Let ℋ be a separable, infinite dimensional, complex Hilbert space and let B(ℋ) denote the algebra of all bounded linear operators on ℋ. An operator T ∈ B(ℋ) is said to be normal if T^*T = TT^*, hyponormal if T^*T ≥ TT^*, and subnormal if there exists a normal operator N on some Hilbert space such that T = N|_ℋ. To discuss gaps between hyponormality and subnormality, several classes of operators have been introduced, for example, k-hyponormal and weakly k-hyponormal operators (cf. [3]), whose definitions will be given below. An n-tuple (T₁, ···, T_n) of operators in B(ℋ) is hyponormal if the operator matrix ([Tj*,Ti])i,j=1n is positive on the direct sum of n copies of ℋ, where [X, Y] = XY − YX for X, Y ∈ B(ℋ). An n-tuple (T₁, ···, T_n) is weakly hyponormal if λ₁T₁ + ··· + λ_nT_n is hyponormal for every λ_i ∈ ℂ, i = 1, ···, n, where ℂ is the set of complex numbers. For k ≥ 1 and T ∈ B(ℋ), T is k-hyponormal if (I, T, ···, T^k) is hyponormal. An operator T ∈ B(ℋ) is weakly k-hyponormal if (T, T², ···, T^k) is weakly hyponormal. It is well known that subnormal ⇒ k-hyponormal ⇒ weakly k-hyponormal, for every k ≥ 1. In particular, weak 2- and weak 3-hyponormality are often referred to as quadratic- and cubic-hyponormality [5, 6, 8, 9, 10].

The weighted shifts have played a fundamental role in studying properties of weak subnormality. Indeed the flatness of weighted shift operators makes an important role to detect the structure of k-hyponormality and weak k-hyponormality of weighted shifts [2, 4, 5, 6]. In 1966, Stampfli [13] proved that if a weighted shift W_α with a weight sequence α={αn}n=0∞ is subnormal with α_k = α_k+1 for some k ≥ 0, then α₁ = α₂ = ···. In 1990, Curto [4] showed that Stampfli’s result on flatness holds in the case of 2-hyponormality. It is well known that the associated weighted shift W_α with a weight sequence α:2/3,2/3,(k+1)/(k+2) (k≥2) is quadratically hyponormal but not 2-hyponormal [3, 4]. Recently Cho-Lee-Li [11] showed such the property of flatness also preserves when W_α is a cubically hyponormal weighted shift. Also, in [7] they proved that a semi-cubically hyponormal weighted shift W_α with α_k = α_k+1 for some k ≥ 1 has α₁ = α₂ = ···, i.e., it has flatness property. This paper is a continuation of such study.

This paper consists of four sections. In Section 2 we recall the definition of cubically hyponormal weighted shifts and obtain a recursive formula of determinants for a pentadiagonal matrix corresponding to its operator. In Section 3 we introduce a new notion between cubic hyponormality and quadratic hyponormality. For 0<θ≤π2, we define a local-cubically hyponormal of order θ, which will serve as a bridge of gaps between quadratic hyponormality and semi-cubic hyponormality. In Section 4, we discuss the flatness of local-cubic hyponormal weighted shifts of some order θ. Finally, we prove that a local-cubically hyponormal weighted shift W_α with first three equal weights of order θ satisfies the flatness property.

Some of the calculations in this paper were obtained through computer experiments using the software tool Mathematica [15].

Let W_α be a weighted shift with a weight sequence α={αi}i=0∞. We recall that a weighted shift W_α is said to be cubically hyponormal if

In particular, if we take s = 0 in (2.1), the weighted shift W_α is semi-cubically hyponormal [1, 7]. Also, if t = 0 in (2.1), then W_α is quadratically hyponormal. Let P_n denote the orthogonal projection onto ∨i=0n{ei}. For n ≥ 0 and s, t ∈ ℂ, define

where

with α₋₃ = α₋₂ = α₋₁ = 0. It is obvious that if W_α is cubically hyponormal if and only if D_n(s, t) ≥ 0 for every s, t ∈ ℂ and n ≥ 0. To detect the positivity of D_n(s, t) for every s, t ∈ ℂ and n ≥ 0, we usually utilize the Sylvester’s criterion, which is called the nested determinants test (see [5, p.213]). We consider the determinant for the matrix D_n, d_n ≡ d_n(s, t): = det D_n. By simple computations, we get

