On 𝑘th-Order Slant Weighted Toeplitz Operator

Abstract

Let 𝛽={𝛽_𝑛}_𝑛∈ℤ be a sequence of positive numbers with 𝛽₀=1, 0<𝛽_𝑛/𝛽_𝑛+1≤1 when 𝑛≥0 and 0<𝛽_𝑛/𝛽_𝑛−1≤1 when 𝑛≤0. A 𝑘th-order slant weighted Toeplitz operator on 𝐿²(𝛽) is given by 𝑈_𝜙=𝑊_𝑘𝑀_𝜙, where 𝑀_𝜙 is the multiplication on 𝐿²(𝛽) and 𝑊_𝑘 is an operator on 𝐿²(𝛽) given by 𝑊_𝑘𝑒_𝑛𝑘(𝑧)=(𝛽_𝑛/𝛽_𝑛𝑘)𝑒_𝑛(𝑧), {𝑒_𝑛(𝑧)=𝑧^𝑘/𝛽_𝑘}_𝑘∈ℤ being the orthonormal basis for 𝐿²(𝛽). In this paper, we define a 𝑘th-order slant weighted Toeplitz matrix and characterise 𝑈_𝜙 in terms of this matrix. We further prove some properties of 𝑈_𝜙 using this characterisation.

1. Introduction

Toeplitz operators and slant Toeplitz operators [1] have been found immensely useful, especially in the study of prediction theory [2], wavelet analysis [3], and solution of differential equations [4]. Originally, these operators were defined and studied on the usual 𝐻² and 𝐿² spaces. During the past few decades, different generalisations of these spaces, like the weighted function spaces 𝐻^𝑃_𝑤 and 𝐿^𝑃_𝑤 and the weighted sequence spaces 𝐻²(𝛽) and 𝐿²(𝛽) have developed. Shields [5] has made a systematic study of the multiplication operator on 𝐿²(𝛽). Lauric [6] has discussed the Toeplitz operator on 𝐻²(𝛽). Motivated by these studies, we introduced and studied the notion of a slant weighted Toeplitz operator [7] on 𝐿²(𝛽). In this paper, we study a 𝑘th-order slant weighted Toeplitz operator on the space 𝐿²(𝛽). We begin with the following preliminaries.

Let 𝛽={𝛽_𝑛}_𝑛∈ℤ be a sequence of positive numbers with 𝛽₀=1, 0<(𝛽_𝑛/𝛽_𝑛+1)≤1 when 𝑛≥0 and 0<(𝛽_𝑛/𝛽_𝑛−1)≤1 when 𝑛≤0.

Throughout the paper, we assume that for a fixed integer 𝑘≥2, 𝛽_𝑘𝑛/𝛽_𝑛≤𝑀<∞. Consider the following spaces [5, 6]: 𝐿²𝛽=𝑓(𝑧)=^∞_𝑛=−∞𝑎_𝑛𝑧^𝑛𝑎_𝑛𝑓∈ℂ,_2𝛽=^∞_𝑛=−∞𝑎_𝑛²𝛽²_𝑛,𝐻<∞₂𝛽=𝑓(𝑧)=^∞_𝑛=0𝑎_𝑛𝑧^𝑛𝑎_𝑛𝑓∈ℂ,_2𝛽=^∞_𝑛=0𝑎_𝑛²𝛽²_𝑛<∞. (1)

