Raffael Hagger

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.

The eigenvalues of tridiagonal sign matrices are dense in the spectra of periodic tridiagonal sign operators

Abstract Chandler-Wilde, Chonchaiya and Lindner conjectured that the set of eigenvalues of finite tridiagonal sign matrices (i.e. plus and minus ones on the first sub- and superdiagonal, zeroes everywhere else) is dense in the set of spectra of periodic tridiagonal sign operators on the usual Hilbert space of square summable bi-infinite sequences. 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.

On Symmetries of the Feinberg–Zee Random Hopping Matrix

In this paper we study the spectrum Σ of the infinite Feinberg–Zee random hopping matrix, a tridiagonal matrix with zeros on the main diagonal and random ±1’s on the first sub- and super-diagonals; the study of this non-selfadjoint random matrix was initiated in Feinberg and Zee (Phys. Rev. E 59 (1999), 6433–6443). Recently Hagger (Random Matrices: Theory Appl., 4 1550016 (2015)) has shown that the so-called periodic part Σπ of Σ, conjectured to be the whole of Σ and known to include the unit disk, satisfies \(p^{-1}(\Sigma_{\pi})\;\subset\;{\Sigma_{\pi}}\) for an infinite class S of monic polynomials p. In this paper we make very explicit the membership of S, in particular showing that it includes \(P_m(\lambda)\;=\;\lambda U_{m-1},\;\mathrm{for}\;m\;\geq 2\), where Un(x) is the Chebychev polynomial of the second kind of degree n. We also explore implications of these inverse polynomial mappings, for example showing that Σπ is the closure of its interior, and contains the filled Julia sets of infinitely many \(p\in\;\mathcal{S}\), including those of P m, this partially answering a conjecture of the second author.

Uniform continuity and quantization on bounded symmetric domains: UC FUNCTIONS AND QUANTIZATION

We consider Toeplitz operatorsTλfwith symbolfacting on the standard weighted Bergmanspaces over a bounded symmetric domain Ω⊂Cn.Hereλ>genus−1 is the weight parameter.The classical asymptotic relation for the semi-commutatorlimλ→∞∥∥∥TλfTλg−Tλfg∥∥∥λ=0,withf,g∈C(Bn),(∗)where Ω =Bndenotes the complex unit ball, is extended to larger classes of bounded andunbounded operator symbol-functions and to more general domains. We deal with operatorsymbols that generically are neither continuous inside Ω nor admit a continuous extension to theboundary. Letβdenote the Bergman metric distance function on Ω. We prove that(∗) remainstrue forfandgin the space UC(Ω) of allβ-uniformly continuous functions on Ω. Note that thisspace contains also unbounded functions. In case of the complex unit ball Ω =Bn⊂Cnwe showthat(∗) holds true for bounded symbols in VMO(Bn), where the vanishing oscillation insideBnismeasured with respect toβ.Atthesametime(∗) fails for generic bounded measurable symbols.We construct a corresponding counterexample using oscillating symbols that are continuousoutside of a single point in Ω

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

We study

Essential pseudospectra and essential norms of band-dominated operators

C^*

Limit operators, compactness and essential spectra on bounded symmetric domains

C∗-algebras generated by Toeplitz operators acting on the standard weighted Bergman space