COMMUTATIVITY OF WEIGHTED SLANT HANKEL OPERATORS

Year 2016, Volume: 4 Issue: 1, 164 - 171, 01.04.2016

Gopal Datt , Anshika Mıttal

Abstract

For a positive integer k  2, the kth-order weighted slant Hankel operator D k; on L2( ) with  2 L1( ) is de ned as D k; = J WkM  , where J is the re ection operator given by J en = e􀀀n for each n 2 Z and Wk is given by Wken(z) = m km em(z) if n = km;m 2 Z and Wken(z) = 0 if n 6= km. The paper discusses the product and commutativity of kth-order weighted slant Hankel operators of di erent order. Compactness and essential commutativity of these operators are also addressed and it is obtained that the commutativity of these operators coincides with the essential commutativity.

Keywords

Hankel operator, weighted Hankel Operator, weighted slant Hankel operator

References

• [1] V.M. Adamjan, D.Z. Arov and M.G. Krein, Innite Hankel matrices and generalized problems of Caratheodory-Fejer and F. Riesz, Functional Anal. Appl., 2, 1968, 1-18.
• [2] S.C. Arora and R. Batra, On generalized slant Toeplitz operators, Indian J. Math., 45(2), 2003, 121-134.
• [3] S.C. Arora and J. Bhola, kth-order slant Hankel operators, Mathematical Sc. Reas. Journal (U.S.A.), 12(3), 2008, 53-63.
• [4] S.C. Arora and R. Kathuria, On kth􀀀order slant weighted Toeplitz operators, The Scientic World Journal, Volume 2013, Article ID. 960853, 1-5.
• [5] Ruben Marti nez-Avenda~no, Essentially Hankel operators, J. London Math. Soc., Vol.66(2), 2002, 741-752.
• [6] G. Datt and N. Ohri, Commutativity of slant weighted Toeplitz operators, Communicated.
• [7] G. Datt and D.K. Porwal, Weighted Hankel operators and matrices, Matematicki Vesnik, 65(3), 2013, 353{363.
• [8] G. Datt and D.K. Porwal, Generalization of weighted slant Hankel operators, To appear in Mathematica Slovaca.
• [9] G. Datt and R. Aggarwal, Essentially 􀀀 Toeplitz operators, General Mathematics, Vol. 21 (2), 2013, 57-69.
• [10] H. Hamburger, Uber eine Erweiterung des Stieltjesschen Momentproblems, I, Math. Ann., 81, 1920, 235-319.
• [11] Chaomei Liu and Yufeng Lu, Product and commutativity of slant Toeplitz operators, J. Math. Reasearch with Applications, 33(1), 2013, 122-126.
• [12] A.L. Shields, Weighted shift operators and analytic function theory, Topics in Operator Theory, Math. Surveys, No.13, American Mathematical Society, Rhode Ireland, 1974, 49{128.