Solution of the Reconstruction-of-the-Measure Problem for Canonical Invariant Subspaces
aa r X i v : . [ m a t h . F A ] S e p SOLUTION OF THE RECONSTRUCTION-OF-THE-MEASUREPROBLEMFOR CANONICAL INVARIANT SUBSPACES
RA ´UL E. CURTO, SANG HOON LEE, AND JASANG YOON
Abstract.
We study the Reconstruction-of-the-Measure Problem (ROMP) for com-muting 2-variable weighted shifts W ( α,β ) , when the initial data are given as the Bergermeasure of the restriction of W ( α,β ) to a canonical invariant subspace, together with themarginal measures for the 0–th row and 0–th column in the weight diagram for W ( α,β ) .We prove that the natural necessary conditions are indeed sufficient. When the initialdata correspond to a soluble problem, we give a concrete formula for the Berger measureof W ( α,β ) . Our strategy is to build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions.This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspacesof ℓ ( Z ). Contents
1. Introduction 12. Statement of Main Results 43. Notation and Preliminaries 54. Generalized One-step Reconstruction-of-the-Measure Problem 75. Two-Step Extensions 86. ROMP for Canonical Invariant Subspaces 12References 161.
Introduction
Let H be a complex Hilbert space and let B ( H ) denote the algebra of bounded linearoperators on H . We say that T ∈ B ( H ) is normal if T ∗ T = T T ∗ , subnormal if T = N | H ,where N is normal and N ( H ) ⊆ H , and hyponormal if T ∗ T ≥ T T ∗ . These notions Mathematics Subject Classification.
Primary 47B20, 47B37, 47A13, 28A50; Secondary 44A60,47-04, 47A20.
Key words and phrases. two-step extension, 2-variable weighted shifts, subnormal pair, Berger mea-sure, canonical invariant subspace.The second author of this paper was partially supported by NRF (Korea) grant No.2020R1A2C1A0100584611.The third named author was partially supported by a grant from the University of Texas System andthe Consejo Nacional de Ciencia y Tecnolog´ıa de M´exico (CONACYT). xtend to commuting n -tuples of Hilbert space operators T ≡ ( T , . . . , T n ); for instance, T is normal if T ∗ i T j = T j T ∗ i for all i, j = 1 , . . . , n , and T is subnormal if T = N | H , where N is normal on a larger Hilbert space K and N i H ⊆ H for all i = 1 , . . . , n .In this paper we will focus on the case of H = ℓ ( Z ) and T a 2-variable weightedshift W ( α,β ) ≡ ( T , T ). As is well known, the subnormality of such pairs is characterizedby the existence of a representing measure (known as the Berger measure of W ( α,β ) ) forthe family of moments of the pair of double-indexed sequences ( α, β ). We will solve theso-called Reconstruction-of-the-Measure Problem (ROMP) when W ( α,β ) admits partialBerger measures; more specifically, when the restriction of W ( α,β ) to a canonical invariantsubspace of ℓ ( Z ) is subnormal. That restriction generates a finite family F of (localized)Berger measures µ , . . . , µ p which satisfy natural compatibility conditions. Two other 1-variable Berger measures, σ and τ (corresponding to the 0–th row and 0–th column in theweight diagram for W ( α,β ) ), are assumed to be part of the initial data. ROMP asks fornecessary and sufficient conditions on the members of F for the subnormality of W ( α,β ) .When soluble, one must also reconstruct the Berger measure of W ( α,β ) from F and themarginal measures σ and τ . Our main result (Theorem 6.5) gives a complete solution ofROMP in the case of canonical invariant subspaces.Besides their relevance for the construction of examples and counterexamples in Hilbertspace operator theory, weighted shifts can also be used to detect properties such as sub-normality, via the Lambert-Lubin Criterion ( [10], [12]): A commuting pair ( T , T ) ofinjective operators acting on a Hilbert space H admits a commuting normal extension ifand only if for every nonzero vector x ∈ H , the 2-variable weighted shift with weights α ( i,j ) := k T i +11 T j x kk T i T j x k and β ( i,j ) := k T i T j +12 x kk T i T j x k has a normal extension.For ω ≡ { ω n } ∞ n =0 ∈ ℓ ∞ ( Z + ) a bounded sequence of positive real numbers (called weights ), let W ω : ℓ ( Z + ) → ℓ ( Z + ) be the associated unilateral weighted shift, defined by W ω e n := ω n e n +1 (all n ≥ { e n } ∞ n =0 is the canonical orthonormal basis in ℓ ( Z + ) . The moments of ω are given as γ k ≡ γ k ( ω ) := (cid:26)
1, if k = 0 ω · · · ω k − , if k > . It is easy to see that W ω ≡ shift( ω , ω , · · · ) is never normal, and that it is hyponormalif and only if ω ≤ ω ≤ · · · . We recall a well known characterization of subnormalityfor single variable weighted shifts, due to C. Berger (cf. [1, II.6.10]) and independentlyestablished by R. Gellar and L.J. Wallen [8]: W ω is subnormal if and only if there existsa probability measure σ supported in [0 , k W ω k ] (called the Berger measure of W ω ) suchthat for k ≥ γ k ( ω ) ≡ γ k ( σ ) = Z t k dσ ( t ) . (1.1)Similarly, consider double-indexed positive bounded sequences α ≡ { α ( k ,k ) } , β ≡{ β ( k ,k ) } ∈ ℓ ∞ ( Z ), ( k , k ) ∈ Z := Z + × Z + and let ℓ ( Z ) be the Hilbert spaceof square-summable complex sequences indexed by Z ; in a canonical way, ℓ ( Z ) isisometrically isomorphic to ℓ ( Z + ) N ℓ ( Z + ). We define the 2-variable weighted shift ( α,β ) ≡ ( T , T ) by T e ( k ,k ) := α ( k ,k ) e ( k ,k )+ ε T e ( k ,k ) := β ( k ,k ) e ( k ,k )+ ε , where ε := (1 ,
