We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if Q is a backward shift invariant subspace of the Hardy space H 2 (D), then the model operator S Q attains its norm. Here S Q = P Q M z | Q , the compression of the shift M z on the Hardy space H 2 (D) to Q.
Let H ∞ denote the Banach algebra of all bounded analytic functions on the open unit disc and denote by B(H ∞ ) the Banach space of all bounded linear operators from H ∞ to itself. We prove that the Bishop-Phelps-Bollobás property holds for B(H ∞ ). As an application to our approach, we prove that the Bishop-Phelps-Bollobás property also holds for operator ideals of B(H ∞ ).
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