Determinacy theory for the Livšic moments problem

34,429-444(1971)

JOUItNALOFMATHEMATICALANALYSISANDAPPLICATIONS

Determinacy

Theory

for the LivSic Moments

JE&S GIL School

of Mathematics,

Institute Minneapolis, Submitted

1,

DE

Problem

LAMADRID*

of Technology, University Minnesota 55455

of Minnesota

by C. L. Dolph

INTRODUCTION

The Liviic moments problem is presentedin terms of a given class.5- of complex functions defined on the real line R = (- 03, + co) and a given positive definite functional p’ on F. We shall refer to 9 as the classof test functions. The problem consistsin finding a measurep on R which gives an integral representationfor p;‘. This problem includesthe classicalHamburger momentsproblem aswell asother well studied problems.In [3], Livlic gave a proof of the existenceof CL,and, sincethen the problem hasbeendiscussedby other Russianauthors [ 1, 21.However, the discussionin theseworks hasbeen limited to the existenceof the solution of the problem. On the other hand, it is known [l 11,even for the classicalHamburger problem, that the solution, in general,is not unique. This haslead to the classicaldeterminacy theory of the Hamburger momentsproblem [l, 11, 121.This theory is concerned with the classificationof solutions, the study of the structure of the set of solutions, and the characterization of those problems which are determinate (have a unique solution). It is the purpose of this work to initiate the study of the determinacy theory for the Livgic problem. Our method, as that of Livgic, is based on the theory of transcendental extensions of symmetric operators (extensions to bigger spaces) due to Naimark [5, 6, 7, 81. Naimark proved the existence of self adjoint transcendental extensions and proceeded to give an analysis of the set of all such extensions.He also pointed to the application of his theory to the moments problem, by establishing[6] a one to one correspondencebetween the set of solutions of the Hamburger problem and the set of equivalence classesof extensions of the operator of multiplication by the independent variable which appearsnaturally in the problem. The work of Liv6c camelater and he * This

work

was partially

supported

by the National

7686). 429

Science

Foundation

(NSF

GP

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usedthe existencetheorem of Naimark to establishthe existenceof a solution of his problem. Our contribution here is to exploit further the parallel (only suggestedby Naimark) between his structure theory for transcendentalextensionsand the determinacy theory of the moments problem. We go beyond the work of LivIic, who consideredonly the existence problem, and not the determinacy problem. We generalize to the Livlic problem the Naimark theorem on the Hamburger problem quoted above. We then consider, for the Livgic problem, other questionsof the determinacy theory which do not appearin either the work of Naimark or that of Livlic, but which are classicalin the earlier literature of the Hamburger problem. These include the concept (and results) of extremal solution in the senseof Nevanlinna, other classificationsof the solutions of the moments problem, and a theorem on the number of each type of solution, in the indeterminate case,which appears in the book of Stone [12], for the Hamburger problem. A consequenceof this approach is what appearsto be a new criterion (Corollary 4.1) for the determinacy of the Hamburger problem. The proof of the generalization of the Stone theorem quoted above is based on a classification (Theorems 2.1 and 2.2 and Corollary 2.1) of self adjoint extensionsof a symmetric operator by unitary equivalence.This classification appears to be new since the main concern in the theories of self adjoint extensions of symmetric operators, both in the work of Naimark and in the classicalsetting [lo], is with individual extension, and no attention is given to which of the extentions are unitarily equivalent. This latter question is our major concern here becausewhat distinguishesbetween two solutions of the moments problem is the equivalence class of the corresponding self adjoint extension, and not the extensionitself. Of lessimportance is a technical improvement made on the presentation of the Livgic problem (Section 3) replacing the positive definite functional v’ on the spaceF of test functions by an inner product on 7. As is well known, a positive definite functional leadsto an inner product, but an inner product is a more general object. In the present situation, however, one seesat the end that the two problems are equivalent; but a priori, the present formulations includesproblems in whose presentation a positive definite functional is not immediately in evidence.

2. PRELIMINARIES Let H be a Hilbert space.We use the term pm-operator in the samesense that the term “operator” is normally used, that is, to refer to a linear transformation T from a vector subspace&- of H into H. In the present discussion it is important to keep in mind that, strictly speaking,a pre-operator consist

