Research Article | Open Access
S. C. Arora, Ritu Kathuria, "On th-Order Slant Weighted Toeplitz Operator", The Scientific World Journal, vol. 2013, Article ID 960853, 5 pages, 2013. https://doi.org/10.1155/2013/960853
On th-Order Slant Weighted Toeplitz Operator
Let be a sequence of positive numbers with , when and when . A th-order slant weighted Toeplitz operator on is given by , where is the multiplication on and is an operator on given by , being the orthonormal basis for . In this paper, we define a th-order slant weighted Toeplitz matrix and characterise in terms of this matrix. We further prove some properties of using this characterisation.
Toeplitz operators and slant Toeplitz operators  have been found immensely useful, especially in the study of prediction theory , wavelet analysis , and solution of differential equations . Originally, these operators were defined and studied on the usual and spaces. During the past few decades, different generalisations of these spaces, like the weighted function spaces and and the weighted sequence spaces and have developed. Shields  has made a systematic study of the multiplication operator on . Lauric  has discussed the Toeplitz operator on . Motivated by these studies, we introduced and studied the notion of a slant weighted Toeplitz operator  on . In this paper, we study a th-order slant weighted Toeplitz operator on the space . We begin with the following preliminaries.
Let be a sequence of positive numbers with , when and when .
Then, , is a Hilbert space  with an orthonormal basis given by and with an inner product defined by Also, is a subspace of . Further, the space is a Banach space with respect to the norm defined by The mapping is the orthogonal projection of onto . For a given , the induced weighted multiplication operator  is denoted by and is given by such that
If we put , then is the operator defined as , where for all , and it is known as a weighted shift .
The slant weighted Toeplitz operator  is an operator on defined as such that .
If is such that then can be alternately defined by
2. th-Order Slant Weighted Toeplitz Operator
Suppose that the operator is such that Then the matrix of is The adjoint of is given by
Definition 1 (see ). For an integer , the th order slant weighted Toeplitz operator is such that for all . Thus, .
The th entry of the matrix of is given by Hence, the matrix of with respect to this basis is as follows:
Theorem 2. The mapping is linear and one to one.
Proof. For linearity, consider
For checking one-to-one, let .
Then, . By linearity, we get .
Hence, This implies that either or the degree of is not divisible by . But since this is true for all , the second possibility is ruled out. Hence, or .
Theorem 3. Consider the following: (i) ; (ii) , .
(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 We therefore conclude that for all ,
(ii) We prove the result by induction on .
For , the result is true by part (i).
Suppose that the result is true for . Then,
Hence by induction, the result is true for all .
For and , For and (a multiple of ), On the other hand, From (23), (24), and (25), we conclude that Hence, the result is true for also. Therefore, by using induction, we can prove it for all negative integers . The case when is trivially true.
Hence the theorem is true.
Corollary 4. Let . Then,
Theorem 5. Consider the following: .
Proof. The proof is as follows:
3. Matrix Characterisation of
Definition 6. Let for all . Then, the th order slant weighted Toeplitz matrix corresponding to the weight sequence is a bilaterally infinite matrix such that
We have proved earlier  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 is a order slant weighted Toeplitz operator is that its matrix with respect to the orthonormal basis is a 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 for all . Thus, the matrix of is a th order slant weighted Toeplitz matrix. Conversely, we assume that is an operator on whose matrix is a th order slant weighted Toeplitz matrix. This means that for all , we have Now, Hence . Therefore, we conclude that is a th order slant weighted Toeplitz operator.
Next, consider the operator given by . Then, .
Now, is bounded as is always positive and bounded.
Lemma 8. Consider the following: .
Proof. The proof is as follows:
Theorem 9. A bounded operator on is a order slant weighted Toeplitz operator if and only if , where and , are the multiplication operators induced by and , respectively.
Proof. Let be a th order slant weighted Toeplitz operator on . Then, from (32) we get that
For the converse part, we assume that is a bounded operator on satisfying for some fixed integer . Then, for all , 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 , the set of all 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 , we get that .
From the previous theorem, this implies that But, for each , .
Hence, . Therefore, is a th order slant weighted Toeplitz operator.
4. Compression to
Definition 11. The compression of a th order slant weighted Toeplitz operator on is denoted by and defined as .
Alternatively, where is the Toeplitz operator on induced by . The matrix of is unilaterally infinite and has the form for a given .
Further, we observe the following.(i)The mapping is linear and one to one. (ii).
- M. C. Ho, “Properties of slant Toeplitz operators,” Indiana University Mathematics Journal, vol. 45, no. 3, pp. 843–862, 1996.
- H. Helson and G. Szegö, “A problem in prediction theory,” Annali di Matematica Pura ed Applicata, vol. 51, no. 1, pp. 107–138, 1960.
- L. Villemoes, “Wavelet analysis of refinement equations,” SIAM Journal on Mathematical Analysis, vol. 25, pp. 1433–1460, 1994.
- T. Goodman, C. Micchelli, and J. Ward, “Spectral radius formula for subdivision operators,” in Recent Advances in Wavelet AnalysisAcademic Press, L. Schumaker and G. Webb, Eds., pp. 335–360, 1994.
- A. L. Shields, Weighted Shift Operators and Analytic Function Theory, Topics in Operator Theory, vol. 13 of Mathematical Survey, American Mathematical Society, Providence, RI, New England, 1974.
- V. Lauric, “On a weighted Toeplitz operator and its commutant,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 6, pp. 823–835, 2005.
- S. C. Arora and R. Kathuria, “On weighted Toeplitz operators,” Australian Journal of Mathematical Analysis and Applications, vol. 8, no. 1, article 11, 2011.
- S. C. Arora and R. Kathuria, “Generalised slant weighted Toeplitz operators,” World Academy of Science, Engineering and Technology, vol. 75, pp. 1103–1106, 2011.
Copyright © 2013 S. C. Arora and Ritu Kathuria. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.