Product and Commutativity of kth-Order Slant Toeplitz Operators

and Applied Analysis 3 Theorem3. Letφ, ψ∈H∞(T) orφ,ψ ∈ H(T), the following statements are equivalent: (1.1) U φ and U ψ commute; (1.2) U φ and U ψ essentially commute; (1.3) φ(zk)ψ(z) = φ(z)ψ(z); (1.4) there exist scalars α andβ, not both zero, such that αφ+ βψ = 0. Nowwe start to study the commutativity of two kth-order slant Toeplitz operators with harmonic symbols. Proposition4. Letφ(z)=∑n p=−n a p z p andψ(z)= ∑n p=−n b p z , where a2 −n +b 2 −n ̸ = 0 and n is a positive integer, then the following statements are equivalent: (1.1) φ(zk)ψ(z) = φ(z)ψ(z); (1.2) there exist scalars α andβ, not both zero, such that αφ+ βψ = 0. Proof. We begin with the easy direction. First, suppose that (1.2) holds and let α ̸ = 0 without lost of generality, so that φ = −(β/α)ψ. Thus, φ(zk)ψ(z) = φ(z)ψ(z). To prove the other direction of the proposition, suppose that (1.1) holds. Since φ(z) = ∑n p=−n a p z p and ψ(z) =