两类无界Hankel算子的对角化
Diagonalizations of two classes of unbounded Hankel operators
摘要：
证明了每个Hankel算子$H$酉等价于作用在空间$L^2 ({\Bbb R}) $中具有特殊结构的伪微分算子$A$.作为一个例子，我们考虑具有核$P (\ln (t+s)) (t+s)^{-1}$的空间$L^2 ({\Bbb R}_{+}) $中的积分算子$H$，其中$P(x)$是$K$的任意实多项式。在这种情况下，$A$是与$K$相同阶的微分算子。这使得我们能够研究具有这种核的Hankel算子$H$的谱性质。特别地，我们证明了$H$的本征谱对于$K$奇数与全轴重合，对于$K$偶数与正半轴重合.在后一种情况下，我们还找到了$H$的积极性的充分必要条件。我们还考虑了Hankel算子，其核在某个正点处具有强奇异性。我们证明了这类算子的谱由无穷重的零本征值和累积到$+\infty$和$-\infty$的本征值组成。我们发现这些特征值的渐近性。
英文摘要：
We show that every Hankel operator $H$ is unitarily equivalent to a pseudo-differential operator $A$ of a special structure acting in the space $L^2 ({\Bbb R}) $. As an example, we consider integral operators $H$ in the space $L^2 ({\Bbb R}_{+}) $ with kernels $P (\ln (t+s)) (t+s)^{-1}$ where $P(x)$ is an arbitrary real polynomial of degree $K$. In this case, $A$ is a differential operator of the same order $K$. This allows us to study spectral properties of Hankel operators $H$ with such kernels. In particular, we show that the essential spectrum of $H$ coincides with the whole axis for $K$ odd, and it coincides with the positive half-axis for $K$ even. In the latter case we additionally find necessary and sufficient conditions for the positivity of $H$. We also consider Hankel operators whose kernels have a strong singularity at some positive point. We show that spectra of such operators consist of the zero eigenvalue of infinite multiplicity and eigenvalues accumulating to $+\infty$ and $-\infty$. We find the asymptotics of these eigenvalues.
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