0) and ε := (0 ,
1) (see Figure 1). Clearly, T commutes with T if andonly if β ( k ,k )+ ε α ( k ,k ) = α ( k ,k )+ ε β ( k ,k ) (cid:0) all ( k , k ) ∈ Z (cid:1) . (1.2)In an entirely similar way one can define multivariable weighted shifts. (0 ,
0) (1 ,
0) (2 , α α α α α α α α α T T (0 , , β β β β β β β β ( k , k )(0 , Figure 1. (left) Weight diagram of a generic 2-variable weighted shift W ( α,β ) ≡ ( T , T ); (right) nondecreasing path used to compute the moment γ ( k ,k ) .For a commuting 2-variable weighted shift W ( α,β ) , and for k , k Z + , the moment of( α, β ) of order ( k , k ) is defined by γ ( k ,k ) ( α, β ) :=
1, if ( k , k ) = (0 , α , · · · α k − , , if k ≥ k = 0 β , · · · β ,k − , if k = 0 and k ≥ α , · · · α k − , · β k , · · · β k ,k − , if k ≥ k ≥
1. (1.3)We remark that, due to the commutativity condition (1.2), γ ( k ,k ) ≡ γ ( k ,k ) ( α, β ) can becomputed using any nondecreasing path from (0 ,
0) to ( k , k ) (see Figure 1(right)).We also recall a well known characterization of subnormality for multivariable weightedshifts T ≡ ( T , · · · , T n ) [9]; for simplicity, we state it in the case n = 2: W ( α,β ) ≡ ( T , T )is subnormal if and only if there is a probability measure µ defined on the rectangle R = [0 , a ] × [0 , a ], where a i := k T i k , such that γ ( k ,k ) = Z R s k t k dµ ( s, t ) , for all ( k , k ) ∈ Z . . Statement of Main Results
For k ∈ Z , let L k := W { e k + p : p ∈ Z } ⊆ ℓ ( Z ). Of special significance in this paperis the closed subspace L ( k,ℓ ) := L (0 ,ℓ ) T L ( k, ; a visual representation of L ( k,ℓ ) appears inFigure 2. A closed subspace R of ℓ ( Z ) is called canonical if R = W { k ∈ Z : e k ∈R} L k . (0 ,
0) ( k,
0) ( k + 1 ,
0) ( k + 2 , · · · · · ·· · · (0 , ℓ )(0 , ℓ + 1)(0 , ℓ + 2) ...... ... L ( k, L (0 ,ℓ ) L (0 ,ℓ ) T L ( k, Figure 2.
The subspaces L (0 ,ℓ ) , L ( k, and L ( k,ℓ ) := L (0 ,ℓ ) T L ( k, Our first main result is Theorem 4.1, which extends the main result in [4] to the caseof arbitrary subspaces L (0 ,ℓ ) and L ( k, . That is, given Berger measures µ (0 ,ℓ ) and µ ( k, satisfying the compatibility condition γ (0 ,ℓ ) s k dµ (0 ,ℓ ) ( s, t ) = γ ( k, t ℓ dµ ( k, ( s, t ) , (2.1)then the subnormality of W ( α,β ) is guaranteed once the natural necessary conditions givenbelow hold:(i) t ℓ ∈ L ( µ (0 ,ℓ ) );(ii) s k ∈ L ( µ ( k, );(iii) γ ( k, (cid:13)(cid:13) s k (cid:13)(cid:13) µ ( k, = λγ (0 ,ℓ ) (cid:13)(cid:13) s k (cid:13)(cid:13) µ ( k, ≤
1; and(iv) γ (0 ,ℓ ) { R Y d µ (0 ,ℓ ) ( s,t ) t ℓ + λ (cid:13)(cid:13) s k (cid:13)(cid:13) µ ( k, dδ ( s ) − λs k R Y dµ ( k, ( s, t ) } ≤ dδ ( s ).Next, let W ( α,β ) be a 2-variable weighted shift such that (i) W ( α,β ) | L ( k,ℓ ) is subnormalwith 2–variable Berger measure ν , (ii) W ( α,β ) restricted to the 0–th row is subnormal with1–variable Berger measure σ , and (iii) W ( α,β ) restricted to the 0–column is subnormal with1–variable Berger measure τ . ur second main result is Theorem 5.2: given the initial data ν , σ and τ , the naturalnecessary conditions for the subnormality of W ( α,β ) are also sufficient. The proof uses anew idea together with appropriate generalizations of results in [3–6].Our third main result provides a complete solution of the reconstruction-of-the-measureproblem for canonical invariant subspaces; we do this in Section 6. Briefly stated, givena finite family of Berger measures ν k associated with a canonical invariant subspace S ,satisfying natural compatibility conditions of the type described in (2.1), and given 1-variable marginal Berger measures σ and τ , the solubility of ROMP for W ( α,β ) is fullydetermined by the solubility of the localized version of ROMP on the pair of subspaces L P and L q , where p and q are two arbitrary lattice points in the so-called foundationset of the subspace S . Once again, the natural necessary conditions for the solubility ofROMP are sufficient. 3. Notation and Preliminaries
For the reader’s convenience, in this section, we gather several well known auxiliaryresults which are needed for the proofs of the main results in this article. We recall someauxiliary facts needed for the proof of our main results. In the single variable case, if W ω is subnormal with Berger measure σ , and if we let L j := W { e n : n ≥ j } denote theinvariant subspace obtained by removing the first j vectors in the canonical orthonormalbasis of ℓ ( Z + ), then the Berger measure of W ω | L j is dσ j ( s ) = s j γ j dσ ( s ) ( j = 1 , , . . . ) . (3.1) Lemma 3.1. (Subnormal backward extension of a -variable weighted shift) (cf. [2, Propo-sition 8], [5, Proposition 1.5]) Let W ω | L be subnormal, with Berger measure µ L . Then W ω is subnormal (with Berger measure µ ) if and only if the following conditions hold:(i) s ∈ L ( µ L ) (ii) ω ≤ (cid:16)(cid:13)(cid:13) s (cid:13)(cid:13) L ( µ L ) (cid:17) − . In this case, dµ ( s ) = ω s dµ L ( s ) + (cid:16) − ω (cid:13)(cid:13) s (cid:13)(cid:13) L ( µ L ) (cid:17) dδ ( s ) , where δ denotes the point-mass probability measure with support the singleton set { } . In particular, W ω is neversubnormal when µ L ( { } ) > . To check the subnormality of 2-variable weighted shifts, we need to introduce somedefinitions.
