Operators induced by weighted Toeplitz and weighted Hankel operators

Abstract

In this paper, the notion of weighted Toep-Hank operator G β φ , induced by the symbol φ ∈ L ∞ (β), on the space H 2 (β), β = {βn} n∈Z being a semi-dual sequence of positive numbers with β0 = 1, is introduced. Symbols are identified for the induced weighted Toep-Hank operator to be co-isometry, normal and hyponormal.

An extension of the notion induced by Toeplitz and Hankel operators

“…We also find the appearance of these operators in the next result. The following findings can be stated for the Toep-Hank operator E φ on L 2 without any extra effort as the proof either holds by using definition or along the same lines for the respective results for Toep-Hank operator G φ on H 2 in [3]. Proposition 2.2.…”

“…Now we prove the equivalency of (1) and (3). To obtain (3) from (1), suppose that A is a slant Toep-Hank operator.…”

“…The Toep-Hank operator G φ = P JM φ K discussed in [3] is a bounded linear operator defined on the space H 2 which can be extended to the space L 2 by defining an operator R from L 2 to L 2 as R(e 2n ) = e n , R(e 2n+1 ) = e −n−1 for n ∈ Z. Clearly R = 1 and the adjoint R * of R, is indeed, R * (e n ) = e 2n + e −2n−1 for n ∈ Z.…”

“…Remark 3.6. We have seen in [3] that if matrix of any bounded linear operator A defined on H 2 is a Toep-Hank matrix, then AC z 2 is a Hankel operator and AM z C z 2 is a Toeplitz operator. It is known that an operator A on H 2 is Hankel (Toeplitz) if U * A = AU (U * AU = A), where U is the unilateral shift operator on H 2 .…”

“…In [3], we discussed Toep-Hank operators on H 2 as those operators whose matrix representation provides a Hankel (Toeplitz) matrix if only even (odd) columns are considered. A structural formula for these operators is given as G φ = P JM φ K, where φ ∈ L ∞ , K being an operator from H 2 to L 2 defined as K(e 2n ) = e n , K(e 2n+1 ) = e −n−1 for all n ≥ 0.…”