On the Toeplitzness of Composition Operators

A consequence of Littlewood’s Subordination Principle [10] is the (not at all obvious) fact that every composition operator restricts to a bounded operator on the Hardy space H2 (see also [5, Theorem 1.7, page 10] or [16, pp. 13–15]), and this in turn has inspired a lively enterprise connecting complex function theory with operator theory, the goal being to understand how properties Cφ are related to those of φ (see [3, 7, 16] for more on this). The work we will describe here has its roots in the paper [1] of Barŕıa and Halmos, who introduced the notion of “asymptotic Toeplitz operator.” One can think of a Toeplitz operator on H2 as a bounded linear operator whose matrix, relative to the orthonormal basis of monomials {zn : n ≥ 0}, has constant diagonals. Such operators T can be characterized by the equation