Relating composition operators on different weighted Hardy spaces. (English) Zbl 0902.47030

Hurst, Paul R.

Arch. Math. 68, No. 6, 503-513 (1997).

Summary: In a 1988 paper, Cowen found a formula expressing the adjoint of any linear fractional composition operator on the Hardy space as a product of Toeplitz operators and another linear fractional composition operator. In this paper, we use Cowen’s adjoint formula to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the \(S_a\) spaces. We find the spectrum of any linear fractional composition operator whose symbol has exactly one fixed point of multiplicity one on the unit circle.

MSC:

47B38	Linear operators on function spaces (general)
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B37	Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Keywords:

linear fractional composition operator; product of Toeplitz operators; weighted Hardy spaces; fixed point of multiplicity one