The commutant of analytic Toeplitz operators

Abstract

In this paper we study the commutant of an analytic Toeplitz operator. For ϕ H ∞ \phi \;\;{H^\infty } , let ϕ = χ F \phi = \chi F be its inner-outer factorization. Our main result is that if there exists λ ϵ C \lambda \;\epsilon \;{\text {C}} such that X factors as χ = χ 1 χ 2 ⋯ χ n \chi = {\chi _1}{\chi _2} \cdots {\chi _n} , each χ i {\chi _i} an inner function, and if F − λ F - \lambda is divisible by each χ i {\chi _i} , then { T ϕ } ′ = { T χ } ′ ∩ { T F } ′ \{ {T_\phi }\} ’ = \{ {T_\chi }\} ’ \cap \{ {T_F}\} ’ . The key step in the proof is Lemma 2, which is a curious result about nilpotent operators. One corollary of our main result is that if χ ( z ) = z n , n ≥ 1 \chi (z) = {z^n},n \geq 1 , then { T ϕ } ′ = { T χ } ′ ∩ { T F } ′ \{ {T_\phi }\} ’ = \{ {T_\chi }\} ’ \cap \{ {T_F}\} ’ , another is that if ϕ ϵ H ∞ \phi \;\epsilon {H^\infty } is univalent then { T ϕ } ′ = { T z } ′ \{ {T_\phi }\} ’ = \{ {T_z}\} ’ . We are also able to prove that if the inner factor of ϕ \phi is χ ( z ) = z n , n ≥ 1 \chi (z) = {z^n},n \geq 1 , then { T ϕ } ′ = { T z s } ′ \{ {T_\phi }\} ’ = \{ {T_{{z^s}}}\} ’ where s is a positive integer maximal with respect to the property that z n {z^n} and F ( z ) F(z) are both functions of z s {z^s} . We conclude by raising six questions.