Let ℋ be a separable infinite dimensional complex Hilbert space and let ℒ(ℋ) be the algebra of all bounded linear operators on ℋ. For A, B ∈ ℒ(ℋ), we set [A,B] := AB–BA. A k-tuple T = (T₁, ..., T_k) of operators on ℋ is called hyponormal if the operator matrix ([Tj*,Ti])i,j=1k is positive on the direct sum of ℋ⊕· · ·⊕ℋ (k copies). Also an operator T is said to be (strongly) k-hyponormal for each positive integer k if (I, T, ..., T^k) is hyponormal. The Bram-Halmos criterion shows that an operator T is subnormal if and only if T is k-hyponormal for all k ≥ 1 ([2], [16]). An operator T is polynomially hyponormal if for every polynomial p, p(T) is hyponormal, and T is weakly k-hyponormal if for every polynomial p of degree k or less, p(T) is hyponormal ([5],[10],[11]). In particular, weak 2-hyponormality (or weak 3-hyponormality) is referred to as quadratically hyponormal (or cubically hyponormal, respectively). For a positive integer k, an operator T ∈ ℒ(ℋ) is called semi-weakly k-hyponormal if T+sT^k is hyponormal for all s ∈ ℂ ([12]). It is obvious that a weakly k-hyponormal operator is semi-weakly k-hyponormal. In particular, weak 2-hyponormality is equivalent to semi-weak 2-hyponormality.

It is well known that k-hyponormality implies weak k-hyponormality for each positive integer k. The following results provide a bridge between subnormal and hyponormal operators: subnormal ⇒ polynomially hyponormal ⇒ ··· ⇒ weakly 3-hyponormal ⇒ weakly 2-hyponormal ⇒ hyponormal. However, one does not yet have concrete examples about the converse implications for n ≥ 3; see [9], [17] and [18] for weak 2- and weak 3-hyponormalities.

J. Stampfli ([21]) proved that a subnormal weighted shift with two equal weights α_n = α_n+1 for some nonnegative n has the property that α₁ = α₂ = · · ·, which is known as the “flatness property.” Stampfli’s result has been used to attempt the construction of nonsubnormal polynomially hyponormal weighted shifts (cf. [1],[3],[4],[7],[12],[15],[17]). In [3], Choi proved that if a weighted shift W_α is polynomially hyponormal with the first two weights equal, then W_α has the flatness property. In [4], Curto obtained a quadratically hyponormal weighted shift with first two weights equal but not satisfying flatness. Also in [17], the authors showed that a weighted shift W_α with weights α:23,23,n+1n+2 (n≥2) is not cubically hyponormal. And in [19], it was shown that if a weighted shift W_α is cubically hyponormal with first two weights equal, then W_α has flatness. However, in [12], it was proved that there exists a semi-cubically hyponormal weighted shift W_α with α₀ = α₁ < α₂ which is not 2-hyponormal. Hence the following problem arises naturally as the analog to the question for quadratically hyponormal weighted shifts.

Problem 1.1

Describe all semi-cubically hyponormal weighted shifts W_α with first two weights equal.

In [12], Do-Exner-Jung-Li characterized the semi-cubical hyponormality of the weighted shift W_α(x) with positive determinant coefficients (p.d.c. – definition reviewed below), where α(x):x,x,k+1k+2 (k≥2) is a weight sequence with Bergman tail. In this paper we describe the semi-cubical hyponormality of the weighted shifts having the p.d.c. property but with recursive tails. More precisely, for a three step backward extended weight sequence α:1,1x,(u,v,w)∧ with 1 < x < u < v < w, where (u,v,w)∧ is the Stampfli (recursively generated) subnormal completion of u, v,w (cf. [21]), we characterize completely the semi-cubical hyponormality of W_α with p.d.c. Note that, by the nature of a recursive tail, the one and two step backward extensions are special cases of the three step backward extension (see Remark 3.5).

For the reader’s convenience, we recall the Stampfli subnormal completion (cf. [6],[21]). For given numbers α₀, α₁, α₂ with 0 < α₀ < α₁ < α₂, define

where Ψ0=-α02α12(α22-α12)α12-α02 and Ψ1=α12(α22-α02)α12-α02. Then we may obtain a weight sequence {αn}n=0∞ generated recursively by (1.1), which is usually denoted by (α₀, α₁, α₂)^∧ (for example, see [21]); the associated shift is subnormal. It follows from [6] that

The organization of this paper is as follows. In Section 2 we recall some terminology concerning semi-cubically hyponormal weighted shifts. In Section 3 we characterize the semi-cubic hyponormality of weighted shifts W_α with p.d.c., where α:1,1,x,(u,v,w)∧ with 1 < x < u < v < w, and then consider a related example.