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of the triplet (.9r, H, T). W e reserve the term operator to refer to closed densely defined pre-operators. We shall say that two pre-operators Tl and T2 on Hilbert spacesHI and H, are equivalent if there exists a linear isometry U of HI onto H, which transforms the triplet (HI , 9r1 , Tl) into the triplet (H, , 9r2 , T2) in the obviousway. For brevity, we simply write T, = UT,U*. Similarly, two indexed families of pre-operators {T,} and (S,}, (the T’s operating on HI , and the S’s on H,), indexed by the sameindices 01,are said to be equivalent if there is a linear isometry U of HI onto H2 such that S, = UT,U*, for every 0~.We say that U implements the equivalence in question, or that the equivalenceis by means of U. By an extension of the preoperator T on H we shall mean a pre-operator Tl on H, together with an isometry U of H onto a minifold (closedvector subspace)HO of HI , such that T,, = UTU* is the restriction of TX to grO . We shall again say that U implements the extension. If HI = H and U is identity, then the extension in question is said to be within H. Let the pre-operator Tl on HI be an extension of T, the extension implemented by an isometry U, , and let T, , Hz , U, be similar objects defining a secondextension of T. We say that the first extension dominates the first, in symbols Tl 1 T2 , if there exists a linear isometry U of Hz into HI , which is said to implement the dominance, such that Tl is an extension of T, , the extension implemented by the isometry U, and U, = UU, . Two extensionsTl , HI , U, and T, , H, , U2are saidto be equivalent if the first (say) dominates the second, and the dominance is implemented by a linear isometry of H, onto HI . In the present work it is equivalence classes of pre-operators, rather than individual pre-operators which interest us. Similarly for equivalence classesof extensions. Let us call an individual object of an equivalence classa realization of the given equivalenceclass.We shall use the term ordinary to describean extension of T which is equivalent to an extension within H. This is, of course, the usual concept of extension. An extension which is not ordinary is saidto be transcendental. Transcendental extensionswere introduced by Naimark [5,6,7,8], with different terminology and in a slightly different form. The existing theory of such extensions is due to him and in the sequelwe shall be quoting someof the relevant parts of that theory. Clearly, a pre-operator Tl is an extension of a pre-operator T if and only if it is an extension of every pre-operator equivalent to T. Thus, it makessenseto speakof an equivalence classof pre-operators as being an extension of an equivalence class of pre-operators, since the concept is invariant under equivalence. Similarly we extend to equivalence classes, without giving details, the concept of dominanceof extensions.The concept of ordinary extension and transcendental extension bear a more careful examination in this context. What has to be shown is that an ordinary and a transcendental extension of the samepre-operator are never equivalent. This is

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not too difficult and is left to the reader, and thus these concepts can be applied to equivalence classes. Let us now consider a symmetric pre-operator T on H, that is, (Th, K) =T (h, Tk), for h, K E 9r . A self adjoint extension of T is one in which the extending operator is self adjoint. One of the main results in the Naimark theory is that every symmetric operator has a self adjoint (possibly transcendental) extension, although, only if the deficiency indices are equal does it have an ordinary self adjoint extension. The property of being a self adjoint extension is invariant under equivalence and is a property of equivalence classes. A minimal self adjoint extension of T is a self adjoint extension Tl of T, defined on a Hilbert space HI such that, if it dominates another self adjoint extension Tz on H, , the isometry U : Hz--f Hi , implementing the dominance, is onto. Clearly minimality is invariant under equivalence. Hence it makes sense to talk about a minimal self adjoint equivalence class of extensions of T. Let T be a symmetric operator on H. In particular, T is closed and densely defined. A manifold H,, of H is said to reduce T, if the orthogonal projection PO onto Ho has the property that TP,, 3 POT. Suppose that Tl , acting on HI , is a self adjoint extension of T. Since the U implementing the extension is an isometry, we may assume that HI 3 H and that C: is the inclusion mapping. Then, by means of the Cayley transform, one shows that Tl is a minimal self adjoint extension of T if and only if no proper manifold of HI , containing H reduces Tl . We note that H does not necessarily reduce Tl . For the remainder of this paper we are going to assume (or, depending on the context, be able to establish) that T is a symmetric operator on a Hilbert space H, and has equal deficiency indices (m, m). This is the only type of operators which interest us, from the point of view of the moments problem. The three main results of the present sections establish a lower bound for the cardinality of sets of various types of minimal self adjoint extensions of I’. Since we are assuming that T has equal deficiency indices, the classical theory (Stone [12]) tells us that T has an ordinary self adjoint extension. Theorem 1.l complements the classical theory by showing that either T is self adjoint, in which case it obviously has a unique minimal equivalence class of self adjoint extensions, (that of T itself), or T is not self adjoint, in which case we describe the cardinality of the set of minimal ordinary self adjoint equivalence classes of extensions of I’. The theory of Naimark [5, Theorem 111 yields the fact that, in addition, T has transcendental minimal self adjoint extensions, if T itself is not self adjoint. We complement the theory of Naimark by providing a lower bound for the set of equivalence classes of these extensions (Theorem 2.1, below). We also give a theorem providing a lower bound for the cardinality of a special type of transcendental extension. One might add that the classical methods of constructing extensions already show that the number of

