The notion of recursively generated weighted shifts, largely studied in the literature, is employed to solve various questions in operator theory. R. Curto and L. Fialkow [3, 4] have used this concept to solve the Subnormal Completion Problem (SCP for short) in one variable. In this paper, we extend this notion to 2-variables, and we use it to provide a new approach to solve the open problem of 2-variable SCP. A concrete solution to the minimal 2-variable SCP (see Section 4) is given as well as an alternative proof for the propagation phenomena for subnormal 2-variable weighted shifts.

First we recall some definitions and notations. A bounded linear operator T ∈ ℬ(ℋ) on a complex Hilbert space ℋ is normal if TT^* = T^*T, subnormal if T = N |_ℋ, where N is normal and N(ℋ) ⊆ ℋ, and hyponormal if T^*T – TT^* ≥ 0. The n-tuple T ≡ (T₁, …, T_n) is said to be normal if T is commuting and each T_i is normal, and T is subnormal if T is the restriction of a normal n-tuple to a common invariant subspace.

For α≡{αn}n=0∞ a bounded sequence of positive real numbers (called weights), let W_α : l²(ℝ₊) → l²(ℝ₊) be the associated unilateral weighted shift, defined by W_αe_n := α_ne_n+1, where {en}n=0∞ is the canonical orthonormal basis in l²(ℝ₊). Given k ∈ ℝ₊, the moments of α of order k is given by

It is easy to see that W_α is never normal. In the case where W_α is subnormal, Stampfli showed in [13] that a propagation phenomenon occurs which forces the flatness of W_α, that is, if α_k = α_k+1 for some k ≥ 0, then α_n = α_n+1 for every n ≥ 0.

Consider double-indexed positive bounded sequences α≡{αk}k∈ℤ+2 and β≡{βk}k∈ℤ+2. We define in a similar way the 2-variable weighted shift T ≡ (T₁, T₂) acting on the Hilbert space l2(ℤ+2), associated with, α≡{αk}k∈ℤ+2 and β≡{βk}k∈ℤ+2 by

where ε₁ := (1, 0) and ε₂ := (0, 1). We recall here that, the vector space l2(ℤ+2) is canonically isometrically isomorphic to l²(ℝ₊)⊗l²(ℝ₊), equipped with its canonical orthonormal basis {ek}k∈ℤ+2.

Clearly,

Given k≡(k1,k2)∈ℤ+2, the moment of (α, β) of order k is

Conversely, one can recover the weights from the moments, by using the following relations:

The presence of consecutive equal weights, of a 2-variable subnormal weighted shift, leads to horizontal or vertical flatness (see Definition 3.4). Explicitly, if, for some k₁, k₂ ≥ 1, α_{(k1, k2)} = α_{(k1+1, k2)} (resp. β_{(k1, k2)} = β_{(k1, k2+1)}), then α_{(k1, k2)} = α_(1,1) (resp. β_{(k1, k2)} = β_(1,1)) for all k₁, k₂ ≥ 1. This result is known as propagation phenomena. We give new short proof of this fact in Theorem 3.4.

A characterization of subnormality for multivariable weighted shifts is given in [10]. More precisely, T admits a commuting normal extension if and only if there is a probability measure μ, which we call the Berger measure of T, defined on the 2-dimensional rectangle R = [0, ||T₁||²] × [0, ||T₂||²] such that

A measure μ satisfies (1.4) is also known as representing measure for {γk}k∈ℤ+2.

With a given bi-sequence γ(2n)≡{γi}i∈ℤ+2,∣i∣≤2n≡{γij}i+j≤2n, we associate the moment matrix M(n) ≡ M(n)(γ²ⁿ), introduced by R. Curto and L. Fialkow [5, 6], build as follows.

where

Considering the following lexicographic order, to denote rows and columns of the moment matrix M(n),

The matrix M(n) detects the positivity of the Riesz functional

on the cone generated by {p² : p ∈ ℝ_n[x, y]}, the sum of squares of polynomials (sometimes abbreviated as SOS), where ℝ_n[x, y] is the vector space of polynomials in two variables with real coefficients and total degree less than or equal to n.

For reason of simplicity, we identify a polynomial p(x, y) ≡ ∑a_ijxⁱy^j with its coefficient vector p = (a_ij) with respect to the basis of monomials of ℝ_n[x, y] in degree-lexicographic order (see (1.7)). Clearly M(n) acts on these coefficient vectors as follows:

Furthermore, let g ∈ ℝ[x, y] be with coefficient vector {g_β} and let g * γ denote the shifted vector in ℝℤ+2 whose α-th entry is (g*γ)α:=∑βgβγβ+α. The moment matrix M(n) (g*γ)=Mg(n+[1+deg g2]) is called the localizing matrix with respect to γ and g. For example, given γ⁽⁴⁾ ≡ {γ_ij}_i+j≤4, then

In the one variable case, the SCP was stated and solved abstractly by J. Stampfli in [13]:

Problem 1.(One-Variable Subnormal Completion Problem)

Given m ≥ 0 and a finite collection of positive numbersΩm={αk}k=0m, find necessary and sufficient conditions on Ω_m to guarantee the existence of a subnormal weighted shift whose initial weights are given by Ω_m.

