Quantum-limited estimation of continuous spontaneous localization

We apply the formalism of quantum estimation theory to extract information about potential collapse mechanisms of the continuous spontaneous localisation (CSL) form.
In order to estimate the strength with which the field responsible for the CSL mechanism couples to massive systems, we consider the optomechanical interaction
between a mechanical resonator and a cavity field. Our estimation strategy passes through the probing of either the state of the oscillator or that of the electromagnetic field that drives its motion. In particular, we concentrate on all-optical measurements, such as homodyne and heterodyne measurements.
We also compare the performances of such strategies with those of a spin-assisted optomechanical system, where the estimation of the CSL parameter is performed
through time-gated spin-like measurements.

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.

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.

On the regularity of homomorphisms between Riesz subalgebras of L^r(X).

We investigate the automatic regularity of continuous algebra homomorphisms between Riesz algebras of regular operators on Banach lattices.