Then, (𝐿²(𝛽), ‖⋅‖_𝛽) is a Hilbert space [6] with an orthonormal basis given by {𝑒_𝑘(𝑧)=𝑧^𝑘/𝛽_𝑘}_𝑘∈ℤ and with an inner product defined by ^∞_𝑛=−∞𝑎_𝑛𝑧^𝑛,_𝑛=−∞𝑏_𝑛𝑧^𝑛=^∞_𝑛=−∞𝑎_𝑛𝑏_𝑛𝛽²_𝑛. (2) Also, 𝐻²(𝛽) is a subspace of 𝐿²(𝛽). Further, the space 𝐿^∞𝛽=𝜙(𝑧)=^∞_𝑛=−∞𝑎_𝑛𝑧^𝑛𝜙𝐿²𝛽⊆𝐿²𝛽,∃𝑐∈𝑅suchthat𝜙𝑓_𝛽≤𝑐𝑓_𝛽∀𝑓∈𝐿²𝛽 (3) is a Banach space with respect to the norm defined by 𝜙_∞=inf𝑐|𝜙𝑓_𝛽≤𝑐𝑓_𝛽∀𝑓∈𝐿²𝛽. (4) The mapping 𝑃∶𝐿²(𝛽)→𝐻²(𝛽) is the orthogonal projection of 𝐿²(𝛽) onto 𝐻²(𝛽). For a given 𝜙∈𝐿^∞(𝛽), the induced weighted multiplication operator [5] is denoted by 𝑀_𝜙 and is given by 𝑀_𝜙∶𝐿²(𝛽)→𝐿²(𝛽) such that 𝑀_𝜙𝑒_𝑘(𝑧)=1𝛽_𝑘^∞_𝑛=−∞𝑎_𝑛𝛽_𝑛+𝑘𝑒_𝑛+𝑘(𝑧)=1𝛽_𝑘^∞_𝑛=−∞𝑎_𝑛−𝑘𝛽_𝑛𝑒_𝑛(𝑧). (5)

If we put 𝜙₁(𝑧)=𝑧, then 𝑀_𝜙1=𝑀_𝑧 is the operator defined as 𝑀_𝑧𝑒_𝑘(𝑧)=𝑤_𝑘𝑒_𝑘+1(𝑧), where 𝑤_𝑘=𝛽_𝑘+1/𝛽_𝑘 for all 𝑘∈ℤ, and it is known as a weighted shift [5].

The slant weighted Toeplitz operator [7] 𝐴_𝜙 is an operator on 𝐿²(𝛽) defined as 𝐴_𝜙∶𝐿²(𝛽)→𝐿²(𝛽) such that 𝐴_𝜙𝑒_𝑘(𝑧)=(1/𝛽_𝑘)∑^∞_𝑛=−∞𝑎_2𝑛−𝑘𝛽_𝑛𝑒_𝑛(𝑧).

If 𝑊∶𝐿²(𝛽)→𝐿²(𝛽) is such that 𝑊𝑒_2𝑛(𝑧)=𝛽_𝑛𝛽_2𝑛𝑒_𝑛(𝑧),𝑊𝑒_2𝑛−1(𝑧)=0∀𝑛∈ℤ, (6) then 𝐴_𝜙 can be alternately defined by 𝐴_𝜙𝑓=𝑊𝑀_𝜙𝑓=𝑊𝜙𝑓∀𝑓∈𝐿²𝛽. (7)

2. 𝑘th-Order Slant Weighted Toeplitz Operator

Suppose that the operator 𝑊_𝑘∶𝐿²(𝛽)→𝐿²(𝛽) is such that 𝑊_𝑘𝑒_𝑛(𝑧)=𝛽_𝑛/𝑘𝛽_𝑛𝑒_𝑛/𝑘(𝑧)if𝑛isdivisibleby0𝑘,otherwise, (8) Then the matrix of 𝑊_𝑘 is ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯𝛽₀𝛽₀⋯0000000⋯𝛽0000¹𝛽_𝑘0⋯⋯⋯_𝑘-times⋯𝛽0000⋯⋯0²𝛽_2𝑘⋯⋯⋯00⋯⋯⋯⋯⋯⋯⋯.⋯⋯⋯⋯⋯⋯⋯⋯⋯ (9) The adjoint of 𝑊_𝑘 is given by 𝑊^∗_𝑘𝑒_𝑛(𝑧)=𝛽_𝑛𝛽_𝑘𝑛𝑒_𝑘𝑛(𝑧)∀𝑛∈ℤ. (10)