When m = 0 or m = 1 the solution can be given by the canonical completion α₀, α₀, α₀, … and α₀ < α₁, α₁, …, with Berger measure μ:=δα02 and μ:=α02α12δα12, respectively. In [13], J. Stampfli showed that given α:a,b,c with a < b < c, there always exists a subnormal completion of α (this solves the case m = 2), but for α:a,b,c,d, with a < b < c < d, such a subnormal completion may not exist. The complete solution of the SCP in one variable was given by R. curto and L. Fialkow [3], the explicit calculation requires recursively generated weighted shifts (such shifts have finite atomic Berger measures).

We now state the 2-variable SCP:

Problem 2.(2-Variable Subnormal Completion Problem)

Given m ≥ 0 and a finite collection of pairs of positive numbers Ω_m = {(α_k; β_k)}_|k|≤m satisfying (1.1) for all | k |≤ m (where | k |:= k₁ + k₂), find necessary and sufficient conditions to guarantee the existence of a subnormal 2-variable weighted shift whose initial weights are given by Ω_m.

In [8], R. Curto, S. H. Lee and J. Yoon have introduced an approach to the 2-variable SCP based on positivity and rank-preserving extension of the moment matrix, developed in [5, 6, 7]. Although this lead to an explicit criterion for SCP with quadratic moment data (i.e., m = 1), the 2-Variable Subnormal Completion Problem remains open. Recently, S. H. Lee and J. Yoon [11] found, by using the recursiveness, a necessary and sufficient conditions for the existence of 2-Variable subnormal completion with minimal Berger measure for the case m = 2 ( i.e., rank M(1)-atomic Berger measure). We will provide, in Theorem 4.2, a complete (and concrete) solution for the 2-Variable SCP with minimal Berger measure.

In the present paper, we will show that if a given, finite, collection of weights Ω_m admits a subnormal completion in 2-variable, then there exists a 2-variable subnormal weighted shift (solution of the SCP for Ω_m) with moment sequence obey to some recurrence relations, we shall refer to such shifts as 2-variable recursively generated weighted shifts (see Definition 2.5). In Theorem 3.1, we will provide necessary and sufficient conditions for the existence of 2-variable recursively generated weighted shift completion, and thus a solution to the SCP in 2-variable. As application, we provide a generalization of a recent result of S. H. Lee and J. Yoon [11, Theorem 2.2]. More precisely, in Theorem 4.2 we give a concrete necessary and sufficient conditions for the existence of a subnormal completion with minimal Berger measure.

This paper is organized as follows. In Section 2, we introduce the recursively generated weighted shifts and we exhibit some useful results. We devote Section 3 to provide a solution to the 2-Variable Subnormal Completion Problem (Theorem 3.1 ) and the phenomena propagation for subnormal weighted shifts. In section 4, we solve the minimal SCP in 2-variable.

We introduce below the notion of 2-variable weighted shifts, which will play a central role in this paper, and we give some useful properties.

Definition 2.1

Let T ≡ (T₁, T₂) be a 2-variable weighted shift and let γ ≡ {γ_ij}_i,j∈ℤ+ be its associated moment sequence. A polynomial p(x,y)=∑i,jpijxiyj∈ℝ[x,y] is said to be characteristic polynomial associated with T, or with γ, if

Remark 2.2

If p is a characteristic polynomial associated with γ, then, for every q ∈ ℝ[x, y], the polynomial pq is also a characteristic polynomial. In particular, the set of all characteristic polynomials associated with γ is an ideal in ℝ[x, y].

The following proposition is an immediate consequence of relations (1.8) and (2.1).

Proposition 2.3

Let γ ≡ {γ_ij}_i,j∈ℤ+be a bi-indexed sequence and let p(x, y) ∈ ℝ[x, y]. Then p is a characteristic polynomial of γ if and only if M(∞)(γ)p = 0.

Definition 2.4

A sequence γ ≡ {γ_ij}_i,j∈ℤ+ is said to be recursive double indexed sequence (RDIS in short) if it has two characteristic polynomials p₁, p₂ ∈ ℝ[x, y],

with

or in the symetric form

Without loss of generality, throughout this paper, we assume that the pair of characteristic polynomials of the RDIS γ ≡ {γ_ij}_i,j∈ℤ+ is given in the form (2.2).

Definition 2.5

A 2-variable weighted shift is said to be recursively generated if its associated moment sequence is a RDIS.

Let us show that RDIS are well defined. Consider a RDIS γ ≡ {γ_ij}_i,j≥0 associated with a pair of characteristic polynomials as in (2.2), that is, for all n, m, e ∈ ℝ₊ with n ≥ r and m ≥ s,

A direct computation shows that

Equation (2.5) gives the compatibility condition of the two relations in (2.4). Hence the sequence γ ≡ {γ_ij}_i,j≥0 is well defined. The other case (i.e., when the characteristic polynomial are defined as in (2.3)) is treated similarly.

Different pair of characteristic polynomials can be associated with the same RDIS, as shown in the following example.