Definition 3.2. ( [5], [6], [13]) (i) Let µ and ν be two positive measures on X ≡ R + . We say that µ ≤ ν on X, if µ ( E ) ≤ ν ( E ) for all Borel subset E ⊆ X ; equivalently, µ ≤ ν if and only if R f dµ ≤ R f dν for all f ∈ C ( X ) such that f ≥ on X .(ii) Let µ be a probability measure on X × Y, with Y ≡ R + , and assume that s ∈ L ( µ ) (resp t ∈ L ( µ ) ). The extremal measure µ ext ; s (resp. µ ext ; t ) (which is also a probability easure) on X × Y is given by dµ ext ; s ( s, t ) := s k s k L µ ) dµ ( s, t ) (resp. dµ ext ; t ( s, t ) := t k t k L µ ) dµ ( s, t ) ).(iii) Given a measure µ on X × Y, the marginal measure µ X is given by µ X := µ ◦ π − X ,where π X : X × Y → X is the canonical projection onto X . Thus µ X ( E ) = µ ( E × Y ) ,for every E ⊆ X ; equivalently, dµ X ( s ) = R Y dµ ( s, t ) . (Observe that µ X is a probabilitymeasure whenever µ is.) The following result is a very special case of the Reconstruction-of-the-Measure Prob-lem.
Lemma 3.3. (Subnormal backward extension of a -variable weighted shift [5]) Considerthe -variable weighted shift whose weight diagram is given in Figure 1(i). Assume that T | L (0 , is subnormal, with associated measure µ (0 , , and that W ≡ shif t ( α , α , · · · ) issubnormal with associated measure σ . Then T is subnormal if and only if(i) t ∈ L ( µ (0 , ) ;(ii) β ≤ ( (cid:13)(cid:13) t (cid:13)(cid:13) L ( µ (0 , ) ) − ;(iii) β (cid:13)(cid:13) t (cid:13)(cid:13) L ( µ (0 , ) (( µ (0 , ) ext ; t ) X ≤ σ .Moreover, if β (cid:13)(cid:13) t (cid:13)(cid:13) L ( µ (0 , ) = 1 then (( µ (0 , ) ext ; t ) X = σ . In the case when T issubnormal, the Berger measure µ of T is given by dµ ( s, t ) = β (cid:13)(cid:13)(cid:13)(cid:13) t (cid:13)(cid:13)(cid:13)(cid:13) L ( µ (0 , ) d ( µ (0 , ) ext ; t ( s, t )+[ dσ ( s ) − β (cid:13)(cid:13)(cid:13)(cid:13) t (cid:13)(cid:13)(cid:13)(cid:13) L ( µ (0 , ) d (( µ (0 , ) ext ; t ) X ( s )] dδ ( t ) . (3.2) Observation 3.4.
Since the extremal measure ( µ (0 , ) ext ; t involves normalization (to ob-tain again a probability measure), it is easy to see that the expression (cid:13)(cid:13) t (cid:13)(cid:13) L ( µ (0 , ) ( µ (0 , ) ext ; t is equivalent to the expression µ (0 , t ; we will often use the latter expression. For example,Lemma 3.3(iii) can be rewritten as β ( µ (0 , t ) X ≤ σ .Recall that given two positive measures µ and ν on R + , µ is said to be absolutelycontinuous with respect to ν (in symbols, µ ≪ ν ) if ν ( E ) = 0 ⇒ µ ( E ) = 0 for every Borelset E on R + .Given a 2-variable weighted shift T = W ( α,β ) such that T T = T T and T i is subnormal( i = 1 , k , k ≥
0, we let W k := shift( α (0 ,k ) , α (1 ,k ) , · · · ) (3.3)be the k -th horizontal slice of T with associated Berger measure ξ α k ; similarly we let V k := shift( β ( k , , β ( k , , · · · ) (3.4)be the k -th vertical slice of T with associated Berger measure η β k . (Clearly, W and V are the unilateral weighted shifts associated with the 0-th row and 0-column in the eight diagram for T , resp.). Lemma 3.5. ( [6, Theorem 3.1]) Let µ be the Berger measure of a subnormal -variableweighted shift, and let σ (resp. τ ) be the Berger measure of the associated -th horizontal(resp. vertical) -variable shift ( α , α , α , · · · ) (resp. shift ( β , β , β , · · · )) . Then σ = µ X (resp. τ = µ Y ). Lemma 3.6. ( [11]) If µ is a positive regular Borel measure defined on Z := X × Y ≡ R + × R + and t ∈ L ( µ ) , then (cid:13)(cid:13)(cid:13)(cid:13) t (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ) = (cid:13)(cid:13)(cid:13)(cid:13) t (cid:13)(cid:13)(cid:13)(cid:13) L ( µ Y ) . Remark 3.7.