Definition 1 (see [8]). For an integer 𝑘≥2, the 𝑘th order slant weighted Toeplitz operator 𝑈_𝜙∶𝐿²(𝛽)→𝐿²(𝛽) is such that 𝑈_𝜙(𝑓)=𝑊_𝑘𝑀_𝜙(𝑓) for all 𝑓∈𝐿²(𝛽). Thus, 𝑈_𝜙𝑒_𝑗(𝑧)=(1/𝛽_𝑗)∑^∞_𝑛=−∞𝑎_{𝑛𝑘−𝑗}𝛽_𝑛𝑒_𝑛(𝑧).
The (𝑖,𝑗)th entry of the matrix of 𝑈_𝜙 is given by 𝑈_𝜙𝑒_𝑗,𝑒_𝑖=1𝛽_𝑗^∞_𝑛=−∞𝑎_{𝑛𝑘−𝑗}𝛽_𝑛𝑒_𝑛(𝑧),𝑒_𝑖(𝑧)=𝑎_{𝑖𝑘−𝑗}𝛽_𝑖𝛽_𝑗. (11) Hence, the matrix of 𝑈_𝜙 with respect to this basis is as follows: ⋯⋯⋯⋯⋯⋯⋯𝑎_−𝑘+1𝛽₋₁𝛽₋₁𝑎_−𝑘𝛽₁𝛽₀𝑎_−𝑘−1𝛽₋₁𝛽₁𝑎_−𝑘−2𝛽₋₁𝛽₂⋯⋯𝑎₁𝛽₀𝛽₋₁𝑎₀𝛽₀𝛽₀𝑎₋₁𝛽₀𝛽₁𝑎₋₂𝛽₀𝛽₂⋯⋯𝑎_𝑘+1𝛽₁𝛽₋₁𝑎_𝑘𝛽₁𝛽₀𝑎_𝑘−1𝛽₁𝛽₁𝑎_𝑘−2𝛽₁𝛽₂⋯⋯𝑎_2𝑘+1𝛽₂𝛽₋₁𝑎_2𝑘𝛽₂𝛽₀𝑎_2𝑘−1𝛽₂𝛽₁𝑎_2𝑘−2𝛽₂𝛽₂⋯⋯⋯.⋯⋯⋯⋯ (12)

Theorem 2. The mapping 𝜙→𝑈_𝜙 is linear and one to one.

Proof. For linearity, consider 𝑈_{𝛼𝜙+𝛽𝜓}=𝑊_𝑘𝑀_{𝛼𝜙+𝛽𝜓}=𝑊_𝑘𝑀_𝛼𝜙+𝑀_𝛽𝜓=𝑊_𝑘𝑀_𝛼𝜙+𝑊_𝑘𝑀_𝛽𝜓=𝛼𝑊_𝑘𝑀_𝜙+𝛽𝑊_𝑘𝑀_𝜓=𝛼𝑈_𝜙+𝛽𝑈_𝜓. (13) For checking one-to-one, let 𝑈_𝜙=𝑈_𝜓.
Then, 𝑈_𝜙−𝑈_𝜓=0. By linearity, we get 𝑈_𝜙−𝜓=0.
Hence, 𝑈_𝜙−𝜓𝑒_𝑛(𝑧)=0,𝑛∈ℤ,⟹𝑊_𝑘𝑀_𝜙−𝜓𝑒_𝑛(𝑧)=0,𝑛∈ℤ,⟹𝑊_𝑘𝜙−𝜓𝑒_𝑛(𝑧)=0,𝑛∈ℤ. (14) This implies that either 𝜙−𝜓=0 or the degree of (𝜙−𝜓)𝑒_𝑛(𝑧) is not divisible by 𝑘. But since this is true for all 𝑛∈ℤ, the second possibility is ruled out. Hence, 𝜙−𝜓=0 or 𝜙=𝜓.

Theorem 3. Consider the following: (i) 𝑀_𝑧𝑊_𝑘=𝑊_𝑘𝑀_𝑧𝑘; (ii) 𝑀_𝑧𝑚𝑊_𝑘=𝑊_𝑘𝑀_𝑧𝑘𝑚, 𝑚∈ℤ.