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extensions, when they are proper extensions, is infinite, but it is not clear how many distinct equivalence classes result from the construction. It is our contribution, in each of the theorems quoted above, to exhibit, in the interesting cases, an infinite number of inequivalent extensions. The number of equivalence classess, rather than the number of individual extensions, is, as we shall see, the relevant number in the theory of the moments problem. Let T be a symmetric operator with equal deficiency indices (m, m). We shall say that T is indeterminate if it is not self adjoint. In the proof of the next theorem we follow fairly closely the notations and results concerning the Cayley transform, deficiency indices and deficiency spaces expounded in the discussion of extensions of symmetric operators given in the book of Naimark [9, pp. 144-1501. THEOREM 2.1. If T is a symmetric operator with equal deficiency indices (m, m), then, the number of ordinary, self adjoint equivalenceclasses of extensions of T is the sameas the numberof elements in the spaceZ,,, of all boundedoperator on an m-dimensionalHilbert space.In particular, if 1’ is indeterminate,this numberis at least of the power of the continuum.

Proof. Let T be a symmetric operator on H, satisfying our hypotheses. In the first place we remark that we are dealing with classicalextensionsof T within H, and that any two suchindividual extensionsA, and A, are equivalent if and only if they are equal. This is because,in this case,the complete description of each extension A, , K = 1,2, involves A, , H and the identity operator I and, from the definition of equivalenceany unitary operator U of H implementing an equivalencebetweenthe extensionswould have the property I = UI, or U = I. We are thus looking for the number of ways of extending the Cayley transform Ur to a unitary operator on H. This is the sameas the number of isometric linear transformations of the deficient spaceNi onto the other deficiency spaceN-, . Since both spaceshave the samedimension m, the number of extensionsis the number of unitary operatorsin 6p,, which is the sameas the cardinality of -E”, . This completesour proof. If the deficiency indices do not agree, T has no ordinary self adjoint extension. At another extreme, if T is self-adjoint, it itself is its only minimal, and its only ordinary self-adjoint extension,up to equivalence.We have thus given a complete discussionof the existenceof ordinary self-adjoint extensions.We now passto a discussionof the existenceof minimal self-adjoint transcendental extensions. THEOREM 2.2. Let T bea symmetricoperator with deJiciencyindices(m, m). Then the number of minimal self-adjoint transcendentalequivalenceclassesof extensionsof T is at leastthe cardinality of L?,,,,.

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Proof. We prove this theorem by constructing a family of the appropriate cardinality consisting of pairwise inequivalent minimal self-adjoint extensions of T which are not ordinary. Let us look again at the Cayley transform Ur of T, with domain Ri and range Rei . Let Ni and N-, be the corresponding deficiency spaces. We now consider a Hilbert space Ha 3 H which is the direct sum of $1 and a Hilbert space N, of dimension m. To obtain a selfadjoint extension of T to E& we simply extend U, to a unitary operator of H,, , which is the Cayley transform of a selfadjoint operator on H,, . We define it as follows. We take an arbitrary isometric transformation Ui of Ni onto N, and an arbitrary isometric transformation Vi of N,, onto N-, . We now set

which is an isometric transformation of H,, = Ri @ Ni @NO onto R-i @ N-i @ Ns = H, , that is, a unitary operator. Clearly U leaves no closed manifold of N, , other than {0}, invariant. It follows from here that 1 is not an eigenvalue of U, for one can show easily that an invariant vector of U must be orthogonal to H, hence must belong to N,, . From this we conclude that U is the Cayley transform of a self-adjoint operator A of HO , which must be minimal because U leaves invariant no non-trivial manifold of N0 . It is also clear that A is a transcendental extension. We must now investigate which, among the extensions A, constructed by means of (2.1), are equivalent. Let A, and A, be obtained from (2.1) by means of unitary operators Ur , U, , each Uj given by Uji and Uii. Any unitary operator of H,, implementing the equivalence of the extensions A, and A,, if they were equivalent, would reduce to identity IH on H, hence leave N,, invariant. Thus, any such unitary operator W,, would be of the form w, = I, @ w,

(2.2)

where W is a unitary operator of N,, . We now compute W in terms of Uji and UF~, by using the relation A,W,, = W,,A, , expressing the equivalence between the extensions. This gives a similar relation between the Cayley transforms: u,w,

= WJJ,.

