Bases of reproducing kernels in de-Branges spaces

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 Tb is the Toeplitz operator, with symbol b, then the de-Branges space, H(b), associated to b, is defined by H(b)=(Id-TbTb¯)1/2H2, where H2 is the Hardy space of the unit disk. It is equipped with the inner product such that (Id-TbTb¯)1/2 is a partial isometry from H2 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.