Essential norms of differences of composition operators on \(H^{\infty}\). (English) Zbl 1100.47022

The authors study essential norms of differences of composition operators \(C_\phi- C_\psi\) on the space of bounded analytic functions \(H^\infty(D)\), where \(D\) is the open unit disk in the complex plane and \(\phi, \psi: D \to D\) are analytic selfmaps. The essential norm of \(C_\phi -C_\psi\) is by definition its distance to the compact operators. The authors obtain several estimates of the essential norm \(| | C_\phi -C_\psi| | _e\) and their results extend previous ones due to B. MacCluer, S. Ohno and R. Zhao [Integral Equations Oper.Theory 40, No. 4, 481–494 (2001; Zbl 1062.47511)] and P. Gorkin, R. Mortini and D. Suárez [Contemp.Math.328, 177–188 (2003; Zbl 1060.47031)].
We mention but the two main theorems proved in the present paper. Firstly, if \(\| \phi\| _\infty=1\) and \(C_\phi, C_\psi\) are in the same connected component, then under the additional assumption that \[ E_{L^\infty}(| \phi| ) =\{ m \in M(L^\infty):| \phi(m)| =1\} \] is a peak set in \(H^\infty(D)\), we have that \[ \| C_\phi - C_\psi\|_e = \lambda(\sigma_\infty(\phi,\psi)), \] where \(\lambda(t) = \frac{2(1 -\sqrt{1 -t^2})}{t}\) and \(\sigma_\infty(\phi,\psi)=\limsup_{| \phi(z)\psi(z)| \to 1} \rho(\phi(z),\psi(z))\) (\(\rho\) is the pseudohyperbolic distance). Secondly, the authors obtain that \(| | C_\phi -C_\psi| | _e =2\) if and only if \(| | \phi \psi| | _\infty=1\) and \(\sigma_\infty(\phi,\psi) =1.\)


47B33 Linear composition operators
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
Full Text: DOI