(2.3)

From this we get, using (2.2)

U,i( up

= w = (lJ,-“)-l lJ;i,

(2.4)

= u;qJ t 2’

(2.5)

or jy,-“,;

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Thus, the mapping VUi : Ni + N-, is constant through the equivalence class containing the extension A of T. We have then succeededin assigning to every equivalence classof extensionsobtained from (2.1) a unique unitary mapping V = .?PiUi of Ni onto N-, . It follows by re.versingthe above steps that this assignmentis one-to-one. It is onto because,if V is a unitary transformation of Ni onto N-, , we may set Ui, unitary from Ni onto N, , arbitrarily and find Wi = V( Ui)ei. Hence the number of equivalenceclassesof minimal self-adjoint extensionsof T, constructed in the above manner is the sameas the cardinality of the set of all unitary transformations of Ni onto NV,, which is the cardinality of Pm. This completesthe proof. A self adjoint extension of a symmetric operator Twill be saidto be extreme if it definesan extreme spectral function for T, in the senseof Naimark [S, pp. 127, 1281.We refer the reader to [8] or to [13] for a discussionof spectral functions and for the fact that a$nite extension Tl , H1 , U, of T, that is, one suchthat U,(H) hasfinite deficiency in Hi , is extreme. In particular ordinary extensions are extreme. That the converse is not true, and much more, is implied by the corollary below. It follows easily from the definition that extremality is invariant under equivalenceof extension, and therefore applies to equivalence classes. COROLLARY 2.1. Let T be an indeterminate symmetricoperator with equal de$ciency indices. Then, the number of extreme self-adjoint equivalence classes of extensions of T which are not ordinary is at least the power of the continuum.

Proof. We may assumethat the deficiency indices of T are finite, for we can extend T to a symmetric operator T’ on the samespaceH with positive finite equal deficiency indices, and any minimal extension of T’ is a minimal extension of T. Now we employ the construction of Theorem 2.2 and obtain the desired extreme extensions,since finite extensionsare extreme.

3.

THE

LIVSIC PROBLEM

We begin by describing the Livgic moments problem in a slightly more generalform than in [3], but preserving its essentialcharacter. We are given a vector space9 of complex functions defined on R, called the spaceof test functions, and a semi-inner product (v, 4) ( a non-negative sesquilinearfrom) on Y, satisfying certain conditions. which we shall state presently. However, we first need some conventions and notations. For v EY, we write 11 v [I2= (p, p)).We shallconsiderasidentical any two functions v, I,!IEY such that 11 p - # 11= 0, and, with deliberate ambiguity, continue to refer to 9 asthe resulting space,after this identification, on which (v, 9) is a true inner

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product. We shall denote by H the Hilbert space obtained by completing Y with respect to this inner product. We define a pre-operator F on ZZ. Its domain 9~ consists of all v ~7, such that the expression @(t), t E R, represents a function in Y, and the operator is multiplication by t. Now, let p E Y, we denote by q* its complex conjugate, i.e., cp*(t) = v(t). We are now ready to state our conditions. (a) The constant function 1 belongs to Y, (b) The space Y of test functions is self-adjoint, (v* E Y-, for g, E Y), and the mapping p --f 91* is isometric on Y (that is, jl p I/ = 11q* II), (c) Every ‘p E 7 is continuously differentiable (i.e., ,Y C C,(R)), (d) The preoperator 5? is hermitean (( ppl, #) = (y, F#), for v, I/JE@) and densely defined, (e) for every s E R, the range (T - sZ) 9~ of (T - sl), Z the identity operator of H, is a hyperplane of 7. (f) For every y ~58~ , p* ~58~ and (TV)* = TV*. The problem is then to find a positive measure p on R such that, for every q E 7, v E L,(p) and (% $1 = /I”, v(t) Nt) 44).

(3.1)

for every v, 9 E .Y. If the original semi-inner product is given (a priori) by a positive functional y’ on 7, we have the Livlic problem [3] by setting (v, #) = ($J*F, v’). Of course, because of (3.1), the inner product of the moments problem, as we have posed it, is always given by a positive functional, hence our formulation is equivalent to that of LivSic. However we know this only after the problem is solved. Thus our formulation has the technical advantage of applying to situations in which the underlying positive functional of Liviic is not immediately in evidence. In any event, the following comments should be made regarding our conditions (a) through (f) and the corresponding ones in [3]. Condition (c) is the subject of some confusion in [3] and in the subsequent literature. It does not appear in [3], and its absence results in a gap in LivSic argument (see the paper of Krein [2]). In the exposition in the book of Akhiezer [l], it is again omitted from the formulation of the problem, but used in the proof of the existence of p. In Condition (e), the constant function 1 does not belong to (T - sZ) 9~ , for any s E R. This is because every $ E (F - sl)B~ vanishes at t = S. Therefore, for every v E 5, and every s E R, there exists a qs E Y such that

v(t) = (t - 4 v%(t)+ F(S),

(3.2)

for every t E R. Condition (e) appears in [3] in the form (3.2), which clearly reduced to the euclidean algorithm, if Y is the space of polynomials, as in the classical Hamburger problem. Consequently, we refer to Condition (e), in general, as the euclidean algorithm.

