Solving the truncated moment problem. solves the full moment problem. Jan Stochel IMUJ PREPRINT 1999/31

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1 Solving the truncated moment problem solves the ull moment problem Jan Stochel IMUJ PEPINT 999/3 Abstract. It is shown that the truncated multidimensional moment problem is more general than the ull multidimensional moment problem. 99 Mathematics Subject Classication. Primary 44A60; Secondary 30E05. Key words and phrases. Truncated multidimensional moment problem, multidimensionalmoment problem, weak-star topology. This work was supported at its nal stage by the KBN grant # 2 P03A

2 2 JAN STOCHEL Introduction Denote by d + (resp. C d ) the set o all d-tuples o nonnegative integers (resp. complex numbers). I = ( ; : : :; d ) 2 d + and z = (z ; : : : ; z d ) 2 C d, then we write jj = + + d, z = z zd d and z = (z ; : : : ; z d ). Let be a nonempty closed subset o C d and let c (n) = c(; ) : ; 2 d + ; jj + jj ng be a nite multisequence o complex numbers (n 0). The truncated (multidimensional and complex) -moment problem o order n consists in determining conditions under which there exists a positive Borel measure on C d such that the closed support supp o is contained in and () c(; ) = z z d(z); ; 2 d + ; jj + jj n: A positive Borel measure on C d satisying () is called a representing measure o c (n), while the numbers z z d(z) are customarily called moments o. Let now c = c(; ) : ; 2 d +g be a multisequence o complex numbers. The ull (multidimensional and complex) -moment problem entails determining whether there exists a positive Borel measure on C d such that supp and (2) c(; ) = z z d(z); ; 2 d + : As above, a positive Borel measure on C d satisying (2) is called a representing measure o c. We say that a multisequence o moments is determinate i it has precisely one representing measure. The literature concerning the ull -moment problem (not necessarily complex) is extensive and it is still growing (see or instance [6, 7, 8, 9,, 23, 3, 22, 4, 25, 6, 30] and [8, 26, 27, 5, 28, 5, 29] where semi-algebraic 's are considered). The truncated -moment problem has intensively been studied since early 90's mostly by Curto and ialkow (c. [2, 20, 2,, 2, 0, 3, 4]). In 994. E. Curto asked a question 2 whether the truncated -moment problem is more general than the ull -moment problem (see also [2, p. 5]). In the same year I answered this question in the armative (see [] or the negative answer to the converse question). The present paper contains the proo o this statement (some ideas involved in it may appear in the literature under dierent circumstances; or reader's convenience we include all the details). An auxiliary result Denote by A 0 the dual Banach space o a normed space A and by (A 0 ; A) the weak-star topology on A 0. Given a locally compact Hausdor space X, we write C 0 (X) or the Banach space (equipped with the supremum norm k k X ) o all continuous complex unctions on X which vanish at innity. C c (X) stands or the set o all 2 C 0 (X) such that the closed support o is compact. The set C c (X) is dense in C 0 (X) (c. [24, Theorem 3.7]). We attach to every complex Throughout the whole paper, we tacitly assume that all the unctions under the integral sign are absolutely integrable. In particular, by (), the measure is nite and consequently it is regular (e.g. see [24, Theorem 2.8]). 2 at the Semester on Linear Operators held in the Stean Banach International Mathematical Center (the organizers: J. Janas,. H. Szaraniec and J. emanek).

