Bases of reproducing kernels in de-Branges spaces

Abstract

This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If b is a nonconstant function in the unit ball of H ∞ , and T b is the Toeplitz operator, with symbol b, then the de-Branges space, H ( b ) , associated to b, is defined by H ( b ) = ( Id - T b T b ¯ ) 1 / 2 H 2 , where H 2 is the Hardy space of the unit disk. It is equipped with the inner product such that ( Id - T b T b ¯ ) 1 / 2 is a partial isometry from H 2 onto H ( b ) . First, following a work of Ahern–Clark, we study the problem of orthogonal basis of reproducing kernels in H ( b ) . Then we give a criterion for sequences of reproducing kernels which form an unconditional basis in their closed linear span. As far as concerns the problem of complete unconditional basis in H ( b ) , we show that there is a dichotomy between the case where b is an extreme point of the unit ball of H ∞ and the opposite case.