Properties of Rationalized Toeplitz Hankel Operators

Abstract

In this paper, we introduce and study the notion of Rationalized Toeplitz Hankel Matrix of order (k₁, k₂) as the two way infinite matrix (α_ij ) such that    α_ij = α_i+k2,j+k1
where k₁ and k₂ are relatively prime non zero integers. It is proved that a bounded linear operator R on L² is a Rationalized Toeplitz Hankel operator [5] of order (k₁, k₂) if and only if its matrix w.r.t. the orthonormal basis { zⁱ : i ∈ Z } is a Rationalized Toeplitz Hankel matrix of the same order. Some algebraic properties of the Rationalized Toeplitz Hankel operator Rφ like normality, hyponormality and
compactness are also discussed.

Key words and phrases. Slant Toeplitz operator, Slant Hankel operator, Toeplitz and Hankel operator.

2010 Mathematics Subject Classification. 47B35.