3 TUNCATED MOMENT POBLEM VESUS MOMENT POBLEM 3 Borel measure on X the unctional b 2 C 0 (X) 0 dened by b() = d; X 2 C 0 (X): Proposition. Let be a nonempty closed subset o C d and let be a nonnegative continuous unction on. Assume that! g is a net o nite positive Borel measures on and is a nite positive Borel measure on such that (i) the net b! g is (C c ( ) 0 ; C c ( ))-convergent to b, (ii) sup d! <. Dene the measures! and on via d! = d! and d = d. Then (iii) ( ) < and the net b! g is (C 0 ( ) 0 ; C 0 ( ))-convergent to b. Moreover, i the set z 2 : (z) rg is compact or some r > 0, then the net b! g is (C 0 ( ) 0 ; C 0 ( ))-convergent to b and d = lim d! or every :! C such that + 2 C 0( ). Proo. Assume that is not compact. S Let n g n= be an increasing sequence o compact subsets o such that = n= n. By [24, Theorem 2.2], or every n there exists n 2 C c ( ) such that 0 n, and n = on n. Applying the Lebesgue monotone convergence theorem, (i) and (ii) we obtain (3) d = lim n! d lim sup n n! = lim sup n! lim n d n d! sup d! < : According to (ii) and (i), the net b! g C 0 ( ) 0 is uniormly bounded and pointwise convergent on a dense subspace C c ( ) o C 0 ( ) to b 2 C 0 ( ) 0. Hence it is (C 0 ( ) 0 ; C 0 ( ))-convergent to b as well 3. Suppose K = z 2 : (z) rg is compact. Let 2 C c ( ) be such that 0, and = on K. Since lim! d! = d, there exists! 0 2 such that d! M = d + or!! 0. This implies that! (K) d! M or!! 0. On the other hand, by (ii), we have! ( n K) r nk d! sup r 2 that the net b! g!!0 d or! 2, so sup!!0! ( ) <. This means is uniormly bounded and pointwise convergent on C c ( ) to b. Consequently, the net b! g is (C 0 ( ) 0 ; C 0 ( ))-convergent to b. Since the net \! +! g is (C 0 ( ) 0 ; C 0 ( ))-convergent to \ +, we get d = or every :! C + d( + ) = lim such that + d(! +! ) = lim d! + 2 C 0( ). This completes the proo. 3 Notice that the -compactness o is not essential in the proo o part (iii) o Proposition ; indeed, the continuity o and the regularity o (c. ootnote ) imply the inner regularity o which, in turn, yields ( ) sup d! < (mimic (3)).

4 4 JAN STOCHEL Corollary 2. Let c(m; n) : m; n 0; m+n 2N?g be a nite sequence o complex numbers (N ) and let be a nonempty closed subset o C. Assume that! g is a net o nite positive Borel measures on and is a nite positive Borel measure on such that (i) the net b! g is (C c ( ) 0 ; C c ( ))-convergent to b, (ii) c(m; n) = zm z n d! (z) or m; n 0 with m + n 2N? and! 2, (iii) sup zn z N d! (z) <. Then (iv) c(m; n) = zm z n d(z) or m; n 0 with m + n 2N?. Proo. Apply Proposition to the unctions (z) = z N z N and (z) = z m z n (z 2 ) with m; n 0 such that m + n 2N? (notice that + 2 C 0( )). igure : Integer lattice points involved in Corollary 2 n 2N 2N? p p p p p p p r (N;N) p p p p p p p p p p p p p p p p p p p p p p p p@ m+n<2n p p p p p p p p p p p p p p p p p p p p p - 0 2N? 2N m We emphasize that Corollary 2 is optimum in a sense. Namely, it may happen that the equality in (ii) holds or all! 2 and or all integer lattice points (m; n) in the convex triangle with vertices (0; 0), (0; 2N) and (2N; 0), though no integer lattice point (m; n) belonging to the edge o joining (0; 2N) and (2N; 0) satises the equality in (iv) (c. igure ). Moreover, the set o all representing measures o a truncated -moment sequence o order 2N may not be (C 0 ( ) 0 ; C 0 ( ))-closed. Example 3 deals with the case N = and = C. P Example 3. Since j= j =, there exists a strictly increasing sequence o positive integers n g n= such that (4) X n+ d u n = j 4 or n and lim m! u m = 4 : P P d Thereore we have a P n = 2 + u n? n+ d j > 0, 2b 2 n = P 2? u n? n+ > 0 j 3=2 d and 2c n = 2? u n + n+ > 0 or n. Because j 3=2 j= <, (4) yields j 3=2 lim n! a n = 3 4 and lim n! b n = lim n! c n = 8. Denote by z the probability Borel measure on C concentrated P at the point z. Set = d ? and d n = a n 0 +b n +c n? + n+ p j 2 j or n. It is a matter o direct verication that d = lim n! dn or every bounded continuous complex unction on C (in particular b n g n= is (C 0 (C ) 0 ; C 0 (C ))-convergent to b). Moreover, or