Proof. (i) It is sufficient to prove that 𝑀_𝑧𝑊_𝑘𝑒_𝑛(𝑧)=𝑊_𝑘𝑀_𝑧𝑘𝑒_𝑛(𝑧)∀𝑛∈ℤ. (15)
Suppose that 𝑛 is not a multiple of 𝑘. Then, 𝑀_𝑧𝑊_𝑘𝑒_𝑛(𝑧)=𝑀_𝑧0=0=𝑊_𝑘𝑀_𝑧𝑘𝑒_𝑛(𝑧). (16)
Now, when 𝑛=𝑝𝑘 (multiple of 𝑘), 𝑀_𝑧𝑊_𝑘𝑒_𝑛(𝑧)=𝑀_𝑧𝛽_𝑝𝛽_𝑝𝑘𝑒_𝑝(𝑧)=𝛽_𝑝+1𝛽_𝑝𝑘𝑒_𝑝+1(𝑧). (17)
On the other hand, 𝑊_𝑘𝑀_𝑧𝑘𝑒_𝑛(𝑧)=𝑊_𝑘𝑀_𝑧𝑘𝑒_𝑝𝑘(𝑧)=𝑊_𝑘𝛽_(𝑝+1)𝑘𝛽_𝑝𝑘𝑒_(𝑝+1)𝑘(𝑧)=𝛽_𝑝+1𝛽_𝑝𝑘𝑒_𝑝+1(𝑧). (18)
Hence from (17) and (18), we get that 𝑀_𝑧𝑊_𝑘𝑒_𝑛(𝑧)=𝑊_𝑘𝑀_𝑧𝑘𝑒_𝑛(𝑧)whenever𝑛isamultipleof𝑘. (19) We therefore conclude that for all 𝑛∈ℤ, 𝑀_𝑧𝑊_𝑘𝑒_𝑛(𝑧)=𝑊_𝑘𝑀_𝑧𝑘𝑒_𝑛(𝑧)⟹𝑀_𝑧𝑊_𝑘=𝑊_𝑘𝑀_𝑧𝑘. (20)
(ii) We prove the result by induction on 𝑚.
For 𝑚=1, the result is true by part (i).
Suppose that the result is true for 𝑚=𝑙. Then, 𝑀_𝑧𝑙𝑊_𝑘=𝑊_𝑘𝑀_𝑧𝑙𝑘. (21)
Now 𝑀_{𝑧(𝑙+1)𝑘}𝑊_𝑘=𝑀_𝑧𝑀_𝑧𝑙𝑊_𝑘=𝑀_𝑧𝑊_𝑘𝑀_𝑧𝑙𝑘=𝑊_𝑘𝑀_𝑧𝑘𝑀_𝑧𝑙𝑘=𝑊_𝑘𝑀_{𝑧(𝑙+1)𝑘}. (22)
Hence by induction, the result is true for all 𝑚∈𝑍⁺.
For 𝑚=−1 and 1≤𝑗≤𝑘−1, 𝑀_𝑧𝑚𝑊_𝑘𝑒_{𝑛𝑘+𝑗}(𝑧)=𝑀_𝑧−10=0=𝑊_𝑘𝑀_𝑧−𝑘𝑒_{𝑛𝑘+𝑗}(𝑧). (23) For 𝑚=−1 and 𝑗=𝑛𝑘 (a multiple of 𝑘), 𝑀_𝑧𝑚𝑊_𝑘𝑒_𝑗(𝑧)=𝑀_𝑧−1𝑊_𝑘𝑒_𝑛𝑘(𝑧)=𝑀_𝑧−1𝛽_𝑛𝛽_𝑛𝑘𝑒_𝑛(𝑧)=𝛽_𝑛𝛽_𝑛𝑘𝑀_𝑧−1𝑒_𝑛(𝑧)=𝛽_𝑛𝛽_𝑛𝑘𝛽_𝑛−1𝛽_𝑛𝑒_𝑛−1(𝑧)=𝛽_𝑛−1𝛽_𝑛𝑘𝑒_𝑛−1(𝑧). (24) On the other hand, 𝑊_𝑘𝑀_𝑧−1𝑒_𝑛𝑘(𝑧)=𝑊_𝑘𝑧⁻¹𝑧^𝑛𝑘𝛽_𝑛𝑘=𝑊_𝑘𝑧^{(𝑛−1)𝑘}𝛽_𝑛𝑘𝛽_{(𝑛−1)𝑘}𝛽_{(𝑛−1)𝑘}=𝛽_{(𝑛−1)𝑘}𝛽_𝑛𝑘𝑊_𝑘𝑒_{(𝑛−1)𝑘}(𝑧)=𝛽_{(𝑛−1)𝑘}𝛽_𝑛𝑘𝛽_𝑛−1𝛽_{(𝑛−1)𝑘}𝑒_𝑛−1(𝑧)=𝛽_𝑛−1𝛽_𝑛𝑘𝑒_𝑛−1(𝑧). (25) From (23), (24), and (25), we conclude that 𝑀_𝑧−1𝑊_𝑘=𝑊_𝑘𝑀_𝑧−𝑘. (26) Hence, the result is true for 𝑚=−1 also. Therefore, by using induction, we can prove it for all negative integers 𝑚. The case when 𝑚=0 is trivially true.
Hence the theorem is true.

