On the truncated multidimensional moment problems in \mathbb{C}^n
aa r X i v : . [ m a t h . F A ] F e b On the truncated multidimensional momentproblems in C n . Sergey M. ZagorodnyukAbstract.
We consider the problem of finding a (non-negative) measure µ on B ( C n ) such that R C n z k dµ ( z ) = s k , ∀ k ∈ K . Here K is an arbitraryfinite subset of Z n + , which contains (0 , ..., s k are prescribed complexnumbers (we use the usual notations for multi-indices). There are two possi-ble interpretations of this problem. At first, one may consider this problemas an extension of the truncated multidimensional moment problem on R n ,where the support of the measure µ is allowed to lie in C n . Secondly, themoment problem is a particular case of the truncated moment problem in C n , with special truncations. We give simple conditions for the solvabilityof the above moment problem. As a corollary, we have an integral repre-sentation with a non-negative measure for linear functionals on some linearsubspaces of polynomials. Throughout the whole paper n means a fixed positive integer. Let us in-troduce some notations. As usual, we denote by R , C , N , Z , Z + the sets ofreal numbers, complex numbers, positive integers, integers and non-negativeintegers, respectively. By Z n + we mean Z + × . . . × Z + , and R n = R × . . . × R , C n = C × . . . × C , where the Cartesian products are taken with n copies. Let k = ( k , . . . , k n ) ∈ Z n + , z = ( z , . . . , z n ) ∈ C n . Then z k means the monomial z k . . . z k n n , and | k | = k + . . . + k n . By B ( C n ) we denote the set of all Borelsubsets of C n .Let K be an arbitrary finite subset of Z n + , which contains := (0 , ..., S = ( s k ) k ∈K be an arbitrary set of complex numbers. We shall considerthe problem of finding a (non-negative) measure µ on B ( C n ) such that Z C n z k dµ ( z ) = s k , ∀ k ∈ K . (1)There are two possible interpretations of this problem. At first, one mayconsider this problem as an extension of the truncated multidimensionalmoment problem on R n , where the support of the measure µ is allowed tolie in C n . Similar situation is known in the cases of the classical Stieltjesand Hamburger moment problems, where the support of the measure lies1n [0 , + ∞ ) and in R , respectively. Secondly, and more directly, the momentproblem (1) is a particular case of the truncated moment problem in C n (see [4, Chapter 7], [9], [8]), with special truncations. These truncations donot include conjugate terms.It is well known that the multidimensional moment problems are muchmore complicated than their one-dimensional prototypes [1], [2], [4], [5],[10], [12]. An operator-theoretical interpretation of the full multidimensionalmoment problem was given by Fuglede in [6]. In general, the ideas of theoperator approach to moment problems go back to the works of Naimarkin 1940–1943 and then they were developed by many authors, see historicalnotes in [15]. In [17] we presented the operator approach to the truncatedmultidimensional moment problem in R n . Other approaches to truncatedmoment problems can be found in [4], [5], [13], [16], [9], [8]. Recent resultscan be also found in [14], [7].In the case of the moment problem (1) we shall need a modification of theoperator approach, since we have no positive definite kernels here. However,this problem can be passed and we shall come to some commuting boundedoperators. We shall provide a concrete commuting extension for this tuple.Then we apply the dilation theory for commuting contractions to get therequired measure. Consequently and surprisingly, we have very simple con-ditions for the solvability of the moment problem (1) (Theorem 1). As acorollary, we have an integral representation with a non-negative measure forlinear functionals L on some linear subspaces of polynomials (Corollary 1). Notations.
Besides the given above notations we shall use the followingconventions. If H is a Hilbert space then ( · , · ) H and k · k H mean the scalarproduct and the norm in H , respectively. Indices may be omitted in obviouscases. For a linear operator A in H , we denote by D ( A ) its domain, by R ( A ) its range, and A ∗ means the adjoint operator if it exists. If A isinvertible then A − means its inverse. A means the closure of the operator,if the operator is closable. If A is bounded then k A k denotes its norm.For a set M ⊆ H we denote by M the closure of M in the norm of H .By Lin M we mean the set of all linear combinations of elements from M ,and span M := Lin M . By E H we denote the identity operator in H , i.e. E H x = x , x ∈ H . In obvious cases we may omit the index H . If H isa subspace of H , then P H = P HH denotes the orthogonal projection of H onto H . 2 Truncated moment problems on C n . A solution to the moment problem (1) is given by the following theorem.
Theorem 1
Let the moment problem (1) with some prescribed S = ( s k ) k ∈K be given. The moment problem (1) has a solution if and only if one of thefollowing conditions holds:(a) s (0 ,..., > ;(b) s k = 0 , ∀ k ∈ K .If one of conditions ( a ) , ( b ) is satisfied, then there exists a solution µ with acompact support. Proof.
