Examples of de Branges–Rovnyak spaces generated by nonextreme functions

Abstract

We describe de Branges-Rovnyak spaces H(bα), α > 0, where the function bα is not extreme in the unit ball of H ∞ on the unit disk D, defined by the equality bα(z)/aα(z) = (1 − z) −α , z ∈ D, where aα is the outer function such that aα(0) > 0 and |aα| 2 + |bα| 2 = 1 a.e. on ∂D.

The orthogonal complement of $\mathcal{M}(a)$ in $\mathcal{H}(b)$

“…However, g ∈ M((1 − z) n+1/2 ). Indeed, if (1 − z) n ∈ (1 − z) n+1/2 H 2 , then 1 ∈ (1 − z) 1/2 H 2 , a contradiction (see, e.g., [9,Remark 3.4]).…”

“…A complement of M(a) in H(b) (not necessarily orthogonal) for the case when admissible pairs (b, a) are rational has been studied for example in [3], [4], [14], and [8]. Also, in [9] this space has been described for concrete nonextreme functions b that are not rational.…”

On kernels of Toeplitz operators

“…If the function b fails to be an extreme point of the unit ball in H ∞ , that is, when log(1 − |b|) ∈ L 1 (∂D), we will say simply that b is nonextreme. In this case one can define an outer function a whose modulus on ∂D equals 1 − |b| 2 [2,3,5,15,16,22], and [23].…”

On kernels of Toeplitz operators

“…If a pair (b, a) is special and f = a 1−b , then M(a) is dense in H(b) if and only if f 2 is rigid ( [18]). Spaces H(b) for nonextreme b have been studied in [1], [2], [4], [13], [14], [20], and [21].…”