Corollary 4. Let 𝜙(𝑧)=∑^∞_𝑛=−∞𝑎_𝑛𝑧^𝑛∈𝐿^∞(𝛽). Then, 𝑀_𝜙(𝑧)𝑊_𝑘=𝑊_𝑘𝑀_{𝜙(𝑧𝑘)}. (27)

Theorem 5. Consider the following: 𝑀_𝜙(𝑧)𝑈_𝜓(𝑧)=𝑈_{𝜙(𝑧𝑘)𝜓(𝑧)}.

Proof. The proof is as follows: 𝑀_𝜙(𝑧)𝑈_𝜓(𝑧)=𝑀_𝜙(𝑧)𝑊_𝑘𝑀_𝜓(𝑧)=𝑊_𝑘𝑀_{𝜙(𝑧𝑘)}𝑀_𝜓(𝑧)=𝑊_𝑘𝑀_{𝜙(𝑧𝑘)𝜓(𝑧)}=𝑈_{𝜙(𝑧𝑘)𝜓(𝑧)}. (28)

3. Matrix Characterisation of 𝑈_𝜙

Definition 6. Let 𝑤_𝑛=𝛽_𝑛+1/𝛽_𝑛 for all 𝑛∈ℤ. Then, the 𝑘th order slant weighted Toeplitz matrix corresponding to the weight sequence ⟨𝑤_𝑛⟩ is a bilaterally infinite matrix ⟨𝜆_𝑖𝑗⟩ such that 𝜆_{𝑖+1,𝑗+𝑘}=𝑤_𝑖𝑤_𝑗𝑤_𝑗+1⋯𝑤_{𝑗+𝑘−1}𝜆_𝑖,𝑗,𝑖,𝑗=0,±1,±2,…. (29)
We have proved earlier [8] that 𝑈 is a 𝑘th order slant weighted Toeplitz operator if and only if 𝑀_𝑧𝑈=𝑈𝑀_𝑧𝑘.
Next, we prove a characterisation of the 𝑘th order slant weighted Toeplitz operator in terms of the matrix previously defined.

Theorem 7. A necessary and sufficient condition that an operator 𝑈 on 𝐿²(𝛽) is a 𝑘th order slant weighted Toeplitz operator is that its matrix with respect to the orthonormal basis {𝑒_𝑘(𝑧)=𝑧^𝑘/𝛽_𝑘}_𝑘∈ℤ is a 𝑘th order slant weighted Toeplitz matrix.