For the reader’s convenience, throughout the paper we adopt the conventionof denoting R + × R + by X × Y , with independent variables s ∈ X and t ∈ Y . Consistentwith this, the marginal Berger measures are denoted by σ (or dσ ( s )) and τ (or dτ ( t )).We have found that doing so keeps a clear distinction, at the level of Berger measures andmarginal measures, between the two components T and T of W ( α,β ) .4. Generalized One-step Reconstruction-of-the-Measure Problem
In [4, Theorem 1.7] we solved the Reconstruction-of-the-Measure Problem (ROMP) for2-variable weighted shifts W ( α,β ) under the assumption of subnormality for the restrictions W ( α,β ) | L (1 , and W ( α,β ) | L (0 , , together with a natural compatibility condition; that is, if µ (1 , and µ (0 , are the respective Berger measures of the restrictions, then s dµ (0 , ( s, t ) = λt dµ (1 , ( s, t )for some λ > Theorem 4.1. (Generalized One-Step Extension; cf. [4, Remark 3.4]) Consider theROMP in Figure 3(left), and let µ ( k, and µ (0 ,ℓ ) be the Berger measures of the restrictionsof W ( α,β ) to L ( k, and L (0 ,ℓ ) , respectively. Assume that s k dµ (0 ,ℓ ) ( s, t ) = λt ℓ dµ ( k, ( s, t ) ,where λ := γ ( k, γ (0 ,ℓ ) > . Then W ( α,β ) is subnormal (with Berger measure µ ) if and only if (i) t ℓ ∈ L ( µ (0 ,ℓ ) ) ; (ii) s k ∈ L ( µ ( k, ) ; (iii) γ ( k, (cid:13)(cid:13) s k (cid:13)(cid:13) L ( µ ( k, ) = λγ (0 ,ℓ ) (cid:13)(cid:13) s k (cid:13)(cid:13) L ( µ ( k, ) ≤ ; and (iv) γ (0 ,ℓ ) { R Y d µ (0 ,ℓ ) ( s,t ) t ℓ + λ (cid:13)(cid:13) s k (cid:13)(cid:13) L ( µ ( k, ) dδ ( s ) − λs k R Y dµ ( k, ( s, t ) } ≤ dδ ( s ) .In the case when W ( α,β ) is subnormal, the Berger measure of W ( α,β ) is given by dµ ( s, t ) = γ (0 ,ℓ ) dµ (0 ,ℓ ) ( s, t ) t ℓ + (cid:18) dσ ( s ) − γ (0 ,ℓ ) Z Y dµ (0 ,ℓ ) ( s, t ) t ℓ (cid:19) dδ ( t ) . . Two-Step Extensions
We will now pose and solve the ROMP when even less information is available.
Problem 5.1.
For k, ℓ > , consider the canonical invariant subspace L ( k,ℓ ) (see Figure3(right)). (When k = ℓ = 1 , L (1 , is the core of W ( α,β ) .) Assume that (i) W ( α,β ) | L ( k,ℓ ) issubnormal with (two-variable) Berger measure ν ; (ii) W ( α,β ) | L (1 , is subnormal with (one-variable) Berger measure σ ; and (iii) W ( α,β ) | L (0 , is subnormal with (one-variable) Bergermeasure τ . Find necessary and sufficient conditions on ν , σ and τ such that W ( α,β ) issubnormal. When W ( α,β ) is indeed subnormal, find its Berger measure µ in terms of theinitial data ν , σ and τ . (0 ,
0) ( k,
0) ( k + 1 ,
0) ( k + 2 , · · ·· · · · · · · · · (0 , ℓ )(0 , ℓ + 1)(0 , ℓ + 2)... ......... L ( k, L (0 ,ℓ ) The measures µ (0 ,ℓ ) and µ ( k, satisfy the compatibilitycondition in L ( k,ℓ ) (0 ,
0) ( k,
0) ( k + 1 ,
0) ( k + 2 , · · ·· · · · · · · · · (0 , ℓ )(0 , ℓ + 1)(0 , ℓ + 2)... ......... ν represents W ( α,β ) on L (0 ,ℓ ) T L ( k, σ represents W ( α,β ) on 0–th row τ represents W ( α,β ) on0–th column Figure 3. (left) The subspaces L (0 ,ℓ ) , L ( k, and L ( k,ℓ ) := L (0 ,ℓ ) T L ( k, used in Theorem 4.1; (right) initial data of two-step ROMP for the subspace L ( k,ℓ ) .Our strategy for solving Problem 5.1 is to build an equivalent problem, whose solutioncan be found using Theorem 4.1. To this