arXiv subjects

Emilio Fedele

Publications and source records attributed to Emilio Fedele.

3 recordsLinked to original sources

arXiv2019

The spectral density of Hankel operators with piecewise continuous symbols

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).

math.SP↗

arXiv2018

On determinants identity minus Hankel matrix

In this note, we study the asymptotics of the determinant $\det(I_N - βH_N)$ for $N$ large, where $H_N$ is the $N\times N$ restriction of a Hankel matrix $H$ with finitely many jump discontinuities in its symbol satisfying $\|H\|\leq 1$. Moreover, we assume $β\in\mathbb C$ with $|β|<1$ and $I_N$ denotes the identity matrix. We determine the first order asymtoptics as $N\to\infty$ of such determinants and show that they exhibit power-like asymptotic behaviour, with exponent depending on the height of the jumps. For example, for the $N \times N$ truncation of the Hilbert matrix $\mathbf{H}$ with matrix elements $π^{-1}(j+k+1)^{-1}$, where $j,k\in \mathbb Z_+$ we obtain $$ \log \det(I_N - β\mathbf{H}_N) = -\frac{\log N}{2π^2} \big(π\arcsin(β)+\arcsin^2(β)+o(1)\big),\qquad N\to\infty. $$

math.FA↗

arXiv2017

Weighted integral Hankel operators with continuous spectrum

Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in $L^2(\mathbb R_+)$. These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel $s^αt^α(s+t)^{-1-2α}$, where $α>-1/2$. Our analysis can be considered as an extension of J.Howland's 1992 paper which dealt with the unweighted case, corresponding to $α=0$.

math.SP↗