Proof. Suppose that 𝑈 is a 𝑘th order slant weighted Toeplitz operator. Then, the corresponding matrix ⟨𝜆_𝑖𝑗⟩ is given by 𝜆_𝑖𝑗=𝑈𝑒_𝑗,𝑒_𝑖=𝑎_{𝑘𝑖−𝑗}𝛽_𝑖𝛽_𝑗. (30) Further, 𝜆_{𝑖+1,𝑗+𝑘}=𝑈𝑒_𝑗+𝑘,𝑒_𝑖+1=𝑎_{𝑘𝑖+𝑘−𝑗−𝑘}𝛽_𝑖+1𝛽_𝑗+𝑘=𝑎_{𝑘𝑖−𝑗}𝛽_𝑖+1𝛽_𝑗+𝑘=𝛽_𝑖+1𝛽_𝑗+𝑘𝛽_𝑗𝛽_𝑖𝜆_𝑖,𝑗=𝑤_𝑖𝑤_𝑗𝑤_𝑗+1⋯𝑤_{𝑗+𝑘−1}𝜆_𝑖,𝑗, (31) where 𝑤_𝑛=𝛽_𝑛+1/𝛽_𝑛 for all 𝑛∈ℤ. Thus, the matrix of  𝑈 is a 𝑘th order slant weighted Toeplitz matrix. Conversely, we assume that 𝑈 is an operator on 𝐿²(𝛽) whose matrix is a 𝑘th order slant weighted Toeplitz matrix. This means that for all 𝑖,𝑗∈ℤ, we have 𝑈𝑒_𝑗,𝑒_𝑖=𝑤_𝑗𝑤_𝑗+1⋯𝑤_{𝑗+𝑘−1}𝑤_𝑖𝑈𝑒_𝑗+𝑘,𝑒_𝑖+1. (32) Now, 𝑀_𝑧𝑈𝑒_𝑗,𝑒_𝑖=𝑈𝑒_𝑗,𝑀^∗_𝑧𝑒_𝑖=𝑈𝑒_𝑗,𝑤_𝑖−1𝑒_𝑖−1=𝑤_𝑖−1𝑈𝑒_𝑗,𝑒_𝑖−1=𝑤_𝑖−1𝑤_𝑗𝑤_𝑗+1⋯𝑤_{𝑗+𝑘−1}𝑤_𝑖−1×𝑈𝑒_𝑗+𝑘,𝑒_𝑖using()=32𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑒_𝑖,∀𝑖,𝑗∈ℤ. (33) Hence 𝑀_𝑧𝑈=𝑈𝑀_𝑧𝑘. Therefore, we conclude that 𝑈 is a 𝑘th order slant weighted Toeplitz operator.

Next, consider the operator 𝑆∶𝐿²(𝛽)→𝐿²(𝛽) given by 𝑆𝑒_𝑗=(1/𝑤_𝑗)𝑒_𝑗+1. Then, 𝑆^∗𝑒_𝑗=(1/𝑤_𝑗−1)𝑒_𝑗−1.

Now, 𝑆 is bounded as ⟨𝑤_𝑛⟩ is always positive and bounded.

Lemma 8. Consider the following: 𝑆^∗=𝑀_𝑧⁻¹.

Proof. The proof is as follows: 𝑆^∗𝑀_𝑧𝑒_𝑗=𝑆^∗𝑤_𝑗𝑒_𝑗+1=𝑤_𝑗𝑤_𝑗𝑒_𝑗=𝑒_𝑗,𝑀_𝑧𝑆^∗𝑒_𝑗=𝑀_𝑧1𝑤_𝑗−1𝑒_𝑗−1=𝑤_𝑗−1𝑤_𝑗−1𝑒_𝑗=𝑒_𝑗. (34)

Theorem 9. A bounded operator 𝑈 on 𝐿²(𝛽) is a 𝑘th order slant weighted Toeplitz operator if and only if  𝑈=𝑀_𝑧⁻¹𝑈𝑀_𝑧𝑘, where 𝑀_𝑧 and 𝑀_𝑧𝑘, are the multiplication operators induced by 𝑧 and 𝑧^𝑘, respectively.

Proof. Let 𝑈 be a 𝑘th order slant weighted Toeplitz operator on 𝐿²(𝛽). Then, from (32) we get that 𝑈𝑒_𝑗,𝑒_𝑖=𝑤_𝑗𝑤_𝑗+1⋯𝑤_{𝑗+𝑘−1}𝑤_𝑖𝑈𝑒_𝑗+𝑘,𝑒_𝑖+1=𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑆𝑒_𝑖=𝑆^∗𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑒_𝑖=𝑀_𝑧⁻¹𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑒_𝑖,𝑖,𝑗=0,±1,±2,…. (35) Thus, 𝑈=𝑀_𝑧⁻¹𝑈𝑀_𝑧𝑘.
For the converse part, we assume that 𝑈 is a bounded operator on 𝐿²(𝛽) satisfying 𝑈=𝑀_𝑧⁻¹𝑈𝑀_𝑧𝑘 for some fixed integer 𝑘≥2. Then, for all 𝑖,𝑗∈ℤ, 𝑈𝑒_𝑗,𝑒_𝑖=𝑀_𝑧⁻¹𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑒_𝑖=𝑆^∗𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑒_𝑖=𝑈𝑀_𝑧𝑘𝑒_𝑗,𝑆𝑒_𝑖𝑤_𝑗𝑤_𝑗+1⋯𝑤_{𝑗+𝑘−1}𝑤_𝑖𝑈𝑒_𝑗+𝑘,𝑒_𝑖+1. (36) The previous equation shows that the matrix corresponding to 𝑈 is a 𝑘th order slant weighted Toeplitz matrix. Hence, by Theroem 7,   𝑈 is a 𝑘th order slant weighted Toeplitz operator.

