More elementary operators that are spectrally bounded

We discuss some necessary and some sufficient conditions for an elementary operator x↦∑ⁿ_i=1a_ixb_i on a Banach algebra A to be spectrally bounded. In the case of length three, we obtain a complete characterisation when A acts irreducibly on a Banach space of dimension greater than three.

Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

Elementary operators-still not elementary?

Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle.

Non-Gaussian distribution of collective operators in quantum spin chains

We numerically analyse the behavior of the full distribution of collective observables in quantum spin chains. While most of previous studies of quantum critical phenomena are limited to the first moments, here we demonstrate how quantum fluctuations at criticality lead to highly non-Gaussian distributions. Interestingly, we show that the distributions for different system sizes collapse on thesame curve after scaling for a wide range of transitions: first and second order quantum transitions and transitions of the Berezinskii–Kosterlitz–Thouless type. We propose and analyse the feasibility of an experimental reconstruction of the distribution using light–matter interfaces for atoms in optical lattices or in optical resonators.

Orbits of semigroups of truncated convolution operators

We prove that a semigroup generated by finitely many truncated convolution operators on $L_p[0, 1]$ with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators T1,T2,…,TN with N⩾2 which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators T1, T2 whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.