Example 1

Let us consider the bi-sequence γ ≡ {γ_ij}_i,j∈ℤ+ defined by γ_n,m = a2ⁿ +b3^m, where a and b are real numbers. The polynomials x2-4x-12y+92 and y + 2x − 5 are two characteristic polynomials of γ. Indeed,

and

On the other hand, we have

Hence (x²−3x+2; y²−4y+3) is, also, a pair of characteristic polynomials associated with γ. (The last characteristic polynomials are analytic, i.e., (x² − 3x + 2; y² − 4y + 3) ∈ ℝ[X] × ℝ[Y ]).

Let γ ≡ {γ_ij}_i,j∈ℤ+ be a RDIS, we denote by ℘_γ (⊆ ℝ[x, y] × ℝ[x, y]) the set of pair of characteristic polynomials associated with γ and we denote by (⊆ ℝ[X]×ℝ[Y ]) the family of analytic, monic, characteristic polynomials associated with γ.

Remark 2.6

The pair of characteristic polynomials (p₁, p₂) ∈ ℘_γ, together with the initial conditions {γij}0≤i≤deg p1-1,0≤i≤deg p2-1., are said to define the sequence γ.

We use structural properties of moment matrices to get the following interesting lemma.

Lemma 2.7

Let γ ≡ {γ_ij}_i,j∈ℤ+be a bi-indexed sequence and let f, g, h ∈ ℝ[x, y], then

Proposition 2.8

Let γ ≡ {γ_ij}_i,j≥0be a bi-indexed sequence and let M(∞)(γ) be the associated infinite moment matrix. If M(∞)(γ) ≥ 0, Then, for any polynomial p ∈ ℝ[x, y] and any integer n ≥ 1, we have

Theorem 2.9.([9, Theorem 1])

Let{yA(n)}n=0+∞be a RDIS and p(x) be the associated polynomial as above, withp(x)=∏i=1k(x-λi)mi, (m₁ + …+ m_k = r). Then the difference equation (2.8) has r independent solutionnjλln(j = 0, …, m_l−1; l = 1, …, k). Moreover, any solution of (2.8) is of the form

where e_l,j are solutions of the following system of r-equations

As observed in [2, Proposition 2.1], among all characteristic polynomials defining S, there exists a unique monic characteristic polynomial p₀ of minimal degree, called the minimal characteristic polynomial, and which, moreover, divides every characteristic polynomial. The next proposition gives a generalization of this result to the 2-variable case.

Proposition 2.10

For every RDIS γ ≡ {γ_ij}_i,j≥0, with, there exists a unique pair of characteristic polynomials(p1γ,p2γ)∈Aγwith minimal degree. Moreover, for every, the polynomialsp1γandp2γdivide Q₁and Q₂, respectively.

Theorem 2.11

Let γ ≡ {γ_ij}_i,j∈ℤ+be a RDIS and let(p1,p2)=(p1γ,p2γ)∈Aγ. If M(∞)(γ) ≥ 0, then the polynomials p₁and p₂have distinct roots.

Lemma 2.12

Let γ ≡ {γ_ij}_i,j∈ℤ+be a RDIS and let, withP1(x)=∏l=1k1(x-λl)mlandP2(y)=∏l=1k2(y-βl)nl, (λ_l, β_l ∈ ℂ). Then

where e_a,b,c,d are determined by using the initial condition {γ_ij}_{0≤i≤r−1,0≤j≤s−1}.

Corollarly 2.13

Let γ ≡ {γ_ij}_i,j∈ℤ+be a RDIS, with M(∞)(γ) ≥ 0, and let. Then

where Z(P₁) := {z ∈ ℂ such that P(z, z̄) = 0} = {λ₁, …, λ_r+1} and Z(P₂) = {β₁, …, β_s+1}.

The next lemma establishes, for a RDIS γ, a link between the positivity of infinite moment matrix and that of the finite one.

Lemma 2.14

Let γ ≡ {γ_ij}_i,j∈ℤ+be a RDIS and let (p₁, p₂) ∈ ℘_γ. Then

Moreover, rankM(∞)(γ) = rankM(deg p₁ +deg p₂ − 2)(γ).

In this section we involve the 2-variable recursively generated weighted shifts to obtain a solution to the SCP in 2-variable. Also, a simple and new proof to the propagation phenomena, for the 2-variable subnormal weighted shift, is given.

Theorem 3.1

Let Ω_n := {(α_{(k1, k2)}, β_{(k1, k2)}); k₁ + k₂ ≤ 2n} be a given collection of weights obeying (1.1) and γ⁽²ⁿ⁺¹⁾be the associated moment sequence, given by (1.2). The following statements are equivalent.

• Ω_n admits a 2-variable subnormal completion,

• Ω_n admits a recursively generated 2-variable subnormal completion,

• There exists a RDIS γ ≡ {γ_ij}_i,j≥0such that γ⁽²ⁿ⁺¹⁾ ⊂ γ and the matrices M(deg p₁+deg p₂−2)(γ), M_x(deg p₁+deg p₂−1)(γ) and M_y(deg p₁+deg p₂− 1)(γ) are positive, where (p₁, p₂) ∈ ℘_γ.