Examples of de Branges-Rovnyak spaces generated by nonextreme functions
Let H2 denote the standard Hardy space in the open unit disk D and let T=∂D. For χ ∈ L∞(T) let Tχ denote the bounded Toeplitz operator on H2, that is, Tχf=P+(χf), where P+ is the orthogonal projection of L2(T) onto H2. Given a function b in the unit ball of H∞, the de Branges-Rovnyak space H(b) is the image of
H2 under the operator (I-TbTb)1⁄2. The space H(b) is given the Hilbert space structure that makes the operator (I-TbTb)1⁄2 a coisometry of H2 onto H(b), namely
<(I-TbTb)1⁄2f,(I-TbTb)1⁄2g>b=<f ,g>2 f,g∈ker((I-TbTb)1⁄2)⊥
Here we assume that b is a non-extreme point of the unit ball of H∞. Then there exists a unique outer function a∈H∞ such that a(0)>0 and |a|2+|b|2=1 a. e. on T. Then we say that (b,a) is a pair. If (b,a) is a pair, then the quotient φ=b/a is in the Smirnov class N+. (Let us recall that the Smirnov class N+ consists of those holomorphic functions in D that are quotients of functions in H∞ in which the denominators are outer functions.) Conversely, for every nonzero function φ ∈ N+ there exists a unique pair (b,a) such that φ=b/a ().
Many properties of H(b) can be expressed in terms of the function φ=b/a in the Smirnov class N+. It is worth noting here that if φ is rational, then the functions a and b in the representation of φ are also rational (see ) and in such a case (b,a) is called a rational pair. H(b) for rational pairs have been studied, e.g., in  and .
Here we describe de Branges-Rovnyak spaces H(bα), α>0, where the corresponding Smirnov functions are bα(z)/aα(z)= (1-z)b-α, α>0
The talk is based on joint work with Bartosz Lanucha.
- C. Costara, T. Ransford: Which de Branges-Rovnyak spaces are Dirichlet spaces (and vice versa)? J. Funct. Anal. 265 (2013), no. 12, 3204-3218.
- E. Fricain, A. Hartmann, W. T. Ross: Concrete examples of H(b) Comput. Methods Func. Theory, 16 (2016), no 2, 287-306.
- D. Sarason: Unbounded Toeplitz operators, Integral Equations Operator Theory,61 (2008), 281-298.