2013/06/16 by D. R. Yafaev, Yafaev, D. R.
Mathematics · #47A40 #47B25 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical functions and polynomials #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.1306.3676
openalex publication_date 2013/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that every Hankel operator H is unitarily equivalent to a pseudo-differential operator A of a special structure acting in the space L2 (\Bbb R) . As an example, we consider integral operators H in the space L2 (\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 +∞ and -∞. We find the asymptotics of these eigenvalues.