end, consider the two closed subspaces L ( k, and L (0 ,ℓ ) . We focus first on L ( k, , and look for necessary conditions for the subnormalityof W ( α,β ) | L ( k, . From Lemma 3.3, it is clear that for ν to extend to the Berger measureof W ( α,β ) | L ( k, , we need(i) 1 t ℓ ∈ L ( ν )(as a simple application of Lemma 3.3, ℓ times, reveals). Similarly, we must require theconditions(ii) γ ( k,ℓ ) γ ( k, ≤ (cid:13)(cid:13)(cid:13) t ℓ (cid:13)(cid:13)(cid:13) − L ( ν ) , and iii) γ ( k,ℓ ) Z Y dν ( s, t ) t ℓ ≤ s k dσ ( s )(again, by repeated application of Lemma 3.3 and the fact that the subnormal unilateralweighted shift associated with the sequence α ( k, , α ( k +1 , , . . . has Berger measure s k σ ,properly normalized (cf. (3.1))).It now follows that W ( α,β ) | L ( k, is subnormal if and only if (i), (ii) and (iii) hold, and inthat case, the Berger measure is reconstructed (using Lemma 3.3) as dµ ( k, ( s, t ) := γ ( k,ℓ ) γ ( k, dν ( s, t ) t ℓ + [ s k dσ ( s ) γ ( k, − γ ( k,ℓ ) γ ( k, Z Y dν ( s, t ) t ℓ ] · dδ ( t ) (5.1)(observe that the measure inside the square brackets is a one-variable measure in s ).Having dealt with the subspace L ( k, , it is now straightforward to list the necessaryand sufficient conditions for W ( α,β ) to be subnormal on the subspace L (0 ,ℓ ) ; these are(i) 1 s k ∈ L ( ν )(ii) γ ( k,ℓ ) γ (0 ,ℓ ) ≤ (cid:13)(cid:13)(cid:13) s k (cid:13)(cid:13)(cid:13) − L ( ν ) , and(iii) γ ( k,ℓ ) Z X dν ( s, t ) s k ≤ t ℓ dτ ( t ).When these conditions hold, W ( α,β ) | L (0 ,ℓ ) is subnormal, with Berger measure given by dµ (0 ,ℓ ) ( s, t ) := γ ( k,ℓ ) γ (0 ,ℓ ) dν ( s, t ) s k + dδ ( s ) · [ t ℓ dτ ( t ) γ (0 ,ℓ ) − γ ( k,ℓ ) γ (0 ,ℓ ) Z X dν ( s, t ) s k ] (5.2)(with the measure inside the square brackets a one-variable measure in t ).We are now ready to state and prove a solution to Problem 5.1. Theorem 5.2.
Let W ( α,β ) ≡ ( T , T ) be a commuting -variable weighted shift, and let k, ℓ > . Assume that the unilateral weighted shifts corresponding to the –th row R and the –th column C are subnormal, with Berger measures σ and τ , respectively. As-sume also that W ( α,β ) | L (0 ,ℓ ) T L ( k, is subnormal with Berger measure ν . Then W ( α,β ) issubnormal (with Berger measure µ ) if and only if (NC1) 1 s k t ℓ ∈ L ( ν )(NC2) γ ( k,ℓ ) γ (0 ,ℓ ) R X dν ( s, t ) s k ≤ dτ ℓ ( t ) := t ℓ dτ ( t ) γ (0 ,ℓ ) (NC3) γ ( k,ℓ ) γ ( k, R Y dν ( s, t ) t ℓ ≤ dσ k ( s ) := s k dσ ( s ) γ ( k, (NC4) R X × Y dν ( s, t ) s k t ℓ = 1 γ ( k,ℓ ) .When W ( α,β ) is subnormal, the Berger measure µ is given by dµ ( s, t ) = γ ( k,ℓ ) dν ( s, t ) s k t ℓ + (cid:18) dσ ( s ) − γ ( k,ℓ ) Z Y dν ( s, t ) s k t ℓ (cid:19) dδ ( t )+ dδ ( s ) (cid:18) dτ ( t ) − γ ( k,ℓ ) Z X dν ( s, t ) s k t ℓ (cid:19) , (5.3) r in the equivalent (succinct) form µ = γ ( k,ℓ ) νs k t ℓ + (cid:18) σ − γ ( k,ℓ ) Z Y νs k t ℓ (cid:19) × δ + δ × (cid:18) τ − γ ( k,ℓ ) Z X νs k t ℓ (cid:19) . (5.4) Remark 5.3.
Observe that each of the one-variable measures inside the parentheses in(5.4) is positive, by (NC3) and (NC2), respectively. As a result, the Berger measure µ isthe sum of three positive measures, involving the initial data ( ν , σ and τ ) together withthe marginal measures ν X and ν Y . Proof of Theorem 5.2.
