Über gewisse Extremumprobleme der Funktionentheorie

A Non-Linear Extremal Problem on the Hardy Space

“…we can use operator theoretic techniques to explicitly determine the constants c n,λ as well as the associated extremal functions. Although this problem has been well-studied [12,20,26,27], we obtain the same results (see Theorem 3) using the new language of truncated Toeplitz operators.…”

“…Although, at first glance, the expression fzΘ in (12) does not appear to correspond to the boundary values of an H 2 function, let alone one in K Θ , a short computation using (7) reveals that Cf, Θh and Cf, zh both vanish for all h ∈ H 2 and f ∈ K Θ , whence Cf indeed belongs to K Θ .…”

“…From the representation (7) we find that K Θ carries a natural isometric, conjugatelinear involution C : K Θ → K Θ defined in terms of boundary functions by (12) Cf := fzΘ.…”

“…In the special case when c 0 = c 1 = • • • = c n = 1, Egerváry [12] provided the explicit formula is an extremal function (In particular, note that this is the square of an H 2 function). Golusin in [21] finds an extremal function for the original (general) Fejér extremal problem as well as some others [20].…”

Eigenvalue estimates for the resolvent of a non-normal matrix

“…Our approach has the advantage that it yields spectral bounds for any ζ ∈ C − σ(A) and that the optimality statement 3 is automatic. Concerning the second point of the theorem we present a technique going back to [7] in order to compute the norm of Toeplitz matrices of the form…”