Multivariable Matrix-valued moment problems David P. Kimsey The - PowerPoint PPT Presentation

Multivariable Matrix-valued moment problems David P. Kimsey The Weizmann Institute of Science Operator theory and operator algebras 2012 May 22, 2012 Partially supported by NSF grant DMS-0901628

The classical moment problem Full moment problem: (i) Given t s m ✉ m P N 0 we wish to determine whether or not there exists σ P m � ♣ R q such that ➺ x m d σ ♣ x q , m P N 0 : ✏ t 0 , 1 , . . . ✉ . s m ✏ (1) R (ii) Describe all σ which obey (1). The cases when supp σ ❸ r 0 , ✽q , R , and r c , d s are due to Stieltjes, Hamburger, and Hausdorff, respectively. Observation: If σ satisfies (1) then ➺ ➳ n ➳ n x a � b d σ ♣ x q z a ¯ z b s a � b ✏ z a ¯ z b R a , b ✏ 0 a , b ✏ 0 ✞ ✞ ➺ 2 ✞ ✞ ➳ n ✞ ✞ z a x a ✏ ✞ ✞ d σ ♣ x q ➙ 0 . ✞ ✞ R a ✏ 0

Hamburger’s theorem Theorem (Hamburger’s theorem, 1921): t s m ✉ m P N 0 has a representing measure if and only if ☎ ☞ s 0 ☎ ☎ ☎ s n ➳ n ✝ ✍ . . ... ñ ♣ s a � b q n . . z a ¯ z b s a � b ➙ 0 ð a , b ✏ 0 : ✏ ✌ ➞ 0 , ✆ . . a , b ✏ 0 s n ☎ ☎ ☎ s 2 n for all finite subsets t z 1 , . . . , z n ✉ ⑨ C . Remarks on proofs of Hamburger’s theorem: ➓ Hamburger’s original proof is around 150 pages. ➓ The operator theory proof is much shorter. [Krein, 1949] generalized this result to the case when one is a given t S m ✉ m P N 0 .

The full multidimensional K -moment problem Full K -moment problem on R d : Given t s m ✉ m P N d 0 and K ❸ R d , we wish to determine whether or not there exists σ such that ➺ ➺ ➺ R d x m 1 ☎ ☎ ☎ x m d R d x m d σ ♣ x q : ✏ d d σ ♣ x 1 , . . . , x d q , m P N d s m ✏ ☎ ☎ ☎ 0 . 1 (2) and supp σ ❸ K . (3) Observation: If t s m ✉ m P N d 0 has a representing measure then ➳ z m ¯ m ➙ 0 , n P N 0 . (4) z r m s m � r 0 ↕⑤ m ⑤ , ⑤ r m ⑤↕ n Caveat: There do exist sequences t s m ✉ m P N d 0 which satisfy (4) yet do not satisfy (2).

Solutions to the full multidimensional K -moment problem ➓ A solution to the full K -moment problem on R in [M. Riesz, 1923]. ➓ A d -variable ( d → 1) generalization was achieved in [Haviland, 1936]. ➓ When K is compact and semi-algebraic, [Schm¨ udgen, 1991] has a solution to the full K -moment problem based on the approach in [Haviland, 1936]. ➓ Subsequently, [Putinar & Vasilescu, 1999] improved upon this approach.

Truncated Hamburger moment problem Problem: Given a real-valued sequence t s k ✉ 0 ↕ k ↕ n , we wish to determine necessary and sufficient conditions on t s k ✉ 0 ↕ k ↕ n so that there exists σ so that ➺ x k d σ ♣ x q , 0 ↕ k ↕ n . s k ✏ R Remarks: When n ✏ 2 m then a solution exists if and only if we can choose s 2 m � 1 and s 2 m � 2 such that ☎ ☞ s 0 ☎ ☎ ☎ s m ✁ 1 s m s m � 1 ✝ . . . . ✍ ... . . . . ✝ ✍ . . . . ✝ ✍ ✝ ✍ ➞ 0 . s m ✁ 1 ☎ ☎ ☎ s 2 m ✁ 2 s 2 m ✁ 1 s 2 m ✝ ✍ ✆ ✌ s m ☎ ☎ ☎ s 2 m ✁ 1 s 2 m s 2 m � 1 ☎ ☎ ☎ s m � 1 s 2 m s 2 m � 1 s 2 m � 2 In this case, one can find σ ✏ ➦ r j ✏ 1 ρ j δ x j , where � ✟ m r ✏ rank i , j ✏ 0 . s i � j