The necessity part of the theorem is obvious. Let moment prob-lem (1) be given and one of conditions ( a ),( b ) holds. If ( b ) holds, then µ ≡ s (0 ,..., >
0. Observe that we can include the set K into the following set: K d := { k = ( k , . . . , k n ) ∈ Z n + : k j ≤ d, j = 1 , , ..., n } , for some large d ≥
1. Namely, d may be chosen greater than the maximumvalue of all possible indices k j in K . We now set s k := 0, for k ∈ K d \K .Consider another moment problem of type (1), having a new set of indices e K = K d . We are going to construct a solution to this moment problem,which, of course, will be a solution to the original problem.Consider the usual Hilbert space l of square summable complex se-quences ~c = ( c , c , c , ... ), k ~c k l = P ∞ j =0 | c j | . We intend to construct asequence { x k } k ∈ e K , of elements of l , such that( x k , x ) l = s k , k ∈ e K . (2)The elements of the finite set e K can be indexed by a single index, i.e., weassume e K = { k , k , . . . , k ρ } , (3)with ρ + 1 = | e K| , and k = (0 , ..., a := √ s (0 ,..., ( > x := a~e , x k j := ~e j + s k j a ~e , j = 1 , , ..., ρ. (4)Here ~e j means the vector ~c = ( c , c , c , ... ) from l , with c j = 1, and 0’s inother places. Observe that for this choice of elements x k , conditions (2) hold3rue. Moreover, it is important for our future purposes that these elements x k are linearly independent.Consider a finite-dimensional Hilbert space H := Lin { x k } k ∈ e K . Set K d ; l := { k = ( k , . . . , k n ) ∈ K d : k l ≤ d − } , l = 1 , , ..., n. Consider the following operator W j on Z n + : W j ( k , . . . , k j − , k j , k j +1 , . . . , k n ) = ( k , . . . , k j − , k j + 1 , k j +1 , . . . , k n ) , (5)for j = 1 , . . . , n . Thus, the operator W j increases the j -th coordinate. Weintroduce the following operators M j , j = 1 , ..., n , in H : M j X k ∈ K d ; j α k x k = X k ∈ K d ; j α k x W j k , α k ∈ C , (6)with D ( M j ) = Lin { x k } k ∈ K d ; j . Since elements x k are linearly independent,we conclude that M j are well-defined operators. Operators M j can be ex-tended to a commuting tuple of bounded operators on H . In fact, considerthe following operators A j ⊇ M j , j = 1 , ..., n : A j X k ∈ K d α k x k = X k ∈ K d ; j α k x W j k , α k ∈ C . (7)Operators A j are well defined linear operators on the whole H . It can bedirectly verified that they pairwise commute. Notice that A k A k ...A k n n x = x ( k ,k ,...,k n ) , k = ( k , ..., k n ) ∈ K d . (8)Relation (8) can be verified using the induction argument. Since H is finite-dimensional, then k A j k ≤ R, j = 1 , , ..., n ;for some R >
0. Set B j := 1 C A j , j = 1 , ..., n, (9)where C is an arbitrary number greater than √ nR . Then n X j =1 k B j k < . (10)4n this case there exists a commuting unitary dilation U = ( U , ..., U n ) of( B , ..., B n ), in a Hilbert space e H ⊇ H , see Proposition 9.2 in [11, p. 37].Namely, we have: (cid:16) P e HH U k U k ...U k n n (cid:17)(cid:12)(cid:12)(cid:12) H = B k B k ...B k n n , k , ..., k n ∈ Z + . (11)Moreover, we can choose U to be minimal, that is, the subspaces U k ...U k n n H will span the space e H (see Theorem 9.1 in [11, p. 36]): e H = span n U k ...U k n n H, k , ..., k n ∈ Z o . Then the Hilbert space e H will be separable. By (9),(8),(2),(11) we maywrite for an arbitrary k = ( k , ..., k n ) ∈ e K : s k = ( x k , x ) l = ( A k A k ...A k n n x , x ) l = C | k | ( B k B k ...B k n n x , x ) l == C | k | ( U k U k ...U k n n x , x ) l = (( CU ) k ( CU ) k ... ( CU n ) k n x , x ) l == ( N k N k ...N k n n x , x ) l , (12)where N j := CU j , j = 1 , ..., n . Applying the spectral theorem for commutingbounded normal operators N j (or,equivalently, to their commuting real andimaginary parts), we obtain that N j = Z C n z j dF ( z , ..., z n ) , j = 1 , ..., n, where F ( z , ..., z n ) is some spectral measure on B ( C n ). Then s k = Z C n z k ...z k n n d ( F ( z , ..., z n ) x , x ) l , k = ( k , ..., k n ) ∈ e K . This means that µ = ( F ( z , ..., z n ) x , x ) l , is a solution of the momentproblem. Since N j were bounded, µ has compact support. ✷ Corollary 1
Let K be an arbitrary finite subset of Z n + , which contains .Let L be a complex-valued linear functional on M = M ( K ) := Lin { z k ...z k n n } k =( k ,...,k n ) ∈K , such that L (1) > . Then L has the following integral representation: L ( p ) = Z C n p ( z , ..., z n ) dµ, ∀ p ∈ M, (13) where µ is a (non-negative) measure µ on B ( C n ) , having compact support. roof. It follows directly from Theorem 1. ✷ Corollary 1 can be compared with a well known theorem of Boas, whichgives a representation for functionals (see [3, p. 74]). It is of interest to con-sider similar problems with infinite truncations and full moment problems.This will be studied elsewhere.
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