We wish to apply the natural generalization of Lemma 3.3 whenthe subspace M is replaced by L ( k, ). It is straightforward to list the three requiredconditions:(i) 1 t ℓ ∈ L ( ν ) ;(ii) γ ( k,ℓ ) γ ( k, ≤ ( (cid:13)(cid:13)(cid:13) t ℓ (cid:13)(cid:13)(cid:13) L ( ν ) ) − ;(iii) γ ( k,ℓ ) γ ( k, R Y dν ( s, t ) t ℓ ≤ dσ k ( s ) .Observe first that (NC1) implies that both s k and t ℓ belong to L ( ν ); for instance, t ℓ = s k · s k t ℓ , and s k ∈ L ∞ ( ν ). Thus, the first hypothesis in Lemma 3.3 is satisfied. To get(ii), we integrate (NC3) with respect to s ; that is, γ ( k,ℓ ) γ ( k, Z X Z Y dν ( s, t ) t ℓ ≤ Z X dσ k ( t ) = Z X s k dσ ( s ) γ ( k, . It follows that γ ( k,ℓ ) γ ( k, (cid:13)(cid:13)(cid:13)(cid:13) t ℓ (cid:13)(cid:13)(cid:13)(cid:13) L ( ν ) ≤ γ ( k, γ ( k, = 1 , as desired. We now observe that (iii) and (NC3) are identical. Having verified (i), (ii)and (iii), we conclude that ν admits a back-step extension µ ( k, to the subspace L ( k, .In a completely similar way, we use (NC1) and (NC2) to obtain a back-step extensionof ν , denoted µ (0 ,ℓ ) , to the subspace L (0 ,ℓ ) .We now refer to the hypotheses for Theorem 4.1 for the case of two subspaces L ( k, and L (0 ,ℓ ) , and Berger measures µ ( k, and µ (0 ,ℓ ) , respectively. First, we need to check thecompatibility condition s k dµ (0 ,ℓ ) ( s, t ) = λt ℓ dµ ( k, ( s, t ), with λ = γ ( k, γ (0 ,ℓ ) . By (5.2), we have s k dµ (0 ,ℓ ) ( s, t ) = γ ( k,ℓ ) γ (0 ,ℓ ) dν ( s, t ) + s k dδ ( s ) · [ d τ ( t ) − γ ( k,ℓ ) γ (0 ,ℓ ) Z X d ν ( s, t )] = γ ( k,ℓ ) γ (0 ,ℓ ) dν ( s, t ) . On the other hand, by (5.1), λt ℓ dµ ( k, ( s, t ) = γ ( k,ℓ ) γ (0 ,ℓ ) dν ( s, t ) + γ ( k, γ (0 ,ℓ ) [ dσ ( s ) − γ ( k,ℓ ) γ ( k, Z X dν ( s, t )] · t ℓ dδ ( t ) = γ ( k,ℓ ) γ (0 ,ℓ ) dν ( s, t ) . It is now clear that the compatibility condition holds. We now calculate the L –norms of t ℓ and s k , using (5.2) and (5.1), respectively; in each case, we use (NC4) in the last step. X × Y t ℓ dµ (0 ,ℓ ) ( s, t ) = γ ( k,ℓ ) γ (0 ,ℓ ) Z X × Y dν ( s, t ) s k t ℓ + Z X dδ ( s )[ Z Y dτ ( t ) γ (0 ,ℓ ) − γ ( k,ℓ ) γ (0 ,ℓ ) Z Y Z X dν ( s, t ) s k t ℓ ]= 1 γ (0 ,ℓ ) < ∞ and Z X × Y s k dµ ( k, ( s, t ) = γ ( k,ℓ ) γ ( k, Z X × Y dν ( s, t ) s k t ℓ + [ Z X dσ ( s ) γ ( k, − γ ( k,ℓ ) γ ( k, Z X Z Y dν ( s, t ) s k t ℓ ] Z Y dδ ( t )= 1 γ ( k, < ∞ . Moreover, λγ (0 ,ℓ ) (cid:13)(cid:13)(cid:13)(cid:13) s k (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ( k, ) = γ ( k, · γ ( k, = 1 . Finally, we need to check condition (iv) in Theorem 4.1: γ (0 ,ℓ ) { Z Y dµ (0 ,ℓ ) ( s, t ) t ℓ + λ (cid:13)(cid:13)(cid:13)(cid:13) s k (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ( k, ) dδ ( s ) − λs k Z Y dµ ( k, ( s, t ) } ≤ dδ ( s ) . In preparation for this, we first calculate the two integrals on the left-hand side of (iv).Observe that Z Y dµ (0 ,ℓ ) t ℓ = γ ( k,ℓ ) γ (0 ,ℓ ) Z Y dν ( s, t ) s k t ℓ + dδ ( s )[ Z Y dτ ( t ) γ (0 ,ℓ ) − γ ( k,ℓ ) γ (0 ,ℓ ) Z Y Z X dν ( s, t ) s k t ℓ ] = γ ( k,ℓ ) γ (0 ,ℓ ) Z Y dν ( s, t ) s k t ℓ (the last step uses both the fact that τ is a probability measure and condition (NC4), andas a result, the quantity in square brackets is 0). Similarly, Z Y dµ ( k, ( s, t ) = γ ( k,ℓ ) γ (0 ,k ) Z Y dν ( s, t ) t ℓ + ( s k dσ ( s ) γ ( k, − γ ( k,ℓ ) γ (0 ,k ) Z Y dν ( s, t ) t ℓ ) Z Y dδ ( t ) = s k dσ ( s ) γ ( k, . We are now ready to establish (iv). γ (0 ,ℓ ) { Z Y dµ (0 ,ℓ ) ( s, t ) t ℓ + λ (cid:13)(cid:13)(cid:13)(cid:13) s k (cid:13)(cid:13)(cid:13)(cid:13) L ( µ ( k, ) dδ ( s ) − λs k Z Y dµ ( k, ( s, t ) } = γ (0 ,ℓ ) { γ ( k,ℓ ) γ (0 ,ℓ ) Z Y dν ( s, t ) s k t ℓ + λ γ ( k, dδ ( s ) − λs k s k dσ ( s ) γ ( k, } = γ ( k,ℓ ) Z Y dν ( s, t ) s k t ℓ + dδ ( s ) − dσ ( s ) (recall that λ = γ ( k, γ (0 ,ℓ ) ) ≤ dσ ( s ) + dδ ( s ) − dσ ( s ) = dδ ( s ) , where we have used (NC3) in the penultimate step. This proves Theorem 4.1(iv). ByTheorem 4.1, we now know that W ( α,β ) is subnormal. To complete the proof, we need toexplicitly compute its Berger measure. y Theorem 4.1, using the subspace L (0 ,ℓ ) , the measure µ (0 ,ℓ ) and the moment γ (0 ,ℓ ) , wecan reconstruct the Berger measure of W ( α,β ) as µ = γ (0 ,ℓ ) µ (0 ,ℓ ) t ℓ + (cid:18) σ − γ (0 ,ℓ ) Z Y µ (0 ,ℓ ) t ℓ (cid:19) × δ . Now observe that γ (0 ,ℓ ) Z Y dµ (0 ,ℓ ) ( s, t ) t ℓ = γ ( k,ℓ ) Z Y dν ( s, t ) s k t ℓ + dδ ( s ) (cid:18) − γ ( k,ℓ ) Z Y Z X dν ( s, t ) s k t ℓ (cid:19) = γ ( k,ℓ ) Z Y dν ( s, t ) s k t ℓ (using (NC4) to obtain the last equality). Then µ = γ (0 ,ℓ ) µ (0 ,ℓ ) t ℓ + (cid:18) σ − γ ( k,ℓ ) Z Y νs k t ℓ (cid:19) × δ . We now use (5.2) to obtain µ = γ ( k,ℓ ) dνs k t ℓ + δ × (cid:18) τ − γ ( k,ℓ ) Z X dνs k (cid:19) + (cid:18) σ − γ ( k,ℓ ) Z Y νs k t ℓ t ℓ (cid:19) × δ , as desired. (cid:3) ROMP for Canonical Invariant Subspaces