Throughout this paper, ℝ₊, ℕ, and ℕ₀ are the sets of nonnegative real numbers, positive integers, and nonnegative integers, respectively.

We recall some standard terminology for semi-cubically hyponormal weighted shifts (cf. [12]). Let ℓ²(ℕ₀) be the space of square summable sequences in ℂ and let {ei}i=0∞ be an orthonormal basis of ℓ²(ℕ₀). For a weight sequence α={αi}i=0∞ in ℝ₊, the associated weighted shift W_α acting on ℓ²(ℕ₀) is semi-cubically hyponormal if

In fact, the condition in (2.1) is equivalent to a simpler one, as in the following proposition whose proof comes from [8, Prop. 1].

Let W_αbe a weighted shift with a weight sequence α={αi}i=0∞ in ℝ₊. Then W_αis semi-weakly n-hyponormal if and only if Wα+tWαn is hyponormal for all t ≥ 0.

Proof

It is sufficient to show the necessity. For any s ∈ ℂ, we may take nonnegative real numbers t and θ such that s = te^i(n–1)θ. Recall that there exists a unitary operator U such that UW_αU^* = e^−iθW_α. Then

so the inequality for all t ≥ 0 suffices to yield (2.1).

By Proposition 2.1, W_α is semi-cubically hyponormal if and only if D(s) ≥ 0 for all s ∈ ℝ₊. Observe that

where for all k ∈ ℕ₀,

with α₋₃ = α₋₂ = α₋₁ = 0. Consider two submatrices

and observe that D(s) = D⁽¹⁾(s) ⊕ D⁽²⁾(s), s ∈ ℝ₊. Define

where t = s². Then W_α is semi-cubically hyponormal if and only if Dn(j)(t)≥0 for all n ≥ 0, j = 1, 2.

To detect the positivity of Dn(j)(t) in (2.2), we consider a matrix with the form below:

where q̌_k : = ǔ_k + ǔ_kt, rˇk:=wˇkt (k≥0), and ǔ_k ≥ 0, v̌_k ≥ 0, w̌_k ≥ 0, t ≥ 0. (We take the approach in [6] for what follows.) Then

and it follows from [6] that

with c(−n,−i) := 0 for all n, i ∈ ℕ. Suppose that ǔ_nv̌_n+1 = w̌_n (n ≥ 2); this yields that

for all n ≥ 3, where

To detect the positivity of Dn(j)(t), j = 1,2, we consider

for j = 1, 2. Note that if c_j(n, n+1) > 0 and c_j(n, i) ≥ 0 for all n ≥ 0 with 0 ≤ i ≤ n, then every matrix Dn(j)(t) is obviously positive for all n ≥ 0 and t > 0. Recall that W_α has positive determinant coefficients (p.d.c.) (and is therefore semi-cubically hyponormal) if all coefficients in dn(j)(t) are nonnegative and the c_j(n, n + 1) are strictly positive for all n ∈ ℤ₊ and j = 1, 2 (cf. [12, Def. 2.2]).

The following is the crucial lemma which can be obtained from the proof of Theorem 4.3 in [6], and which we will apply in succession to Dn(1)(t) and Dn(2)(t).

Lemma 2.2

Under the notation above, we suppose that ǔ_nv̌_n+1 = w̌_n (n ≥ 2). Then the determinant of M_n(t) in (2.3) has non-negative coefficients c(n, i) for all n and i if and only if the following conditions hold:

• c(1, 1), c(2, 1), c(2, 2), c(3, 2) are all positive,

• Γn:=vˇ2vˇ1vˇ0+vˇnuˇnρ≥0, for all n ≥ 3,

• Ωn:=vˇ2vˇ1vˇ0+vˇn-1uˇn-1ρ+vˇn-1uˇn-1vˇnuˇnτ≥0, for all n ≥ 4.

Let α:1,1,x,(u,v,w)∧ with 1 < x < u < v < w. To determine when W_α is a semi-cubically hyponormal weighted shift with p.d.c., we will use Lemma 2.2 for j = 1, 2 considering conditions (i), (ii), and (iii) in Lemma 2.2. We will denote the coefficients of the determinants c₁(n, i) and c₂(n, i) with the obvious meaning, and distinguish other quantities between j = 1 and j = 2 with superscripts (for example, ρ⁽¹⁾ and ρ⁽²⁾). Note first that it follows from Lemma 3.1 of [20] that we have both un(1)vn+1(1)=wn(1) and un(2)vn+1(2)=wn(2) for n ≥ 2.

