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In this paper, we obtain sufficient and necessary conditions for big truncated Hankel operators on model spaces to be zero, or of finite rank or compact. Our main tools are the properties of Hardy Hankel operators and function algebras.

We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions, for which a dual extremal function exists in terms of the existence of a maximizing vector of the corresponding Hankel operator and in terms of certain special factorizations that involve thematic matrix functions.

We show that on the harmonic Bergman spaces, the Hankel operators with nonconstant harmonic symbol cannot be of finite rank.

We show that if G G is an operator valued analytic function in the open right half plane such that the Hankel operator H G H_G with symbol G G is of trace-class, then G G has continuous extension to the imaginary axis, \[ G ( ∞ ) := lim r → ∞ r ∈ R G ( r ) G(\infty ):=\lim \limits _{r \rightarrow \infty \atop r \in \mathcal {R}} G(r) \] exists in the trace-class norm, and t r ( H G ) = 1 2 t r ( G ( 0 ) − G ( ∞ ) ) \mathrm {tr}(H_G)={1\over 2}\, \mathrm {tr}(G(0)-G(\infty )) .

In this note we discuss a strengthened version of a theorem due to Parrott [8] in operator dilation theory. We relate our result to the one-step extension procedure of Adamjan-Arov-Krein [1].

The theorem in the title is proved and used to give a new proof that elements of L ∞ {L^\infty } have best approximations in H ∞ + C {H^\infty } + C .

Let (H?)b be the Bourgain algebra of HOO c LI?. We prove (H?)b = HI + C. In particular if f E LI then the Hankel operator Hf is a compact map of HOO into BMO iff whenever f, -0 weakly in HI?, then dist(ffn, HOO) , O .