Bivariate Truncated Moment Problems with Algebraic Variety in the - PowerPoint PPT Presentation

Bivariate Truncated Moment Problems with Algebraic Variety in the Nonnegative Quadrant in R 2 (joint work with Sang Hoon Lee and Jasang Yoon) Ra´ ul E. Curto, University of Iowa IWOTA 2019; Lisbon, Portugal Truncated Moment Problems , July 25, 2019 Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

Overview 1 Notation and Preliminaries 2 Multivariable Weighted Shifts 3 The Subnormal Completion Problem 4 One-Step Extensions of Subn. 2 -Var. Weighted Shifts 5 An application Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

Hyponormality and Subnormality L ( H ): algebra of operators on a Hilbert space H T ∈ L ( H ) is normal if T ∗ T = TT ∗ quasinormal if T commutes with T ∗ T subnormal if T = N | H , where N is normal and N H ⊆ H hyponormal if T ∗ T ≥ TT ∗ normal = ⇒ quasinormal = ⇒ subnormal = ⇒ hyponormal Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

For S , T ∈ B ( H ), [ S , T ] := ST − TS . An n -tuple T ≡ ( T 1 , ..., T n ) is (jointly) hyponormal if   [ T ∗ [ T ∗ [ T ∗ 1 , T 1 ] 2 , T 1 ] · · · n , T 1 ]  [ T ∗ [ T ∗ [ T ∗  · · · 1 , T 2 ] 2 , T 2 ] n , T 2 ]   [ T ∗ , T ] := ≥ 0 .   . . . . . .   . . · · · .     [ T ∗ [ T ∗ [ T ∗ 1 , T n ] 2 , T n ] · · · n , T n ] For k ≥ 1, an operator T is k -hyponormal if ( T , ..., T k ) is (jointly) hyponormal, i.e.,  [ T ∗ , T ] [ T ∗ k , T ]  · · · . . ... . .    ≥ 0 . .    [ T ∗ , T k ] [ T ∗ k , T k ] · · · (Bram-Halmos): T subnormal ⇔ T is k -hyponormal for all k ≥ 1. Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

Unilateral Weighted Shifts α ≡ { α k } ∞ k =0 ∈ ℓ ∞ ( Z + ), α k > 0 (all k ≥ 0) W α : ℓ 2 ( Z + ) → ℓ 2 ( Z + ) W α e k := α k e k +1 ( k ≥ 0) When α k = 1 (all k ≥ 0), W α = U + , the (unweighted) unilateral shift In general, W α = U + D α (polar decomposition) � W α � = sup k α k Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

Weighted Shifts and Berger’s Theorem Given a bounded sequence of positive numbers (weights) α ≡ α 0 , α 1 , α 2 , ... , the unilateral weighted shift on ℓ 2 ( Z + ) associated with α is W α e k := α k e k +1 ( k ≥ 0) . The moments of α are given as � � 1 if k = 0 γ k ≡ γ k ( α ) := . α 2 0 · ... · α 2 if k > 0 k − 1 W α is never normal W α is hyponormal ⇔ α k ≤ α k +1 (all k ≥ 0) Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

Berger Measures (Berger; Gellar-Wallen) W α is subnormal if and only if there exists a positive Borel measure ξ on [0 , � W α � 2 ] such that � t k d ξ ( t ) (all k ≥ 0) . γ k = ξ is the Berger measure of W α . The Berger measure of U + is δ 1 . For 0 < a < 1 we let S a := shift ( a , 1 , 1 , ... ). The Berger measure of S a is (1 − a 2 ) δ 0 + a 2 δ 1 . The Berger measure of B + (the Bergman shift) is Lebesgue measure � n +1 n +2 ( n ≥ 0). on the interval [0 , 1]; the weights of B + are α n := Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

Multivariable Weighted Shifts α k , β k ∈ ℓ ∞ ( Z 2 + ) , k ≡ ( k 1 , k 2 ) ∈ Z 2 + := Z + × Z + + ) ∼ ℓ 2 ( Z 2 = ℓ 2 ( Z + ) � ℓ 2 ( Z + ) . We define the 2-variable weighted shift T ≡ ( T 1 , T 2 ) by T 1 e k := α k e k + ε 1 T 2 e k := β k e k + ε 2 , where ε 1 := (1 , 0) and ε 2 := (0 , 1). Clearly, T 1 T 2 = T 2 T 1 ⇐ ⇒ β k + ε 1 α k = α k + ε 2 β k (all k ) . α k 1 ,k 2 +1 ( k 1 , k 2 + 1) ( k 1 + 1 , k 2 + 1) β k 1 ,k 2 β k 1 +1 ,k 2 α k 1 ,k 2 Ra´ ( k 1 , k 2 ) ( k 1 + 1 , k 2 ) ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

. . . . . . . · · · . · · · . · · · (0 , 3) β 02 β 12 β 22 α 02 α 12 α 22 · · · T 2 (0 , 2) β 01 β 11 β 21 · · · α 01 α 11 α 21 (0 , 1) β 00 β 10 β 20 · · · α 00 α 10 α 20 (0 , 0) (1 , 0) (2 , 0) (3 , 0) T 1 Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

