2017/06/15 by Eschmeier, Jörg · 1 citation
#47A13 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1706.04874
Let Hm(\mathbb B,\mathcal D) be the \mathcal D-valued functional Hilbert space with reproducing kernel Km(z,w) = (1-⟨ z,w⟩)-m1\mathcal D. A Km-inner function is by definition an operator-valued analytic function W: \mathbb B → L(\mathcal E, \mathcal D) such that ‖Wx‖Hm(\mathbb B,\mathcal D) = ‖x‖ for all x ∈ \mathcal E and (W\mathcal E) ⊥ Mzα(W\mathcal E) for all α∈ \mathbb Nn ∖ \0\. We show that the Km-inner functions are precisely the functions of the form W(z) = D + C ∑mk=1(1 - ZT^*)-kZB, where T ∈ L(H)n is a pure m-hypercontraction and the operators T^*, B, C,D form a 2 × 2-operator matrix satisfying suitable conditions. Thus we extend results proved by Olofsson on the unit disc to the case of the unit ball \mathbb B ⊂ \mathbb Cn.