Some Explicit Formulae for the Singular Values of Certain Hankel Operators with Factorizable Symbol

1988 ◽  
Vol 19 (5) ◽  
pp. 1081-1089 ◽  
Author(s):  
Ciprian Foias ◽  
Allen Tannenbaum ◽  
George Zames
Keyword(s):  
Singular Values ◽  
Hankel Operators ◽  
Explicit Formulae

scholarly journals Sharp Estimates for Singular Values of Hankel Operators

2015 ◽  
Vol 83 (3) ◽  
pp. 393-411 ◽  
Author(s):  
Alexander Pushnitski ◽  
Dmitri Yafaev
Keyword(s):  
Singular Values ◽  
Hankel Operators ◽  
Sharp Estimates

Inverse spectral problems for compact Hankel operators

2013 ◽  
Vol 13 (2) ◽  
pp. 273-301 ◽  
Author(s):  
Patrick Gérard ◽  
Sandrine Grellier
Keyword(s):  
Explicit Formula ◽  
Singular Values ◽  
Hankel Operators ◽  
Compact Operators ◽  
Unique Sequence ◽  
Complex Numbers ◽  
Inverse Spectral Problems ◽  
Real Numbers ◽  
Sigma 1 ◽  
Image Position

AbstractGiven two arbitrary sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $ of real numbers satisfying $$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$ we prove that there exists a unique sequence $c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $, real valued, such that the Hankel operators ${\Gamma }_{c} $ and ${\Gamma }_{\tilde {c} } $ of symbols $c= ({c}_{n} )_{n\geq 0} $ and $\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $, respectively, are selfadjoint compact operators on ${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have the sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $, respectively, as non-zero eigenvalues. Moreover, we give an explicit formula for $c$ and we describe the kernel of ${\Gamma }_{c} $ and of ${\Gamma }_{\tilde {c} } $ in terms of the sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $. More generally, given two arbitrary sequences $({\rho }_{j} )_{j\geq 1} $ and $({\sigma }_{j} )_{j\geq 1} $ of positive numbers satisfying $$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$ we describe the set of sequences $c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $ of complex numbers such that the Hankel operators ${\Gamma }_{c} $ and ${\Gamma }_{\tilde {c} } $ are compact on ${\ell }^{2} ({ \mathbb{Z} }_{+ } )$ and have sequences $({\rho }_{j} )_{j\geq 1} $ and $({\sigma }_{j} )_{j\geq 1} $, respectively, as non-zero singular values.

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