Hankel算子是符号定算子的判定
Criteria for Hankel operators to be sign-definite
摘要：
我们证明了具有核$h(t)$的自伴Hankel算子$H$的负谱和正谱的重数与实函数乘算子$s(x)$的重数是一致的.特别地，$\pm H\geq 0$当且仅当$\pm s(x)\geq 0$。核$h(t)$和它的“符号函数”$s(x)$由一个显式公式关联。$h(t)$以$s(x)$表示导致$h(t)$的指数表示。我们的方法直接适用于各类汉克尔算子。特别地，对于有限秩的Hankel算子，我们找到了它们的负谱和正谱的总重数的一个显式公式。
英文摘要：
We show that total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ with kernel $h(t)$ and of an operator of multiplication by some real function $s(x)$ coincide. In particular, $\pm H\geq 0$ if and only if $\pm s(x)\geq 0$. The kernel $h(t)$ and its "sign-function" $s(x)$ are related by an explicit formula. An expression of $h(t)$ in terms of $s(x)$ leads to an exponential representation of $h(t)$. Our approach directly applies to various classes of Hankel operators. In particular, for Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra.
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