The Fredholm property of Toeplitz operators on the p-Fock spaces F p α on C n is studied. A general Fredholm criterion for arbitrary operators from the Toeplitz algebra Tp,α on F p α in terms of the invertibility of limit operators is derived. This paper is based on previous work, which establishes corresponding results on the unit balls B n [10].

In this paper we study the Fredholm properties of Toeplitz operators acting on weighted Bergman spaces A p ν (B n ), where p ∈ (1, ∞) and B n ⊂ C n denotes the n-dimensional open unit ball. Let f be a continuous function on the Euclidean closure of B n . It is well-known that then the corresponding Toeplitz operator T f is Fredholm if and only if f has no zeros on the boundary ∂B n . As a consequence, the essential spectrum of T f is given by the boundary values of f . We extend this result to all operators in the algebra generated by Toeplitz operators with bounded symbol (in a sense to be made precise down below). The main ideas are based on the work of Suárez et al. ([17,24]) and limit operator techniques coming from similar problems on the sequence space ℓ p (Z) ([13, 15, 19] and references therein).

We study the symmetries of the spectrum of the Feinberg-Zee Random Hopping Matrix introduced in [6] and studied in various papers therafter (e.g.[1], [2], [3], [7], [9]). In [3], 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 in [1]). 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 and the result of [3], 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.

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.