On kth-Order Slant Weighted Toeplitz Operator

Let β = {β n}n∈ℤ be a sequence of positive numbers with β 0 = 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 L 2(β) is given by U ϕ = W k M ϕ, where M ϕ is the multiplication on L 2(β) and W k is an operator on L 2(β) given by W k e nk(z) = (β n/β nk)e n(z), {e n(z) = z k/β k}k∈ℤ being the orthonormal basis for L 2(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.


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 2 and 2 spaces. During the past few decades, different generalisations of these spaces, like the weighted function spaces and and the weighted sequence spaces 2 ( ) and 2 ( ) have developed. Shields [5] has made a systematic study of the multiplication operator on 2 ( ). Lauric [6] has discussed the Toeplitz operator on 2 ( ). Motivated by these studies, we introduced and studied the notion of a slant weighted Toeplitz operator [7] on 2 ( ). In this paper, we study a th-order slant weighted Toeplitz operator on the space 2 ( ). We begin with the following preliminaries.
Throughout the paper, we assume that for a fixed integer ≥ 2, / ≤ < ∞. Consider the following spaces [5,6]: Then, ( 2 ( ), ‖ ⋅ ‖ ) is a Hilbert space [6] with an orthonormal basis given by { ( ) = / } ∈Z and with an inner product defined by 2 The Scientific World Journal Also, 2 ( ) is a subspace of 2 ( ). Further, the space is a Banach space with respect to the norm defined by The mapping : 2 ( ) → 2 ( ) is the orthogonal projection of 2 ( ) onto 2 ( ). For a given ∈ ∞ ( ), the induced weighted multiplication operator [5] is denoted by and is given by If we put 1 ( ) = , then 1 = is the operator defined as ( ) = +1 ( ), where = +1 / for all ∈ Z, and it is known as a weighted shift [5].
The slant weighted Toeplitz operator [7] is an operator on 2 ( ) defined as then can be alternately defined by

th-Order Slant Weighted Toeplitz Operator
Suppose that the operator : 2 ( ) → 2 ( ) is such that Then the matrix of is ] . (9) The adjoint of is given by * ( ) = ( ) ∀ ∈ Z. (10) Definition 1 (see [8]). For an integer ≥ 2, the th order slant weighted Toeplitz operator Hence, the matrix of with respect to this basis is as follows: For checking one-to-one, let = . Then, − = 0. By linearity, we get − = 0.
This implies that either − = 0 or the degree of ( − ) ( ) is not divisible by . But since this is true for all ∈ Z, the second possibility is ruled out. Hence, − = 0 or = .
The Scientific World Journal 3 Theorem 3. Consider the following: Proof. (i) It is sufficient to prove that Suppose that is not a multiple of . Then, Now, when = (multiple of ), On the other hand, Hence from (17) and (18), we get that ( ) = ( ) whenever is a multiple of .
We therefore conclude that for all ∈ Z, (ii) We prove the result by induction on . For = 1, the result is true by part (i). Suppose that the result is true for = . Then, On the other hand, From (23), (24), and (25), we conclude that Hence, the result is true for = −1 also. Therefore, by using induction, we can prove it for all negative integers . The case when = 0 is trivially true. Hence the theorem is true. Proof. The proof is as follows: We have proved earlier [8] that is a th order slant weighted Toeplitz operator if and only if = . Next, we prove a characterisation of the th order slant weighted Toeplitz operator in terms of the matrix previously defined.

Theorem 7.
A necessary and sufficient condition that an operator on 2 ( ) is a th order slant weighted Toeplitz operator is that its matrix with respect to the orthonormal basis { ( ) = / } ∈Z is a th order slant weighted Toeplitz matrix.
Proof. Suppose that is a th order slant weighted Toeplitz operator. Then, the corresponding matrix ⟨ ⟩ is given by Further, where = +1 / for all ∈ Z. Thus, the matrix of is a th order slant weighted Toeplitz matrix. Conversely, we assume that is an operator on 2 ( ) whose matrix is a th order slant weighted Toeplitz matrix. This means that for all , ∈ Z, we have Now, Hence = . Therefore, we conclude that is a th order slant weighted Toeplitz operator.
Thus, = −1 . For the converse part, we assume that is a bounded operator on 2 ( ) satisfying = −1 for some fixed integer ≥ 2. Then, for all , ∈ Z, The previous equation shows that the matrix corresponding to is a th order slant weighted Toeplitz matrix. Hence, by Theroem 7, is a th order slant weighted Toeplitz operator.
Corollary 10. For a fixed integer ≥ 2, the set of all th order slant weighted Toeplitz operators is weakly closed and hence strongly closed.
Proof. We assume that for each positive integer , is a th order slant weighted Toeplitz operator and → weakly. Then, for , ∈ 2 ( ), we get that ⟨ , ⟩ → ⟨ , ⟩.
From the previous theorem, this implies that But, for each , . Therefore, is a th order slant weighted Toeplitz operator.

Compression to
where : 2 ( ) → 2 ( ) is the Toeplitz operator on 2 ( ) induced by ∈ ∞ ( ). The matrix of is unilaterally infinite and has the form