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# On the Singular Values of the Four-Block Operator and Certain Generalized Interpolation Problems

DOI link for On the Singular Values of the Four-Block Operator and Certain Generalized Interpolation Problems

On the Singular Values of the Four-Block Operator and Certain Generalized Interpolation Problems book

# On the Singular Values of the Four-Block Operator and Certain Generalized Interpolation Problems

DOI link for On the Singular Values of the Four-Block Operator and Certain Generalized Interpolation Problems

On the Singular Values of the Four-Block Operator and Certain Generalized Interpolation Problems book

## ABSTRACT

In the paper [1], Adamjan-Arov-Krein consider the problem of giving an interpolation theoretic interpretation to the singular values of a Hankel operator. More precisely, let wE L00 • (All of our Hardy and Lebesgue spaces will be defined on the unit circle in the standard way.) Let H00(k) := {q E L00 : q = r + t with r rational and having::; k poles in the unit disc D and t E H 00 } . Set

where w(U) : L2 -+ L2 denotes the operator induced by multiplication by w. (U : L2 -+ L2 is the bilateral shift.) Let sk(T), k = 0, ... , oo denote the kth singular value of a bounded linear operator T, so(T) 2:: s1(T) 2:: · · · 2:: sk(T) 2:: sk+t(T) 2:: · · · . Note by definition so(T) = IITJI. Then in [1], it is proven that Sk(Hw) = Uk . For k = 0, this is the Nehari theorem [10] .