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Unless the contrary is indicated, we shall refer to the above problem of Livgic, simply asthe moments problem. By applying the methodsof LivGc, one concludesimmediately that there existsa measurep on R satisfying (3.1). Any such p will be called a solution of the moments problem. Our purpose here is to bring out the almostexact parallel betweenthe study of the set of all solutionsof the momentsproblem, and the theory of Naimark of transcendental extensionsof symmetric operators. By doing this we succeedin extending to the LivSic problem some of the classical determinacy theory for the Hamburger problem. Let usconsiderthe pre-operator f. It follows from the theory of symmetric operatorsthat ?‘can be closed,thereby obtaining a symmetric denselydefined operator T, which is an ordinary extension of T within H. Let il be a (possibly transcendental)self adjoin extension of T with spectral resolution of the identity EA . Let the isometry U implement the extension. We set

(3.3) for every Bore1B C R. LivSic proved [3] that pa is a solution of the moments problem. We are going to seethat, conversely, every such solution is given by (3.3) for a suitable self-adjoint extension A of T. Let us begin with a solution p and let us construct the appropriate selfadjoint extension A. It follows from (3.1) that H can be imbedded isometrically in&(p). We define the operator A, whosedomain consistsof all 0 EL.&) such that the expressionto(t) representsa function in L,(p), asmultiplication by t on gn, . It is well known that A is a self-adjoint operator whosespectral resolution of the identity is given by

E,(B) 0 = x&J, for every 0 E&(P), where xe is the characteristic function of the Bore1set B. The operator A on L&) will be said to be given by multiplication by the independent variable. Clearly A is an extension of T. Now we are ready to analyze the self-adjoint extensionsof the operator T of the momentsproblem, given in terms of a solution CL,by multiplication by the independent variable. We note first that, since 1 E Y CL,(p), every solution p of the moments problem is finite, that is p(R) < + co. THEOREM 3.1. Let p be a solution to the moments problem. Then the selfadioint extensionA of r definedon L&L) by multiplication by the independent variable is a minimal self-adjoint extensionof T, from which p is obtainedby meansof (3.3).

Proof. To show that A is a minimal self adjoint extension of T it suffices to show that if a closed manifold H,, contains H, is contained in L&) and reducesA, then HO = L&). Now, for every Bore1set B C R, xe = E(B) 1,

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by (3.4) and, since 1 EY C H,, , and H0 is invariant, we get xs E El,, , hence L&L)

= H,, .

The fact that p = /.L~ as in (3.3), with U representing the imbedding H CL,(p), follows immediately from (3.4). This completesour proof. The next theorem is a converseof Theorem 3.1. THEOREM 3.2. Every minimal self-adjoint extension of the operator T of the moments problem is given, up to equivalence, by multiplication by the independent variable on the space L&L) of a solution p of the moments problem.

Proof. Let the operator A on a Hilbert space HA be a minimal selfadjoint extension of T. We may assumethat H C HA , this inclusion implementing the extension. Let E be the spectral resolution of the identity for A. We show that A is cyclic in H, , with cyclic vector 1, that is, the span of vectors of the form E(B) 1, B a Bore1set in R, is densein HA . Let HO be the closureof the spanof the vectors E(B) 1. One can seeeasilythat 15sreducesA. Since A is minimal, it sufficesto show that HO contains H. This and the fact that p = pA , as in (3.3), is a solution of the momentsproblem follow from the work of LivZc [3] (seealso[13]), and we omit the details. From the fact that A is cyclic, and the generaltheory of self-adjoint operatorswe have that A is unitarily equivalent to multiplication by the independent variable on the spaceL&L). It remainsto show that the transformation implementing the equivalence in question actually implementsan extension of T. It actually sufficesto show that v E.Y goes into g, EL&). Now under the unitary equivalence in question, the cyclic vector 1 for A goeson to the function 1 in L.&J), and the spectral projection E(B) goesinto multiplication by characteristic functions xs . From this it follows readily that the vector v E.Y C Hk goes on to the function v EL&). This completesour proof. We can now combinethe above two theoremsto obtain the following result exhibiting the complete equivalence between the momentsproblem and the problem of finding all the minimal self-adjoint extensionsof the symmetric operator T. We note first that the set A?=of all the solutions to the moments problem is a convex set of measures.On the other hand, Naimark [8] introduced a convex structure on the set of all minimal self adjoint extensionsof a symmetric operator Ton a Hilbert spaceH, in such a way that, if A, and A, are two such extensionswith respective resolutions of the identity E1 , E, , a convex combination olA, + /3A, hasa resolution of the identity E such that (assumingthat all extensions in question are implemented by the inclusion mapping) (Eh, 4 = 4W, 4 + B(E,h, 4 (3.5) for h, h E H.

