**Examples of de Branges-Rovnyak spaces generated by nonextreme functions**

Let H^{2} denote the standard Hardy space in the open unit disk D and let T=∂D. For χ ∈ L^{∞}(T) let T_{χ} denote the bounded Toeplitz operator on H^{2}, that is, T_{χ}f=P_{+}(χf), where P_{+} is the orthogonal projection of L^{2}(T) onto H^{2}. Given a function b in the unit ball of H^{∞}, the *de Branges-Rovnyak space *H(b) is the image of

H^{2} under the operator (I-T_{b}T_{b})^{1⁄2}. The space H(b) is given the Hilbert space structure that makes the operator (I-T_{b}T_{b})^{1⁄2} a coisometry of H^{2} onto H(b), namely

<(I-T_{b}T_{b})^{1⁄2}f,(I-T_{b}T_{b})^{1⁄2}g>_{b}=<f ,g>_{2} f,g∈ker((I-T_{b}T_{b})^{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 ([3]).

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 [3]) and in such a case (b,a) is called a rational pair. H(b) for rational pairs have been studied, e.g., in [1] and [2].

Here we describe de Branges-Rovnyak spaces H(b_{α}), α>0, where the corresponding Smirnov functions are b_{α}(z)/a_{α}(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.