In order to find the recursive formula of the determinant d_n for any n ≥ 4, we consider the following pentadiagonal submatrices in (2.2) as follows

where q_i, r_i and z_i (i ≥ 0) are given in (2.3). In particular, we give initial matrices of Φ_i and Ψ_i for i = 0, 1 as follows: Φ₀: = (r̄₀), Ψ₀: = (r₀) (which are considered as 1 × 1 matrices for our convenience),

Now we denote φ_j and ψ_j for the determinants of the matrices Φ_j and Ψ_j (j ≥ 0) in (2.5) and (2.6), respectively. Then it is obvious that ψ_j = φ̄_j for all j ≥ 0. Moreover we can have the following Lemma 2.1.

Lemma 2.1

For any j ≥ 0, it holds that:

where d_n denotes for the determinant of the matrix D_n in (2.2).

We now give the recursive formula of determinant of pentadiagonal matrix in (2.2) which improves the main theorem in [14]; see Corollary 2.3 below.

Theorem 2.2

Let the matrix D_n be given by (2.2) and let d_n = det D_n (n ≥ 1). Then the following recursive formula holds:

with d_i ≡ 1 and φ_i ≡ 1 for i = −3, −2, −1.

Corollary 2.3

Let the matrix D_n be given by (2.2) and let n ≥ 1. Then

where d_i ≡ 1 for i = −4, −3, −2, −1.

Recall that cubic hyponormality implies semi-cubic hyponormality, and semi-cubic hyponormality and quadratic hyponormality are independent one from another [12]. We now define a new notion, namely local-cubic hyponormality between cubic hyponormality and quadratic hyponormality.

Definition 3.1

Let α={αi}i=0∞ be a sequence of positive real numbers and let W_α be the associated weighted shift with a sequence α. For θ∈[0,π2], a weighted shift W_α is called a local-cubically hyponormal of order θ if Wα+s(cosθ)Wα2+s(sinθ)Wα3 is hyponormal for all s ∈ ℂ, i.e.,

Remark 3.2

Let W_α be the weighted shift with a weight sequence α={αi}i=0∞. Then the following assertions hold.

• W_α is local-cubically hyponormal of order 0 if and only if it is quadratically hyponormal.

• W_α is local-cubically hyponormal of order π2 if and only if it is semi-cubically hyponormal.

Proposition 3.3

Letα:23,23,{n+1n+2}n=2∞and let W_α be the associated weighted shift. Then there exists a subinterval J of(0,π2)such that, for any θ ∈ J, W_α can not be local-cubically hyponormal of order θ.

Problem 3.4

Let α:23,23,{n+1n+2}n=2∞ and let W_α be the associated weighted shift. For arbitrary θ∈(0,π2), is it true that W_α is not a local-cubically hyponormal weighted shift of order θ?

Recall that if α:2/3,2/3,(k+1)/(k+2) (k≥2), then it is well-known that the associated weighted shift W_α is quadratically hyponormal, i.e., the local-cubical hyponormality with order 0 does not preserve the flatness. It follows from [7, Lemma 3.10] that if W_α is a local-cubically hyponormal weighted shift of order π2 with α₁ = α₂, then α₁ = α₂ = ···. Hence the following natural question arises.

Question: Forθ∈(0,π2), suppose that W_α is local-cubically hyponormal weighted shift of order θ such that α₀ = α₁ = α₂. Is it true α₀ = α₁ = α₂ = ···?

The answer of this question is affirmative (see Theorem 4.2). Before proving Theorem 4.2, we give an elementary lemma of the outer propagation for reader’s convenience.

Lemma 4.1

Letα={αi}i=0∞be a sequence of positive real numbers and let W_α be the associated weighted shift with a sequence α. If W_α is local-cubically hyponormal of order θ, then the weighted shift W_α′with a weight sequenceα′={αn}n=k∞is also local-cubically hyponormal of order θ, k ≥ 1.

Note that Lemma 4.1 can be extended to the case of any weakly n-hyponormal weighted shifts.