Multivariable THMPs Truncated Hamburger moment problem (even total degree): Given t s m ✉ 0 ↕⑤ m ⑤↕ 2 n we wish to determine whether or not there exists σ such that ➺ ➺ ➺ R d x 1 m 1 ☎ ☎ ☎ x d m d d σ ♣ x 1 , . . . , x d q . R d x m d σ ♣ x q : ✏ s m ✏ ☎ ☎ ☎ Given t s m ✉ 0 ↕⑤ m ⑤↕ 2 n we can construct the following moment matrix: � ✟ M ♣ n q : ✏ s m � r m ⑤↕ n . m 0 ↕⑤ m ⑤ , ⑤ r When d ✏ 2 and n ✏ 1, ☎ ☞ s 00 s 01 s 10 ✆ ✌ M ♣ 1 q : ✏ s 01 s 02 s 11 s 10 s 11 s 20

Flat extension theory of Curto and Fialkow Given t s m ✉ 0 ↕⑤ m ⑤↕ 2 n , we call M ♣ n � 1 q a flat extension of M ♣ n q when there exist new data t s r m ✉ 2 n � 1 ↕⑤ r m ⑤↕ 2 n � 2 such that 1. M ♣ n � 1 q ✏ ♣ s m � r m q 0 ↕⑤ m ⑤ , ⑤ r m ⑤↕ n � 1 ➞ 0 2. rank M ♣ n � 1 q ✏ rank M ♣ n q . For instance, when we are given t s 00 , s 01 , s 10 , s 02 , s 11 s 20 ✉ so that M ♣ 1 q ➞ 0 , we wish to find t s 03 , s 12 , s 21 , s 30 , s 04 , s 13 , s 22 , s 31 , s 40 ✉ so that ☎ s 00 s 01 s 10 s 02 s 11 s 20 ☞ s 01 s 02 s 11 s 03 s 12 s 21 ✝ ✍ ✝ ✍ s 10 s 11 s 20 s 12 s 21 s 30 ✝ ✍ M ♣ 2 q : ✏ ➞ 0 and rank M ♣ 1 q ✏ rank M ♣ 2 q . ✝ ✍ s 02 s 03 s 12 s 04 s 13 s 22 ✝ ✍ ✝ ✍ s 11 s 12 s 21 s 13 s 22 s 31 ✆ ✌ s 20 s 21 s 30 s 22 s 31 s 40 Theorem (Curto and Fialkow, 1996): The even total degree HMP has a solution if and only if M ♣ n q eventually admits a flat extension.

Truncated matrix-valued Hamburger moment problem Problem: Given t S m ✉ 0 ↕ m ↕ n , we wish to find Σ : ✏ ♣ σ jk q p j , k ✏ 1 such that ➺ �➩ ✟ p x m d Σ ♣ x q : ✏ R x m d σ jk ♣ x q S m ✏ j , k ✏ 1 , 0 ↕ m ↕ n . R This problem has been studied by ➓ [Dym, 1989]; ➓ [Bolotnikov, 1996]; ➓ [Bakonyi & Woerdeman, 2011]. Under analogous conditions, one can find σ ✏ ➦ r j ✏ 1 T j δ x j , where � ✟ n ➦ r j ✏ 1 rank T j ✏ rank S i � j i , j ✏ 0 .

An illustrative example Given the sequence t s ♣ k ,ℓ q ✉ 0 ↕ k � ℓ ↕ 3 , suppose ☎ ☞ ☎ ☞ 3 1 2 s 00 s 01 s 10 ✆ ✌ ✏ ✆ ✌ → 0 , Φ ✏ s 01 s 02 s 11 1 1 1 2 1 2 s 10 s 11 s 20 ☎ ☞ ☎ ☞ s 10 s 11 s 20 2 1 2 ✆ ✌ ✏ ✆ ✌ , Φ 1 ✏ 1 1 1 s 11 s 12 s 21 s 20 s 21 s 30 2 1 2 and ☎ ☞ ☎ ☞ s 01 s 02 s 11 1 1 1 ✆ ✌ ✏ ✆ ✌ . Φ 2 ✏ s 02 s 03 s 12 1 1 1 1 1 1 s 11 s 12 s 21

An illustrative example continued Realize that ☎ ☞ 0 0 0 ✆ ✌ Θ 1 ✏ Φ ✁ 1 Φ 1 ✏ 0 1 0 1 0 1 and ☎ ☞ 0 0 0 Θ 2 ✏ Φ ✁ 1 Φ 2 ✏ ✆ ✌ 1 1 1 0 0 0 commute. Also the eigenvalues of Θ 1 and Θ 2 are t 0 , 1 , 1 ✉ and t 0 , 0 , 1 ✉ , respectively. Put ♣ x 1 , y 1 q ✏ ♣ 0 , 0 q , ♣ x 2 , y 2 q ✏ ♣ 1 , 0 q and ♣ x 3 , y 3 q ✏ ♣ 1 , 1 q . Then σ ✏ ➦ 3 j ✏ 1 δ ♣ x j , y j q is a solution!