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One should note that the convex combination orA, + /IAs does not have any immediate “concrete” interpretation. It is defined in an abstract manner, without imbedding the set of extensions into a vector space. The concept of extremality of extensions, used here earlier is defined in terms of this convex structure. We refer the reader to the work of Naimark [8], or to [13] for details. It follows readily from the definition of the convex structure, that this structure is consistent with the equivalence of extensions and defines, therefore, a convex structure on the set A& of equivalence classes of minimal self adjoint extensions of T. This is the convex structure used in the next theorem. THEOREM 3.3. There exists a one to one correspondence between the set J$, of all minimal self adjoint equivalence classes of extensions of the operator T of the moments problem and the set JY~ of all solutions of the moments problem. This correspondence preserves the convex structure of the two set, and if the equivalence class A of the minimal self adjoint extension A of T corresponds to the solution TV of the moments problem, then A and p are related by (3.3). There exists a realization A of the equivalence class which is multiplication by the independent variable

on -Md. Proof. To a solution p of the momentsproblem, we assignthe equivalence classA of the minimal self adjoint extensionA given by multiplication by the independent variable on L&), as in Theorem 3.2. The assignmentis one to one, sincep can be recovered from A, asindicated above. To show that p can be obtained, via (3.3) from any realization of the equivalence classA, it sufficesto show that any two equivalent minimal self adjoint extensionsA, and A, of T yield the samesolution of the momentsproblem. Let the extension A, be implemented by the isometry IT, and the equivalencebetweenthe two extensionsby the unitary transformation I’. Supposethat A, yields the solution pk to the moments problem. Then, we have, for each Bore1 set BCR, /1I(B) = (E,(B)

VII,

i&l)

= (E,(B)

VUJ,

VUJ)

= (Y*E,(B)

v%l,

&I)

In (3.6), Ek standsfor the spectral resolution of A,. . The fact that the assignment p -+ k is onto is the contents of Theorem 3.1. The fact that the correspondence preservesthe convex structures of the two setsfollows from (3.5). This completesour proof. For the Hamburger problem Theorem 3.3, was establishedby Naimark [8, p. 3411.Ours is an extension to the LivSic problem. We have thus reduced the study of the structure of the set of all solutions of the momentsproblem

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to the correspondingproblem for the set of self-adjoint extensionsof a given symmetric operator. We now proceedto the classificationof solutions. Given a minimal self-adjoint extensionA of the operator T of the momentsproblem, let us denote by pa the solution of the moments problem assignedto A in Theorem 3.3. Similarly, given a solution CL,we denote by A, the corresponding minimal self-adjoint extension of T, which is multiplication by the independent variable on L&). We shall call a solution CL,N-extremal (Extremal in the senseof Nevanlinna) if H = La(p). This term was usedin [l, p. 431in connection with the Hamburger momentsproblem. Clearly, p is N-extremal if and only if the self-adjoint extension il,, is ordinary. We shall show the existence of N-extremal solutions. First we need the following lemma. LEMMA

3.1.

The operator

T has equal deficiency indices.

Proof. Since the mapping v + q* (see Condition (b) of the moments problem) is isometric, it can be extended from the spaceF of test functions to the entire H, and, from Condition (f) we get that gr is invariant under conjugation, and that (TV)* = TV*, for v ~9~ . It now follows from a theorem of Stone [I 2, p. 361, Theorem 9.141,that the deficiency indices of T are equal. We need two more commentson terminology. We have seenthat the set AL of all solutionsof the momentsproblem is a convex set, and by the convex character of the correspondencebetween the set A” and the set ~88~ , we have that a solution p EA, is extreme (i.e., an extreme point of Use,)if and only if A, is an extreme minimal self-adjoint extension of T (seeSection 2). In the classicalliterature the moments problem is called determinate if and only if it hasonly one solution. Otherwise it is called indeterminate. Now we can state our classificationtheorem. THEOREM 3.4. The moments problem is determinate or indeterminate according as the symmetric operator T is self-adjoint or indeterminate. If the problem is indeterminate, then the number of N-extremal solutions, the number of extreme solutions (i.e., extreme points of A?=) which are not N-extremal and the number qf non-extreme solutions are all equal to the power of the continuum. Proof. We have seenthat if T is self-adjoint, it is its only minimal selfadjoint extension, up to equivalence, and is, of course, N-extremal; hence, extreme. By Theorem 3.3, &XLconsistsexactly of one point, the solution to the momentsproblem. Conversely, if T is indeterminate, then, by Theorem 2.1, the number of N-extremal self-adjoint equivalenceclassesof extensionsof T is at least of the power of the continuum, and Corollary 2.1 yields a similar conclusion for extreme non-N-extremal solutions. The statement for nonextreme solutions is obvious, since A+& contains the entire open segment

