Treffer: Interpolating functions of minimal norm, star-invariant subspaces and kernels of Toeplitz operators: Interpolating functions of minimal norm, star-invariant subspaces, and kernels of Toeplitz operators

It is proved that for each inner function θ \theta there exists an interpolating sequence { z n } \left \{ {{z_n}} \right \} in the disk such that sup n | θ ( z n ) | > 1 {\sup _n}|\theta ({z_n})| > 1 , but every function g g in H ∞ {H^\infty } with g ( z n ) = θ ( z n ) ( n = 1 , 2 , … ) g({z_n}) = \theta ({z_n})(n = 1,2, \ldots ) satisfies | | g | | ∞ ≥ 1 ||g|{|_\infty } \geq 1 . Some results are obtained concerning interpolation in the star-invariant subspace H 2 ⊖ θ H 2 {H^2} \ominus \theta {H^2} . This paper also contains a "geometric" result connected with kernels of Toeplitz operators.