The following (n + 1) × (n + 1) matrix is the expression of Dn(1):

Since c₁(1, 1) = (uv − 1)x > 0 and positivity conditions for the coefficients c₁(2, 1), c₁(2, 2), c₁(3, 2) can be obtained by some direct computations, we will concentrate our consideration on (ii) and (iii) in Lemma 2.2.

Set ηn=vnun. Then we may prove that η_n ↗ U; see [20, Lemma 3.3], where

where the values Ψ₀ and Ψ₁ are those associated with the Stampfli completion.

Lemma 3.1

With the notation above, we have that Γn(1):=v4v2v0+η2nρ(1)≥0 for all n ≥ 3, if and only if one of the following conditions holds:

• ρ⁽¹⁾ ≥ 0,

• ρ⁽¹⁾ < 0 and v₄v₂v₀ + Uρ⁽¹⁾ ≥ 0,

where ρ⁽¹⁾ = v₄c₁(1, 1) − w₂c₁(0, 1) is as in (2.5).

Lemma 3.2

With above notation, we have that Ωn(1):=v4v2v0+η2n-2ρ(1)+η2n-2η2nτ(1)≥0 for n ≥ 4 if and only if one of the following conditions holds:

• s ≥ 0 and v₄v₂v₀ + η₆ρ⁽¹⁾ + η₆η₈τ⁽¹⁾ ≥ 0,

• s < 0 and v₄v₂v₀ + Uρ⁽¹⁾ + U²τ⁽¹⁾ ≥ 0,

where τ⁽¹⁾ = v₄c₁(1, 0) − w₂c₁(0, 0) as in (2.5) and

Proof. To consider the positivity of Ωn(1), we first define ΔΩn(1)=Ωn+1(1)-Ωn(1) for any n ∈ ℕ. Then

For brevity, we set

We will claim that s_n(x, u, v,w) is constant in n ≥ 4, i.e., s_n is independent of n for n ≥ 4. The original idea for the proof comes from [14, Lemma 4.3]; see more recently [13]. Define

Observe that with α:1,1,x,(u,v,w)∧,

Then it follows from (3.1) and (3.2) that

A direct computation shows that

in fact, we can confirm that its value is

Put p=α62, where α₆ is the 6th term of α. Then it is well known that

which implies that

If we mimic the proof of [14, Lemma 4.3], we get

Repeating this argument, we obtain

Hence Q_n is constant in n, n ≥ 8, and so s_n(x, u, v,w) is constant in n for n ≥ 4.

Now, we set s = s(x, u, v,w) := s₄(x, u, v,w). By (3.1), obviously

Repeating the method in the proof of Lemma 3.1, we obtain that Ωn(1)≥0 for n ≥ 4 if and only if either (i) s ≥ 0 and Ω4(1)=v4v2v0+η6ρ(1)+η6η8τ(1)≥0 or (ii) s < 0 and limn→∞ Ωn(1)=v4v2v0+Uρ(1)+U2τ(1)≥0.

Hence the proof is complete.

3.2. The p.d.c. condition for Dn(2)

The following (n + 1) × (n + 1) matrix is Dn(2):

As in the case of Dn(1), conditions for the positivity of the coefficients c₂(1, 1), c₂(2, 1), c₂(2, 2), c₂(3, 2) can be obtained by some direct computations. One may also compute that

It is straightforward to verify that u2(w-u)3w(v-u)(uv-2uw+w2) is positive, and it results from the above that c₂(2, 1) and c₂(1, 1) have the same sign, as do c₂(3, 2) and c₂(2, 2).

Lemma 3.3

With the above notation, the following conditions are equivalent:

• Γn(2):=v5v3v1+η2n+1ρ(2)≥0for all n ≥ 3,

• Ωn(2):=v5v3v1+η2n-1ρ(2)≥0for all n ≥ 4,

• one of the following conditions holds:

  • (iii-a) ρ⁽²⁾ ≥ 0,

  • (iii-b) ρ⁽²⁾ < 0 and v₅v₃v₁ + Uρ⁽²⁾ ≥ 0,

  where ρ⁽²⁾ = v₅c₂(1, 1) − w₃c₂(0, 1) is as in (2.5).