Extremal functions as divisors for kernels of Toeplitz operators.
The paper studies the extremal functions for kernels of Toeplitz operators on the classical Hardy spaces \(H^p\) when \(1<p<\infty\). Such kernels are special cases of the so-called nearly invariant subspaces of the backward shift. It is a theorem of Hitt that the extremal function of a nearly invariant subspace of the backward shift on \(H^2\) acts as an isometric divisor. The present paper shows that the extremal function of the kernel of a Toeplitz operator on \(H^p\) is a contractive divisor when \(p<2\) and an expansive divisor when \(p>2\), modulo \(p\)-dependent multiplicative constants. Examples are also supplied to show that the extremal function is generally not expansive when \(p<2\) and not contractive when \(p<2\) and not contractive when \(p>2\).
Invariant subspaces with respect to the backward shift
scientific article; zbMATH DE number 1974802
scientific article