We now recall the notion of moment of order k for a commuting pair ( α, β ). Given k ∈ Z 2 + , the moment of ( α, β ) of order k is γ k ≡ γ k ( α, β )  1 if k = 0     α 2 (0 , 0) · ... · α 2 if k 1 ≥ 1 and k 2 = 0   ( k 1 − 1 , 0) := β 2 (0 , 0) · ... · β 2 if k 1 = 0 and k 2 ≥ 1  (0 , k 2 − 1)    α 2 (0 , 0) · ... · α 2 ( k 1 − 1 , 0) · β 2 ( k 1 , 0) · ... · β 2 if k 1 ≥ 1 and k 2 ≥ 1 .   ( k 1 , k 2 − 1) By commutativity, γ k can be computed using any nondecreasing path from (0 , 0) to ( k 1 , k 2 ). ( k 1 , k 2 ) Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

(Jewell-Lubin) k 1 − 1 k 2 − 1 � α 2 � β 2 W α is subnormal ⇔ γ k := ( i , 0) · ( k 1 − 1 , j ) i =0 j =0 � t k 1 1 t k 2 = 2 d µ ( t 1 , t 2 ) (all k ≥ 0 ) . Thus, the study of subnormality for multivariable weighted shifts is intimately connected to multivariable real moment problems. Ra´ ul E. Curto, Univ. of Iowa Bivariate Truncated Moment Problems () Lisbon, July 25, 2019

The Subnormal Completion Problem Consider the following completion problem: Given √ d √ e be = af (commutativity) √ √ b f √ a √ c Figure: The initial family of weights Ω 1 we wish to add infinitely many weights and generate a subnormal 2-variable weighted shift, whose Berger measure interpolates the initial family of weights.

Strategy: The initial family needs to satisfy an obvious necessary condition, that is,     γ 00 γ 01 γ 10 1 a b     M (Ω 1 ) :=  ≡  ≥ 0 . γ 01 γ 02 γ 11 a ac be       γ 10 γ 01 γ 20 b be bd We use tools and techniques from the theory of TMP to solve SCP in the foundational case of six prescribed initial weights; these weights give rise to the linear and quadratic moments. For Ω 1 , the natural necessary conditions for the existence of a subnormal completion are also sufficient.

To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion. In this case, solving the SCP consists of finding a probability measure µ supported on R 2 + such that � y i x j d µ ( x , y ) = γ ij ( i , j ≥ 0 , i + j ≤ 2) . R 2 + To ensure that the support of µ remains in R 2 + we will use the localizing matrices M x (2) and M y (2); each of these matrices will need to be appropriately defined and positive semidefinite.

Definition Given m ≥ 0 and a finite family of positive numbers Ω m ≡ { ( α k , β k ) } | k |≤ m , we say that a 2-variable weighted shift T ≡ ( T 1 , T 2 ) with weight sequences α T k and β T k is a subnormal completion of Ω m if (i) T is subnormal, and (ii) ( α T k , β T k ) = ( α k , β k ) whenever | k | ≤ m . Definition Given m ≥ 0 and Ω m ≡ { ( α k , β k ) } | k |≤ m , we say that ˆ α k , ˆ α k , ˆ Ω m +1 ≡ { (ˆ β k ) } | k |≤ m +1 is an extension of Ω m if (ˆ β k ) = ( α k , β k ) whenever | k | ≤ m . When m = 1, we say that Ω 1 is quadratic.

Recall that a commuting pair ( T 1 , T 2 ) is 2 -hyponormal if the 5-tuple ( T 1 , T 2 , T 2 1 , T 1 T 2 , T 2 2 ) is (jointly) hyponormal. For 2-variable weighted shifts and m = 2, this is equivalent to the condition M u (2) := ( γ u +( m , n )+( p , q ) ) 0 ≤ p + q ≤ 2 , 0 ≤ m + n ≤ 2 ≥ 0 ( all u ∈ Z 2 + ); that is,   γ u γ u +(0 , 1) γ u +(1 , 0) γ u +(0 , 2) γ u +(1 , 1) γ u +(2 , 0)   γ u +(0 , 1) γ u +(0 , 2) γ u +(1 , 1) γ u +(0 , 3) γ u +(1 , 2) γ u +(2 , 1)      γ u +(1 , 0) γ u +(1 , 1) γ u +(2 , 0) γ u +(1 , 2) γ u +(2 , 1) γ u +(3 , 0)    ≥ 0   γ u +(0 , 2) γ u +(0 , 3) γ u +(1 , 2) γ u +(0 , 4) γ u +(1 , 3) γ u +(2 , 2)       γ u +(1 , 1) γ u +(1 , 2) γ u +(2 , 1) γ u +(1 , 3) γ u +(2 , 2) γ u +(3 , 1)     γ u +(2 , 0) γ u +(2 , 1) γ u +(3 , 0) γ u +(2 , 2) γ u +(3 , 1) γ u +(4 , 0) (all u ∈ Z 2 + ). M u (2) detects the 2-hyponormality of ( T 1 , T 2 ).

Recall the initial weight diagram: √ d √ e be = af (commutativity) √ √ b f √ a √ c Figure: The initial family of weights Ω 1 with its associated moment matrix   1 a b   M (Ω 1 ) :=  . a ac be    b be bd

Notation We also need the notion of localizing matrix ; in this case, these are M x (2) and M y (2). Giving a 6 × 6 moment matrix M (2), with rows and columns labeled 1, X , Y , X 2 , XY and Y 2 , the localizing matrix M x (2) is a 3 × 3 matrix obtained from M (2) by selecting the three columns X , X 2 and XY and the top three rows. Similarly, the localizing matrix M y (2) consists of the columns Y , XY and Y 2 and the top three rows.