Indexing sets ➓ We say a finite set Γ ⑨ N d 0 is a lattice set when for all γ P Γ there exist γ 1 ✏ 0 d , γ 2 , . . . , γ k P Γ and j 1 , . . . , j k P t 1 , . . . , d ✉ so that γ 2 ✏ γ � e j 1 , . . . , γ ✏ γ k � e j k , where k ✏ ⑤ γ ⑤ . ➓ We say a finite set Γ ⑨ N d 0 is lower inclusive when for any m ✏ ♣ m 1 , . . . , m d q P N d 0 and γ ✏ ♣ g 1 , . . . , g d q P Γ with m j ↕ g j , 1 ↕ j ↕ d , we have that m P Γ. ➓ Note that t♣ 0 , 0 q , ♣ 0 , 1 q , ♣ 1 , 1 q✉ ⑨ N 2 0 is a lattice set but not a lower inclusive set. Also t♣ 0 , 0 q , ♣ 1 , 1 q✉ ⑨ N 2 0 is not a lattice set.

Truncated matrix-valued K -moment problem on R d Problem: Let K ❸ R d , Γ ⑨ N d 0 be a lattice set and t S γ ✉ γ P Γ be given. We wish to determine whether or not there exists Σ so that ➩ R d x γ d Σ ♣ x q , for all γ P Γ; (i) S γ ✏ (ii) supp Σ ❸ K . ➓ When Γ ✏ t m P N d 0 : 0 ↕ ⑤ m ⑤ ↕ 2 n ✉ and t S γ ✉ γ P Γ is scalar-valued then [Curto and Fialkow, 2000 & 2005] analyzed the K -moment problem on R d and its multidimensional complex analogue. ➓ [Stochel, 2001] showed that the full K -moment problem on R d or C d has a solution if and only if the truncated K -moment problem, where the moments given have indices which are of total degree at most 2n, has a solution for every n P N .

Construction of moment matrices from given data ➓ Given a lattice set Λ ⑨ N d 0 , we define Λ � Λ ✏ t λ � r λ : λ, r λ P Λ ✉ and Λ � Λ � e j ✏ t λ � r λ � e j : λ, r λ P Λ ✉ , 1 ↕ j ↕ d . ➓ Put Γ ✏ ♣ Λ � Λ q ❨ ♣ Λ � Λ � e 1 q ❨ ☎ ☎ ☎ ❨ ♣ Λ � Λ � e d q , which will serve as the indexing set for t S γ ✉ γ P Γ . ➓ Introduce Φ , Φ 1 , . . . , Φ d as follows. Index the rows and columns of Φ by Λ. Let the entry in the row indexed by λ and the column indexed by r λ be given by S λ � r λ . That is, Φ ✏ ♣ S λ � r λ q λ, r λ P Λ ➞ 0 . ➓ Similarly, index the rows and columns of Φ j by Λ. That is, Φ j ✏ ♣ S λ � r λ � e j q λ, r λ P Λ , 1 ↕ j ↕ d .

An example of the construction Let t S γ ✉ γ P Γ , where Γ ✏ t γ P N d 0 : 0 ↕ ⑤ γ ⑤ ↕ 3 ✉ . Put Λ ✏ t♣ 0 , 0 q , ♣ 0 , 1 q , ♣ 1 , 0 q✉ and so then we get the following matrices: ☎ ☞ S 00 S 01 S 10 ✆ ✌ ➞ 0 , Φ ✏ ♣ S λ � ˜ λ q λ, ˜ λ P Λ ✏ S 01 S 02 S 11 S 10 S 11 S 20 ☎ ☞ S 10 S 11 S 20 ✆ ✌ , Φ 1 ✏ ♣ S λ � ˜ λ � e 1 q λ, ˜ λ P Λ ✏ S 11 S 12 S 21 S 20 S 21 S 30 and ☎ ☞ S 01 S 02 S 11 ✆ ✌ . Φ 2 ✏ ♣ S λ � ˜ λ � e 2 q λ, ˜ λ P Λ ✏ S 02 S 03 S 12 S 11 S 12 S 21