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441

PROBLEM

joining two different extreme solutions. Now all the sets of solutions described above are at most of the power of the continuum, because this is the case for the entire set AL . This is because, AL is a subset of the spaces C,,‘(R) of all finite Bore1 measures on R. The latter is of the power of the continuum, because it is the dual (conjugate) space of the separable space C,,(R) of all continuous functions vanishing at co. The above theorem is an extension to the Livshits problem of a part of the classical determinacy theory of the Hamburger moments problem [12, p. 610, Theorem IO&l]. We know that an element TVof A%?~is N-extremal if and only if F is dense in L,(p). The next theorem, due to Naimark [6, p. 342, Theorem 43 in the case of the Hamburger moments problem, gives a similar characterization of extreme solutions of the moments problem. We denote by F the linear span of all products 4, q, (GE F. Clearly .F C&(p). 3.5. Let p be an element of AL . Then p is an extreme point of AL if and only if the space 9 is dense in L&).l THEOREM

Proof. Supposethat TVis an extreme point of AL , but that .F is not dense in &f&L). Then there is an element BEL&) CL,(p), 6 + 0 such that (5, 0) = 0, for every [ E .F. We may assume that 0 is real and that 0 <./I 0 /jm< 1. We define, for each Bore1 set B C (- CO, co)

~o(Bl= j, 'W 4-N

(3.7)

and obtain a measurep,, on (- co, + co). We define Pl

=cL

+

PO f/4

P2=P-Po#P

(3.8)

=

(3.9)

and verify easily that CL1 + _I_

CL2

2

p'

Clearly pi and p2, which are positive, are solutionsto the momentsproblem; hencep is not an extreme point of AYE. Supposeconversely, that % is densein L,(p), and that (3.9) is satisfied.It suffices to show that p1 = II. We set p. = p - pr , and see that also p. = (,u~- pr)/2. It follows from this and (3.9) that - TV< p. < 2~. In particular p. is absolutely continuous with respect to CL,hence, its RadonNikodym derivative is a function in L,(p) which is orthogonal to F, hence must be 0, and ,U = t.~r. This completesour proof. this

1 I am indebted theorem.

to R. Langer

for pointing

out a mistake

in the original

version

of

442

GIL DE LAMADRID 4. THE HAMBURGER PROBLEM

For the Hamburger momentsproblem the above abstract operator methods yield easily someof the sharper classicalresults. Before specializing directly to the Hamburger problem, let us still consider the LivSic problem discussed in the previous section,but with the extra hypothesis: (g) For someimaginary number z,, , the image (T - zOZ)9~ is a hyperplane of .Y. Thus there exists an element#,, E7, whosenorm may be assumedto be 1, suchthat for every q~EY, there is qz, E7 for which the following euclidean algorithm holds

We have alsothe following linear functional #i, defined on 7 by the formula

CT,9x”, = L .

(4.2)

Clearly, (91,$k,) can be interpreted asthe “value” of IJJat a+,. In the caseof the Hamburger problem Y is the spaceof all complex polynomials which do have values at z0 , and by taking I/J~,to be the polynomial 1, (q~,#i,) turns out to be ~(2,). THEOREM 4.1. Zf the LivSic moments problem satisfies condition (g), then the dejiciency indices of the operator T are (0,O) or (1. l), according as the problem is determinate or indeterminate, and according as the linear functional $J:, on F is unbounded or bounded with respect to the Hilbert space norm on F. Proof. We already know that the deficiency indicescoincide (Lemma 3.1) and that the problem is determinate or not depending on whether the deficiency indices are (0,O) or otherwise. It remainsto show that under the present hypotheses,the only choicesare (0, 0) or (1, 1) and that this is controlled by the bounded character of $b, . The bounded character of $6, is determined by wheter or not the hyperplane (T - zOZ)9~ is denseor closed in Y. If it is dense,then #:, is unbounded, and (T - z,,Z) Br is alsodensein H. But, sincez,, is imaginary, we have actually (T - zoZ)CST = H [9, p. 145, Theorem 31, hence the Cayley transform U,(x,) is everywhere defined and unitary, from which we get that T is self-adjoint, and, by Theorem 4.5, the momentsproblem is determinate. On the other hand, if ( T - z,,Z)9~ is closed in 9, then (T - q,Z) Sr is a closedhyperplane in H, #‘,, is a bounded linear functional on H, and 9t’r(~ = (T - q,Z) 9T is a proper closed manifold in H, hence T is not self-adjoint and the momentsproblem is indeterminate. From this discussion,it follows readily that the only possible choices of deficiency indices of T are (0, 0) or (1, 1), depending on whether