5 TUNCATED MOMENT POBLEM VESUS MOMENT POBLEM 5 every n, the ollowing conditions hold true z z 0 d n (z) = z 2 z 0 d n (z) = z 0 z 0 d n (z) = z 0 z d n (z) = z z d n (z) = z 2 z 0 d(z) = z z d(z) = z 0 z 0 d(z) = ; z z 0 d(z) = z 0 z d(z) = 0; z 0 z 2 d n (z) = 2 ; z 0 z 2 d(z) = 4 : The main result Theorem 4. Let be a nonempty closed subset o C d and let c(; )g ;2d + be a multisequence o complex numbers. I or every n 0 there exists a positive Borel measure n on such that (i) c(; ) = z z d n (z) or all ; 2 d + with jj + jj n, then there exists a positive Borel measure on such that c(; ) = z z d(z) or all ; 2 d +. Proo. Assume that is not compact (the other case is simpler). Since is locally compact metrizable and separable, one can see { applying [9, Theorem V.6.6] to the one-point compactication 4 o { that 5 (5) C 0 ( ) is a separable Banach space: Given ; 2 d +, we dene the unction ' ; : C d! C ' ; (z) = by z z Q d j= ( + jz jj 2 ) j +j+ ; z 2 C d : Since the unctions z 7! (+jzj 2 ) (m; n 0) are in C m+n+ 0 (C ) and the d-old tensor product o C 0 unctions is again a C 0 unction, we conclude that z m z n (6) It ollows rom (i) that jbn()j ' ; 2 C 0 ( ); ; 2 d + : z 0 z 0 d n (z)kk = c(0; 0)kk or every 2 C 0 ( ), so b n belongs to c(0; 0)B, where B is the closed unit ball in C 0 ( ) 0. By (5), the set c(0; 0)B is weak-star metrizable and weak-star compact (c. [9, Theorems V.3. and V.5.]). Hence there exists a subsequence b kn g n=0 o b n g n=0 which is weak-star convergent to a unctional 2 c(0; 0)B. Notice that i 2 C 0 ( ) and 0, then () = lim n! b kn () 0, so by the iesz representation theorem (c. [24, Theorems 2.4 and 6.9] or [7, x56]) there exists a nite positive Borel measure on such that = b. I n() 2 + is chosen so that k n() 2jj, then, by (i), we have z z d kn (z) = c(; ) or n n() ( 2 d +). Applying Proposition to (z) = z z gives us z z d(z) < or 2 d + and (7) lim n! (z)z z d kn (z) = (z)z z d(z); 2 C 0 ( ); 2 d + : 4 One can show that i X is a locally compact Hausdor space which is not compact, then the one-point compactication o X is metrizable i and only i X is metrizable and separable. 5 The separability o C 0 ( ) can also be deduced rom the Stone-Weierstrass theorem.

6 6 JAN STOCHEL It ollows rom (i), (6) and (7) that c(; ) = lim n! = lim n! = = which completes the proo. ' ; (z) z z d kn (z) ' ; (z) dy j= dy j= ( + jz j j 2 ) j +j+ d kn (z) ( + jz j j 2 ) j+j + d(z) z z d(z); ; 2 d + ; emark 5. In act, we have proved that i or every n 0 there exists a positive Borel measure n on satisying condition (i) o Theorem 4, then there exists a positive Borel measure on with all its moments nite and a subsequence kn g n=0 o ng n=0 such that d kn g n=0 is (C 0( ) 0 ; C 0 ( ))-convergent to c or 2 d + ; here d (z) = z z d(z) or = ; n. This, in turn, has enabled us to show that is a representing measure o c(; )g ;2d +. It is clear that all representing measures o c(; )g ;2d + can be obtained by way o this limit procedure. In case is unique we can prove more. Theorem 6. Let c(; )g ;2d +,, n and be as in Theorem 4. I, moreover, the multisequence c(; )g ;2d + is determinate, then the sequence c n g n=0 is (C 0 ( ) 0 ; C 0 ( ))-convergent to c or every 2 d +. Proo. Analysis similar to that in the proo o Theorem 4 (c. emark 5) shows that or every subsequence kn g n=0 o n g n=0 there exists a subsequence kln g n=0 o kn g n=0 such that d kln g n=0 is (C 0 ( ) 0 ; C 0 ( ))-convergent to c or 2 d + (use the act that the representing measure is unique). Hence the general topological characterization o convergent sequences yields the conclusion. It is worth while to notice that i the multisequence o moments c(; )g ;2d + is not determinate, then b n g n=0 may not be (C 0 ( ) 0 ; C 0 ( ))-convergent. Indeed, i 6= are two representing measures o c(; )g ;2d + and the sequence n g n=0 is dened by 2k = and 2k+ = or k 0, then n g n=0 satises condition (i) o Theorem 4 but b n g n=0 is not (C 0 ( ) 0 ; C 0 ( ))-convergent (indeed, otherwise it must be b = b which, by the iesz representation theorem (see also ootnote ), gives us =, a contradiction). Theorems 4 and 6 can easily be adopted to the context o other classical moment problems, in particular to the multidimensional Hamburger moment problem. Acknowledgement. The author would like to thank Proessor. E. Curto or his encouragement to write this paper. eerences [] N. I. Akhiezer, The classical moment problem, Haner Publ. Co., New York, 965. [2] T. Ando, Truncated moment problems or operators, Acta Sci. Math. (Szeged) 3 (970), 39{333.

