Essentially Slant Toeplitz Operators

The notion of an essentially slant Toeplitz operator on the space L is introduced and some of the properties of the set ESTO(L), the set of all essentially slant Toeplitz operators on L, are investigated. In particular the conditions under which the product of two operators in ESTO(L) is in ESTO(L) are discussed. The notion is generalized to kth-order essentially slant Toeplitz operators. The notion of Toeplitz operators was introduced by O. Toeplitz [8] in the year 1911. Subsequently many researchers like Devinatz [4], Abrahamse [1], Barria and Halmos [3] came up with various generalizations of the notion of Toeplitz operators. The essential commutant of the unilateral forward shift has been the object of study for several years for its far reaching applications to various branches like probability, statistics, oscillation signal processing etc. Barria and Halmos [3] brought much attention to this set and mooted an idea of deriving ways to characterize completely this set. The essential commutant of the forward shift has sometimes been referred to as the set of essentially Toeplitz operators. Ho [7], in the year 1995, began a systematic study of yet another class of operators having the property that the matrices of such operators with respect to the standard orthonormal basis could be obtained from those of Toeplitz operators just by eliminating every other row. Such operators were termed as slant Toeplitz operators [7]. Villemoes [9] associated the Besov regularity of solutions of the refinement equation with the spectral radius of an associated slant Toeplitz Date: Received: 13 June 2008; Accepted: 2 January 2009. ∗ Corresponding author. 2000 Mathematics Subject Classification. Primary 47B35; Secondary 47B20.