Theorem 4.2

Forθ∈(0,π2), suppose that a weighted shift W_α is local-cubically hyponormal of order θ such that α₀ = α₁ = α₂. Then α₀ = α₁ = α₂ = α₃ = ···, i.e. W_α satisfies the flatness property.

The authors would like to express their gratitude to the referee for careful reading of the paper and helpful comments.

Expressions of polynomials c_n in the proof of Theorem 4.2.

• For n = 2, c2=-h(2+h) tan2 θ∑i=02c2i tan2i θ,

  where c20=6h2(2+h)2(1+h+k)2, c21=∑j=02φ21jℓj, c22=(1+h+k)2∑j=02φ22jmj,

  φ210:=Σj=06φ210jkj, ϕ₂₁₁ := −2(1 + h + k)³(2h + h² + 2k + 2hk)(4 + 6h + 3h² + 2k + 2hk),

  ϕ₂₁₂ := −(1 + h + k)²(2h + h² + 2k + 2hk)(4 + 6h + 3h² + 2k + 2hk),

  φ220:=Σi=04φ220iℓi, ϕ₂₂₁ := −2(2h + h² + k + hk)(2 + 2h + h² + k + hk)(1 + h + k + ℓ)³,

  ϕ₂₂₂ := −(2h + h² + k + hk)(2 + 2h + h² + k + hk)(1 + h + k + ℓ)²,

  ϕ₂₁₀₀ := −h²(1 + h)²(2 + h)²(7 + 6h + 3h²), ϕ₂₁₀₁ := −4h(1 + h)³(2 + h)(8 + 10h + 5h²),

  ϕ₂₁₀₂ := −2(1 + h)²(4 + 6h + 3h²)(4 + 18h + 9h²), ϕ₂₁₀₃ := −4(1 + h)(16 + 72h + 112h² + 76h³ + 19h⁴),

  ϕ₂₁₀₄ := −56 − 232h − 352h² − 236h³ − 59h⁴, ϕ₂₁₀₅ := −24(1 + h)³, ϕ₂₁₀₆ := −4(1 + h)²,

  φ2200:=Σj=06φ2200jkj,

  ϕ₂₂₀₁ := 2(1 + h + k)³(−6h−3h² + 8h³ + 12h⁴ + 6h⁵ + h⁶ −4k −12hk −12h²k −4h³k −2k² −4hk² −2h²k²),

  ϕ₂₂₀₂ := (1 + h + k)²(−22h−27h²−8h³ + 8h⁴ + 6h⁵ + h⁶−12k−36hk−36h²k−12h³k−6k²−12hk²−6h²k²),

  ϕ₂₂₀₃ := −4(1 + h + k)(2h + h² + k + hk)(2 + 2h + h² + k + hk), ϕ₂₂₀₄ := −(2h + h² + k + hk)(2 + 2h + h² + k + hk),

  ϕ₂₂₀₀₀ := h³(1 + h)²(2 + h)³(2 + 2h + h²), ϕ₂₂₀₀₁ := 2(1 + h)³(−1 − 8h + 20h³ + 25h⁴ + 12h⁵ + 2h⁶),

  ϕ₂₂₀₀₂ := 3(1 + h)²(3 + 4h + 2h²)(−1 − 4h + 2h² + 4h³ + h⁴),

  ϕ₂₂₀₀₃ := 4(1 + h)(−4 − 18h − 21h² − 4h³ + 9h⁴ + 6h⁵ + h⁶),

  ϕ₂₂₀₀₄ := −14 − 58h − 81h² − 44h³ − h⁴ + 6h⁵ + h⁶, ϕ₂₂₀₀₅ := −6(1 + h)³, ϕ₂₂₀₀₆ := −(1 + h)².

