Singular Values of Hankel Operators
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In this chapter we study singular values of Hankel operators. The main result of the chapter is the fundamental theorem of Adamyan, Arov, and Krein. This theorem says that if Γ is a Hankel operator, then to evaluate the nth singular value s n (Γ) of r, there is no need to consider all operators of rank at most n, s n (Γ) is the distance from Γ to the set of Hankel operators of rank at most n. In §1 we prove the Adamyan—Arov—Krein theorem in the special case when s n (Γ) is greater than the essential norm of r. In §2 we reduce the general case to the case treated in §1. In §1 we also prove the uniqueness of the corresponding Hankel approximant of rank at most n under the same condition s n (Γ) > ||Γ||e, and we obtain useful formulas for multiplicities of singular values of related Hankel operators. We prove a generalization of the Adamyan—Arov—Krein theorem to the case of vectorial Hankel operators in §3. We also obtain in §3 a formula for the essential norm of vectorial Hankel operators.
KeywordsHilbert Space Toeplitz Operator Dense Range Separable Hilbert Space Hankel Operator
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