arXiv subjects

Raffael Hagger

Publications and source records attributed to Raffael Hagger.

22 records · Page 2Linked to original sources

arXiv2015

Essential pseudospectra and essential norms of band-dominated operators

An operator $A$ on an $l^p$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of $A$ in the Calkin algebra determines, for example, the Fredholmness of $A$, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of $A$. This coset can be identified with the collection of all so-called limit operators of $A$. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent norms (hence pseudospectra). In fact we work with a generalization of the ideal of compact operators, so-called $\mathcal{P}$-compact operators, allowing for a more flexible framework that naturally extends to $l^p$-spaces with $p\in\{1,\infty\}$ and/or vector-valued $l^p$-spaces.

math.FA↗

arXiv2015

On the Spectrum and Numerical Range of Tridiagonal Random Operators

In this paper we derive an explicit formula for the numerical range of (non-self-adjoint) tridiagonal random operators. As a corollary we obtain that the numerical range of such an operator is always the convex hull of its spectrum, this (surprisingly) holding whether or not the random operator is normal. Furthermore, we introduce a method to compute numerical ranges of (not necessarily random) tridiagonal operators that is based on the Schur test. In a somewhat combinatorial approach we use this method to compute the numerical range of the square of the (generalized) Feinberg-Zee random hopping matrix to obtain an improved upper bound to the spectrum. In particular, we show that the spectrum of the Feinberg-Zee random hopping matrix is not convex.

math.SP↗

arXiv2015

Symmetries of the Feinberg-Zee Random Hopping Matrix

We study the symmetries of the spectrum of the Feinberg-Zee Random Hopping Matrix. Chandler-Wilde and Davies proved that the spectrum of the Feinberg-Zee Random Hopping Matrix is invariant under taking square roots, which implied that the unit disk is contained in the spectrum (a result already obtained slightly earlier by Chandler-Wilde, Chonchaiya and Lindner). In a similar approach we show that there is an infinite sequence of symmetries at least in the periodic part of the spectrum (which is conjectured to be dense). Using these symmetries, we can exploit a considerably larger part of the spectrum than the unit disk. As a further consequence we find an infinite sequence of Julia sets contained in the spectrum. These facts may serve as a part of an explanation of the seemingly fractal-like behaviour of the boundary.

math.SP↗

arXiv2015

The Eigenvalues of Tridiagonal Sign Matrices are Dense in the Spectra of Periodic Tridiagonal Sign Operators

Chandler-Wilde, Chonchaiya and Lindner conjectured that the set of eigenvalues of finite tridiagonal sign matrices ($\pm 1$ on the first sub- and superdiagonal, $0$ everywhere else) is dense in the set of spectra of periodic tridiagonal sign operators on $\ell^2(\mathbb{Z})$. We give a simple proof of this conjecture. As a consequence we get that the set of eigenvalues of tridiagonal sign matrices is dense in the unit disk. In fact, a recent paper further improves this result, showing that this set of eigenvalues is dense in an even larger set.

math.SP↗