• For n = 3, c3=-2h(2+h) tan θ∑i=02c3i tan2i θ,

  where c30=-2h2(2+h)2(1+h+k)2, c31=∑j=02φ31jℓj, c32=(1+h+k)2∑i=02φ32imi,

  φ310:=Σj=06φ310jkj,

  ϕ₃₁₁ := 2(1 + h + k)³(14h + 27h² + 20h³ + 5h⁴ + 12k + 36hk + 36h²k + 12h³k + 6k² + 12hk² + 6h²k²),

  ϕ₃₁₂ := (1 + h + k)²(14h + 27h² + 20h³ + 5h⁴ + 12k + 36hk + 36h²k + 12h³k + 6k² + 12hk² + 6h²k²),

  φ320:=Σj=04φ320jℓj,

  ϕ₃₂₁ := 4(2h + h² + k + hk)(2 + 2h + h² + k + hk)(1 + h + k + ℓ)³,

  ϕ₃₂₂ := 2(2h + h² + k + hk)(2 + 2h + h² + k + hk)(1 + h + k + ℓ)²,

  ϕ₃₁₀₀ := h²(1 + h)²(2 + h)²(11 + 10h + 5h²), ϕ₃₁₀₁ := 2h(1 + h)³(2 + h)(25 + 32h + 16h²),

  ϕ₃₁₀₂ := (1 + h)²(48 + 298h + 485h² + 336h³ + 84h⁴), ϕ₃₁₀₃ := 4(1 + h)(24 + 110h + 171h² + 116h³ + 29h⁴),

  ϕ₃₁₀₄ := 84 + 350h + 531h² + 356h³ + 89h⁴, ϕ₃₁₀₅ := 36(1 + h)³, ϕ₃₁₀₆ := 6(1 + h)²,

  φ3200:=Σi=06φ3200iki, φ3201:=2(1+h+k)Σi=04φ3201iki,φ3202:=Σi=04φ3202iki,

  ϕ₃₂₀₀₀ := h²(1 + h)²(2 + h)²(3 + 2h + h²), ϕ₃₂₀₀₁ := 2h(1 + h)³(2 + h)(7 + 8h + 4h²),

  ϕ₃₂₀₀₂ := (1 + h)²(8 + 86h + 139h² + 96h³ + 24h⁴), ϕ₃₂₀₀₃ := 4(1 + h)(6 + 34h + 53h² + 36h³ + 9h⁴),

  ϕ₃₂₀₀₄ := 26 + 114h + 173h² + 116h³ + 29h⁴, ϕ₃₂₀₀₅ := 12(1 + h)³, ϕ₃₂₀₀₆ := 2(1 + h)²,

  ϕ₃₂₀₁₀ := 3h(1 + h)²(2 + h)(2 + 2h + h²), ϕ₃₂₀₁₁ := 2(1 + h)(2 + 26h + 41h² + 28h³ + 7h⁴),

  ϕ₃₂₀₁₂ := 18 + 90h + 137h² + 92h³ + 23h⁴, ϕ₃₂₀₁₃ := 16(1 + h)³, ϕ₃₂₀₁₄ := 4(1 + h)²,

  ϕ₃₂₀₂₀ := 11h(1 + h)²(2 + h)(2 + 2h + h²), ϕ₃₂₀₂₁ := 2(1 + h)(10 + 90h + 137h² + 92h³ + 23h⁴),

  ϕ₃₂₀₂₂ := 58 + 282h + 425h² + 284h³ + 71h⁴, ϕ₃₂₀₂₃ := 48(1 + h)³, ϕ₃₂₀₂₄ := 12(1 + h)².

• For n = 4, c4=∑i=04c4i tan2i θ,

  where

  • c₄₀ = h(2 + h)(2h + h² + k + hk)(2 + 2h + h² + k + hk),

  • c₄₁ = h(1 + h)²(2 + h)(1 + h + k)²(h + k + ℓ)(2 + h + k + ℓ),

  • c₄₂ = (1 + h)²(h + k)(2 + h + k)τ₁τ₂,

  • c₄₃ = (1 + h)⁴(1 + h + k)²(2h + h² + k + hk)(2 + 2h + h² + k + hk)(1 + h + k + ℓ)²,

  • c₄₄ = (1 + h)⁴(1 + h + k)²τ₁τ₂,

  τ₁ := 3h + 3h² + h³ + 2k + 4hk + 2h²k + k² + hk² + ℓ + 2hℓ + h²ℓ + kℓ + hkℓ,

  τ₂ := 2 + 3h + 3h² + h³ + 2k + 4hk + 2h²k + k² + hk² + ℓ + 2hℓ + h²ℓ + kℓ + hkℓ.

For cases n = 5 and n = 6, we can have some polynomials c₅ and c₆ in h, k, ℓ, m. But we leave these expressions to the interested readers.