We discuss three open problems in operator and function theory: Crouzeix’s conjecture, Sendov’s conjecture, and the invariant subspace problem. We include the full conjecture, prior work, and related questions.
Abstract. We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces K 1 B generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with L 1 -norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition B ∈ L ∞ (R) for the boundedness of the differentiation on K 1 B . Our main tool is a construction of an unconditional basis of rational fractions in K 1 B .
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