On kernels of Toeplitz operators
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On kernels of Toeplitz operators M. T. Nowak1
· P. Sobolewski1 · A. Sołtysiak2 · M. Wołoszkiewicz-Cyll1
Received: 10 September 2020 / Revised: 30 October 2020 / Accepted: 31 October 2020 / Published online: 12 November 2020 © The Author(s) 2020
Abstract We apply the theory of de Branges–Rovnyak spaces to describe kernels of some Toeplitz operators on the classical Hardy space H 2 . In particular, we discuss the kernels of the operators T f¯/ f and TI¯ f¯/ f , where f is an outer function in H 2 and I is inner such that I (0) = 0. We also obtain a result on the structure of de Branges– Rovnyak spaces generated by nonextreme functions. Keywords Toeplitz operators · de Branges–Rovnyak spaces · Nearly invariant subspaces · Rigid functions · Nonextreme functions · Kernel functions Mathematics Subject Classification 47B32 · 46E22 · 30H10
1 Introduction Let H 2 denote the standard Hardy space on the unit disk D. For ϕ ∈ L ∞ (∂D) the Toeplitz operator on H 2 is given by Tϕ f = P+ (ϕ f ), where P+ is the orthogonal projection of L 2 (∂D) onto H 2 . We will denote by M(ϕ) the range of Tϕ equipped with the range norm, that is, the norm that makes the operator Tϕ a coisometry of H 2 onto M(ϕ). For a nonconstant function b in the unit ball of H ∞ the de Branges–
B
M. T. Nowak [email protected] P. Sobolewski [email protected] A. Sołtysiak [email protected] M. Wołoszkiewicz-Cyll [email protected]
1
Institute of Mathematics, Maria Curie-Skłodowska University, pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland
2
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Pozna´nskiego 4, 61-614 Pozna´n, Poland
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Rovnyak space H(b) is the image of H 2 under the operator (1 − Tb Tb¯ )1/2 with the corresponding range norm. The norm and the inner product in H(b) will be denoted by · b and ·, ·b . The space H(b) is a Hilbert space with the reproducing kernel b kw (z) =
1 − b(w)b(z) 1 − wz
(z, w ∈ D).
In the case when b is an inner function the space H(b) is the well-known model space K b = H 2 bH 2 . If the function b fails to be an extreme point of the unit ball in H ∞ , that is, when log(1 − |b|) ∈ L 1 (∂D), we will say simply that b is nonextreme. In this case one can 1/2 define an outer function a whose modulus on ∂D equals 1 − |b|2 . Then we say that the functions b and a form a pair (b, a). By the Herglotz representation theorem there exists a positive measure μ on ∂D such that 1 + b(z) = 1 − b(z)
∂D
1 + e−iθ z 1 + b(0) dμ(eiθ ) + i Im , z ∈ D. −iθ 1−e z 1 − b(0)
(1)
a 2 Moreover the function 1−b is the Radon-Nikodym derivative of the absolutely continuous component of μ with respect to the normalized Lebesgue measure. If the measure μ is absolutely continuous the pair (b, a) is called special. Recall that a function f ∈ H 1 is called rigid if and only if no other functions in 1 H , except for positive scalar multiples of f have the same argument as f a.e. on ∂D. If (b, a)
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