Corollary 10. For a fixed integer 𝑘≥2, the set of all 𝑘th order slant weighted Toeplitz operators is weakly closed and hence strongly closed.

Proof. We assume that for each positive integer 𝑛, 𝑈_𝑛 is a 𝑘th order slant weighted Toeplitz operator and 𝑈_𝑛→𝑈 weakly. Then, for 𝑓,𝑔∈𝐿²(𝛽), we get that ⟨𝑈_𝑛𝑓,𝑔⟩→⟨𝑈𝑓,𝑔⟩.
From the previous theorem, this implies that 𝑀_𝑧⁻¹𝑈_𝑛𝑀_𝑧𝑘𝑓,𝑔=𝑆^∗𝑈_𝑛𝑀_𝑧𝑘=𝑈𝑓,𝑔_𝑛𝑀_𝑧𝑘=𝑈𝑓,𝑆𝑔_𝑛𝑧^𝑘⋅𝑓,𝑆𝑔⟶𝑈𝑧^𝑘=𝑀⋅𝑓,𝑆𝑔_𝑧−1𝑈𝑀_𝑧𝑘𝑓,𝑔. (37) But, for each 𝑛, 𝑈_𝑛=𝑀_𝑧⁻¹𝑈_𝑛𝑀_𝑧𝑘.
Hence, 𝑈=𝑀_𝑧⁻¹𝑈𝑀_𝑧𝑘. Therefore, 𝑈 is a 𝑘th order slant weighted Toeplitz operator.

4. Compression to 𝐻²(𝛽)

Definition 11. The compression of a 𝑘th order slant weighted Toeplitz operator on 𝐻²(𝛽) is denoted by 𝑉_𝜙 and defined as 𝑉_𝜙𝑒_𝑗(𝑧)=(1/𝛽_𝑗)∑^∞_𝑛=0𝑎_{𝑛𝑘−𝑗}𝛽_𝑛𝑒_𝑛(𝑧).𝑗=0,1,2,….

Alternatively, 𝑉_𝜙=𝑃𝑈_{𝜙|𝐻2(𝛽)}=𝑃𝑊_𝑘𝑀_{𝜙|𝐻2(𝛽)}=𝑊_𝑘𝑃𝑀_{𝜙|𝐻2(𝛽)}since𝑃reduces𝑊_𝑘[8]=𝑊_𝑘𝑇_𝜙, (38) where 𝑇_𝜙∶𝐻²(𝛽)→𝐻²(𝛽) is the Toeplitz operator on 𝐻²(𝛽) induced by 𝜙∈𝐿^∞(𝛽). The matrix of 𝑉_𝜙 is unilaterally infinite and has the form 𝑎₀𝛽₀𝛽₀𝑎₋₁𝛽₀𝛽₁𝑎₋₂𝛽₀𝛽₂⋯𝑎_𝑘𝛽₁𝛽₀𝑎_𝑘−1𝛽₁𝛽₁𝑎_𝑘−2𝛽₁𝛽₂⋯𝑎_2𝑘𝛽₂𝛽₀𝑎_2𝑘−1𝛽₂𝛽₁𝑎_2𝑘−2𝛽₂𝛽₂⋯⋯⋯⋯⋯ (39) for a given 𝜙=∑^∞_𝑛=−∞𝑎_𝑛𝑧^𝑛.

Further, we observe the following.(i)The mapping 𝜙→𝑉_𝜙 is linear and one to one. (ii)𝑇_𝜙𝑊_𝑘=𝑃𝑀_𝜙𝑊_𝑘=𝑃𝑊_𝑘𝑀_𝜙(𝑧^𝑘)=𝑃𝑈_{𝜙(𝑧𝑘)}=𝑉_𝜙(𝑧^𝑘).