7 TUNCATED MOMENT POBLEM VESUS MOMENT POBLEM 7 [3] A. Atzmon, A moment problem or positive measures on the unit disc, Pacic J. Math. 59 (975), 37{325. [4] C. Berg, J. P.. Christensen, C. U. Jensen, A remark on the multidimensional moment problem, Math. Ann. 243 (979), 63{69. [5] T. M. Bisgaard, Extensions o Hamburger's theorem, Semigroup orum 57 (998), 397{429. [6] T. Carleman, Sur le probleme des moments, C.. Acad. Sci. Paris 74 (922), 680{682. [7] T. Carleman, Sur les equations integrales singulieres a noyau reel et symetrique, Uppsala Universitets Arsskrit, Uppsala, 923. [8] G. Cassier, Probleme des moments sur un compact de n et decomposition de polynomes a plusieurs variables, J. unct. Anal. 58 (984), 254{266. [9] J. B. Conway, A course in unctional analysis, Springer-Verlag, New York, 985. [0]. E. Curto, An operator-theoretic approach to truncated moment problems, in: Linear operators, J. Janas,. H. Szaraniec and J. emanek (eds.), Banach Center Publications 38, Warszawa, 997, 75{04. []. E. Curto, L. A. ialkow, ecursiveness, positivity, and truncated moment problems, Houston J. Math. 7 (99), 603{635. [2]. E. Curto, L. A. ialkow, Solution o the truncated complex moment problem or at data, Mem. Amer. Math. Soc. 9 (996). [3]. E. Curto, L. A. ialkow, lat extensions o positive moment matrices: ecursively generated relations, to appear in Mem. Amer. Math. Soc. [4]. E. Curto, L. A. ialkow, The truncated complex K-moment problem, preprint 998. [5]. E. Curto, M. Putinar, Nearly subnormal operators and moment problems, J. unct. Anal. 5 (993), 480{497. [6] B. uglede, The multidimensional moment problem, Expo. Math. (983), 47{65. [7] P.. Halmos, Measure theory, Springer-Verlag, New York, 974. [8] E. K. Haviland, On the momentum problem or distributions in more than one dimension, Amer. J. Math. 57 (935), [9] E. K. Haviland, On the momentum problem or distributions in more than one dimension, II, Amer. J. Math. 58 (936), [20] M. G. Krein, A. Nudel'man, The Markov moment problem and extremal problems, Transl. Math. Monographs 50, Amer. Math. Soc., Providence, I, 977. [2] H. Landau, Classical background o the moment problem, in: Moments in Mathematics, Proc. Sympos. Appl. Math. 37 (987), -5. [22] P. H. Maserick, Moments o measures on convex bodies, Pacic J. Math. 68 (977), 35{52. [23] A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6 (967), 79{9. [24] W. udin, eal and Complex Analysis, McGraw-Hill, New York, 974. [25] K. Schmudgen, An example o a positive polynomial which is not a sum o squares o polynomials. A positive, but not strongly positive unctional, Math. Nachr. 88 (979), 385{390. [26] K. Schmudgen, The K-moment problem or compact semi-algebraic sets, Math. Ann. 289 (99), 203{206. [27] J. Stochel, Moment unctions on real algebraic sets, Ark. Mat. 30 (992), 33{48. [28] J. Stochel,. H. Szaraniec, Algebraic operators and moments on algebraic sets, Portugal. Math. 5 (994), [29] J. Stochel,. H. Szaraniec, The complex moment problem and subnormality: a polar decomposition approach, J. unct. Anal. 59 (998), [30]. H. Szaraniec, Moments on compact sets, in: Prediction Theory and Harmonic Analysis, V. Mandrekar and H. Salehi (eds.), North-Holland, Amsterdam, 983, 379{385. Instytut Matematyki, Uniwersytet Jagiellonski, ul. eymonta 4, PL Krakow address: stochel@im.uj.edu.pl

arxiv: v1 [math.fa] 11 Jul 2012

arxiv: v1 [math.fa] 11 Jul 2012 arxiv:1207.2638v1 [math.fa] 11 Jul 2012 A multiplicative property characterizes quasinormal composition operators in L 2 -spaces Piotr Budzyński, Zenon Jan Jab loński, Il Bong Jung, and Jan Stochel Abstract.