0, then T γ is not a Stieltjes moment sequence, ϑ,a (cid:0) (cid:1) where c:=ψ infsupp(β) . ϑ,a