LIV.?IC

MOMENTS

(T--x,1)~3,=Hor(T--xd)~~ cludes our proof. Now for the classical Hamburger characterization of determinacy. 4.1. satigying

COROLLARY

problem,

is a closed hyperplane of 22. This conmoments problem we can give a simple

Given the moments I+,, v, ,... in the Hamburgu

j;l

cizkvj+k

a necessary and su$icient condition

sup

443

PROBLEM

3

moments

(4.3)

0,

that the problem be detuminate

is that

1[go (-l>i b]+i[i (-l)ibzj+;l I2 j=o

:

=+a

(4.4)

&bk”j+k

i,k=l

where the sup is taken over all complex polynomials p(t) = zi:, bjtj, with the possibility b,, = 0.

p, written

in the form

Proof. In (4.4) we adopt the convention that O/O= 0. The momentsdefine a positive functional q’ on the space.F of all complex polynomials p on (- co, + co), and this q’, in turn, definesan inner product on F, given by (v, $) = (#*q~, #>. Clearly the resulting system consistingof 5, T and the inner product satisfiesthe conditions (a) through (f) of the LivGc problem. It also satisfiesCondition (g), stated at the beginning of this section. Only, in this case,the polynomial p is defined in the entire complex plane, and then, if in (4.1) we take z, = i, and +z, identically 1, (clearly the function 1 does not belong to (T - il) .&.), then I,&’becomes,strictly speaking,evaluation at i. Applying Theorem 4.1, we get that the problem is determinate if and only if &’ is unbounded, which saysthat sup 1p(i)/“/(p*p, ~JJ’)= + co, but this is (4.4). This concludesour proof. Added in Proof. It has been pointed out by R. Langer that the pre-operator p defined in the first paragraph of Section 3 may not make sense on the space .Y obtained after identifying pairs of functions p and I,G with 11‘p - 4 11 = 0, since II pq 11may not be 0, if 11q II = 0. The simplest way out if the dilemma is the following. ?‘is defined on Y before the identification. Let us interpret the assumptions (a) through (f) as valid, before this identification. Then 11pp 11= 0 follows from II c 11 = 0, as seen from the following argument which we owe again to Langer. For I/ E 9p , /(TV, $)I = [(v, p$)I ,< II T II II T# II = 0, from the symmetry of p and the Schwartz inequality. Since 9~ is dense in 9, 11Tv I/ = 0.

444

GIL

DE LAMADRID REFERENCES

1. N. I. AKHIEZER, “The Classical Moments Problem,” Russian edition: Isdat. Fiz. Mat. Lit., Moscow, 1961. English edition: Oliver and Boyd, Edinburgh and London, 1965. 2. M. G. KREIN, On hermitean operators with directed functionals, Z&r. Pvats. inst. Matem. Akad Nauk Ukran. URSR (1948). 3. M. S. LIVSIC, An application of the theory of hermitean operators to the generalized moments problem, Dokl. Akad. Nauk. SSSR 44 (1944), 3-7. 4. B. SZ.-NACY, “Prolongements des transformations de l’espace de Hilbert qui sortent de cet espace” (Appendice au livre: “Lecons d’analyse fonctionnelle” par F. Riesz et B. Sz.-Nagy), Adademia Kiado, Budapest, 1955. 5. M. A. NAIMARK, On self-adjoint extensions of the second kind of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat. 4 (1940), 53-104. 6. M. A. NAIMARK, Spectral functions of a symmetric operator, 1.~. Akad. Nauk.

SSSR Ser. Matem. 4 (1940), 277-318. 7.

M.

A. NAISIARK,

On extremal

spectral

functions

of a symmetric

operator,

Dokl.

Akad. Nauk. SSSR N. S. 54 (1946), 7-9. 8. M.

A. NAIMARK,

Extremal

spectral

functions of a symmetric operator, 327-344. Differentialoperatoren,” Akademie-Verlag,

Izv. Akad.

Nauk. SSSR Ser. Mat. 11 (1947),

9. M. A. NAIMARK, “Linearen Berlin, 1963. 10. F. Rmsz AND B. SZ.-NAGY, “Lecons d’analyse fonctionnelle,” AkadCmia Kiado, Budapest, 1953. 11. J. A. SHOHAT AND J. D. TAMARKIN, “The Problem of Moments,” Amer. Math. Sot. Surveys, No. 1, New York, 1943. 12. M. H. STONE, Linear transformations in Hilbert space and their applications, Amer. Math. Sot. Colloq. Publ. XV, New York, 1932. 13. J. GIL DE LAMADRID, The Naimark transcendental extension of symmetric operators and the LivSic moments problem, Technical Report, University of Minnesota, School of Mathematics.