Let P ⊆ Z be such that P + Z = P , and let L P be the closed subspace of ℓ ( Z )generated by the orthonormal basis vectors e k , where k ∈ P . The subspace L P isinvariant under T and T , for any 2-variable weighted shift W ( α,β ) ≡ ( T , T ). Definition 6.1.
A set P such that P + Z = P is said to be full . A closed subspace S of ℓ ( Z ) invariant under T and T is said to be canonical if S = L P for some full set P . Remark 6.2.
Typical examples of full and non-full sets are given in Figure 4. Observethat for any nonempty set P ⊆ Z , the set P + Z is full. Observe also that anyfull set P is the union of the (full) sets k + Z , where k is a point in P ; in symbols, P = S k ∈ P ( k + Z ). Thus, L P = W k ∈ P L { k } (cf. the left diagram in Figure 4). As aconsequence, if P and Q are full sets, and if L P = L Q , then P = Q . (With slight abuseof notation, we will often write L k instead of L { k } ). Definition 6.3.
Let P be a full set in Z . Amongst all subsets Q of P satisfying Q + Z = P , there is a smallest one; i.e., one that is contained in any other subset R of P such that R + Z = P . We will call this minimal set the foundation of P , and denoteit by F ( P ) . We now define S ( P ) to be the unique nonincreasing path contained in P and including all points in F ( P ) . Remark 6.4. (i) The points of F ( P ) are characterized by the following property: k ∈F ( P ) if and only k ∈ P , k − ε / ∈ P and k − ε / ∈ P .(ii) For P a full set, observe that F ( P ) is always a finite set, and that F ( P ) ⊆ S ( P ) ⊆ P (by definition). Also, F ( P ) + Z = S ( P ) + Z = P .(iii) It is easy to see that S ( P ) can be represented by a descending staircase; for a visual epiction, see the green staircase in the left diagram of Figure 4.(iv) To list the points in S ( P ), we will follow the descending staircase from left to right;that is, p ≡ ( p , p ) will come before q ≡ ( q , q ) if and only if p ≤ q and p ≥ q .(v) If P is full, one can recover P from S ( P ) by selecting all points in Z which arelocated above and to the right of the descending staircase representing S ( P ). (0 ,
0) (1 , , · · · ( k, , k ) · · · ... · · · · · · T T ...... ...... T (0 ,
0) (1 , · · · ( k, · · · ... · · · ... Figure 4. (left) Diagram of a full set P ( P + Z = P ), its foundation F ( P ), and its descending staircase S ( P ); (right) diagram of a non-full set Q ( Q + Z = Q ). ( Color codes : a light-blue disk denotes a point in P , a redsquare denotes a point not in P , and a blue square denotes a point in F ( P ).) Consider now a 2-variable weighted shift W ( α,β ) and a canonical invariant subspace S ,which is necessarily of the form L P for some P full. Let F ( P ) be the foundation P ; itfollows that S = W k ∈F ( P ) L k . Assume now W ( α,β ) | L k is subnormal. It immediately followsthat, for each k ∈ F ( P ), the restriction of W ( α,β ) to the subspace L k must be subnormal.Now, it is clear that W ( α,β ) | L k is a 2-variable weighted shift, so it has a Berger measure,which we will denote by ν k . Thus, the subnormality of W ( α,β ) | L P translates into theexistence of a finite family of Berger measures ν k , one for each point k ∈ F ( P ).There is a compatibility condition, however. If we consider two points p , q ∈ F ( P ),we know that the intersection of the canonical invariant subspaces L p and L q is L (max { p ,q } , max { p ,q } ) . Without loss of generality, assume that p ≤ q and q ≤ p . Then R := L p T L q = L ( q ,p ) . By (3.1), the Berger measure of W ( α,β ) | R is given by twoexpressions, namely γ ( p ,p ) γ ( q ,p ) s q − p ν ( p ,p ) and γ ( q ,q ) γ ( q ,p ) t p − q ν ( q ,q ) . herefore, as a necessary condition for the solubility of ROMP we must require γ ( p ,p ) γ ( q ,p ) s q − p ν ( p ,p ) = γ ( q ,q ) γ ( q ,p ) t p − q ν ( q ,q ) , that is, γ p s q − p ν p = γ q t p − q ν q . (6.1)We will denote by ν pq the Berger measure of W ( α,β ) restricted to L p T L q . Similarly, wewill let σ p and τ q denote the restrictions of σ to L p ( ⊆ ℓ ( Z + )) and τ to L q ( ⊆ ℓ ( Z + )),respectively. With the above mentioned compatibility condition in mind, we are readyto state and prove the solution of ROMP for arbitrary canonical invariant subspaces. Theorem 6.5.
