Inverse spectral problems for positive Hankel operators
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908968456028160 |
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| author | Pushnitski, Alexander Treil, Sergei |
| author_facet | Pushnitski, Alexander Treil, Sergei |
| contents | A Hankel operator $Γ$ in $L^2(\mathbb{R}_+)$ is an integral operator with the integral kernel of the form $h(t+s)$, where $h$ is known as the kernel function. It is known that $Γ$ is positive semi-definite if and only if $h$ is the Laplace transform of a positive measure $μ$ on $\mathbb{R}_+$. Thus, positive semi-definite Hankel operators $Γ$ are parameterised by measures $μ$ on $\mathbb{R}_+$. We consider the class of $Γ$ corresponding to \emph{finite} measures $μ$. In this case it is possible to define the (scalar) spectral measure $σ$ of $Γ$ in a natural way. The measure $σ$ is also finite on $\mathbb{R}_+$. This defines the \emph{spectral map} $μ\mapstoσ$ on finite measures on $\mathbb{R}_+$. We prove that this map is an involution; in particular, it is a bijection. We also consider a dual variant of this problem for measures $μ$ that are not necessarily finite but have the finite integral \[ \int_0^\infty x^{-2}\mathrm{d}μ(x); \] we call such measures \emph{co-finite}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse spectral problems for positive Hankel operators Pushnitski, Alexander Treil, Sergei Spectral Theory 47B35 A Hankel operator $Γ$ in $L^2(\mathbb{R}_+)$ is an integral operator with the integral kernel of the form $h(t+s)$, where $h$ is known as the kernel function. It is known that $Γ$ is positive semi-definite if and only if $h$ is the Laplace transform of a positive measure $μ$ on $\mathbb{R}_+$. Thus, positive semi-definite Hankel operators $Γ$ are parameterised by measures $μ$ on $\mathbb{R}_+$. We consider the class of $Γ$ corresponding to \emph{finite} measures $μ$. In this case it is possible to define the (scalar) spectral measure $σ$ of $Γ$ in a natural way. The measure $σ$ is also finite on $\mathbb{R}_+$. This defines the \emph{spectral map} $μ\mapstoσ$ on finite measures on $\mathbb{R}_+$. We prove that this map is an involution; in particular, it is a bijection. We also consider a dual variant of this problem for measures $μ$ that are not necessarily finite but have the finite integral \[ \int_0^\infty x^{-2}\mathrm{d}μ(x); \] we call such measures \emph{co-finite}. |
| title | Inverse spectral problems for positive Hankel operators |
| topic | Spectral Theory 47B35 |
| url | https://arxiv.org/abs/2503.22189 |