Let W ( α,β ) be a -variable weighted shift, let L P be a canonical invariantsubspace, and assume that W ( α,β ) | L P is subnormal. Let F ( P ) be the foundation of P (with points listed in descending-staircase order), and let { ν k } k ∈F ( P ) be the finite familyof Berger measures for the restrictions of W ( α,β ) to the subspaces L k ( k ∈ F ( P )) . Supposethat the compatibility condition (6.1) holds for every p , q ∈ F ( P ) . In addition, let σ and τ be the Berger measures of the –th row and –th column in the weigh diagram of W ( α,β ) .The following statements are equivalent. (i) W ( α,β ) is subnormal. (ii) For every p , q ∈ F ( P ) , the ROMP with initial data ν pq , σ p and τ q is soluble. (iii) For every consecutive pair of points p , q ∈ F ( P ) (in the descending-staircaseorder), the ROMP with initial data ν pq , σ p and τ q is soluble. (iv) For every p , q ∈ F ( P ) , with p = q and p = q , the ROMP with initial data ν pq , σ p and τ q is soluble.Proof. Looking at the staircase diagram in Figure 4(left), we can easily see that four basicand distinct descending-staircase types for F ( P ) arise. We exhibit these types in Figures5 and 6. We now analyze ROMP for each descending-staircase type. Type I : S ( P ) T (0 × Z + ) = ∅ and S ( P ) T ( Z + ×
0) = ∅ .We refer the reader to the left staircase in Figure 5, and recall that we denote arbitrarypoints in Z as k ≡ ( k , k ). Observe that the points p , q , r determine a weight diagramas in the generalized one-step extension case of ROMP (Theorem 4.1). As a result,assuming that the natural necessary conditions for solubility are satisfied, we can extend ν pr to the subspace L ( p ,r ) . We therefore reduce the original ROMP to a new ROMPwhose descending staircase starts at ( p , r ), and continues with the points s , t , u and v .The situation is then almost identical to what we had before, so with the natural necessaryconditions for solubility, and using Theorem 4.1 once again, we extend the measure to thesubspace L ( p ,t ) . This yields a new descending staircase, connecting ( p , t ) to u and v .Another application of Theorem 4.1 leads to an extension to the subspace L ( p ,v ) . Tofinish, we now use the back-step extension result (Lemma 3.3) applied p times. Type II : S ( P ) T (0 × Z + ) = ∅ and S ( P ) T ( Z + × = ∅ .This case is completely analogous to the previous one, so after successive instances of , · · · ( k, · · · p qr st uv ...... ... · · · · · ·· · · T T T (0 , · · · ( k, · · · p qr st uv ... · · ·· · · · · · ...... Figure 5.
Weight diagrams of the 2-variable weighted shifts for Type I(left) and Type II (right). (0 , · · · ( k, · · · p qr st uv wx ... ... ... · · ·· · ·· · · T T T (0 , · · · ( k, · · · p qr st u ... ... · · ·· · · · · · Figure 6.
Weight diagrams of the 2-variable weighted shifts for Type III(left) and Type IV (right).Theorem 4.1 we end up with the subspace L ( p , , and we then apply Lemma 3.3, p times, to obtain the Berger measure µ of W ( α,β ) . ype III : S ( P ) T (0 × Z + ) = ∅ and S ( P ) T ( Z + × = ∅ .Here repeated application of Theorem 4.1 does the job. Type IV : S ( P ) T (0 × Z + ) = ∅ and S ( P ) T ( Z + ×
0) = ∅ .In this case, and after a few instances of Theorem 4.1, we end up with the subspace L ( p ,t ) . This case fits well within the scope of Theorem 5.2. As a result, the naturalnecessary conditions are sufficient for the existence of the Berger measure µ .The proof is now complete. (cid:3) References [1] J. Conway,
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Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242
E-mail address : [email protected] URL : Department of Mathematics, Chungnam National University, Daejeon, 34134, Republicof Korea
E-mail address : [email protected] School of Mathematical and Statistical Sciences, The University of Texas Rio GrandeValley, Edinburg, Texas 78539, USA
E-mail address : [email protected]@utrgv.edu