Spectrally isometric elementary operators

We present criteria for unital elementary operators (of small length) on unital semisimple Banach algebras to be spectral isometries. The surjective ones among them turn out to be algebra automorphisms.

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.

Enzyme-induced Aggregation of Plasmonic Nanoparticles for the Development of Label-free and Ultrasensitive Detection of Bacterial DNA

Sensitive detection of pathogens is critical to ensure the safety of food supplies and to prevent bacterial disease infection and outbreak at the first onset. While conventional techniques such as cell culture, ELISA, PCR, etc. have been used as the predominant detection workhorses, they are however limited by either time-consuming procedure, complicated sample pre-treatment, expensive analysis and operation, or inability to be implemented at point-of-care testing. Here, we present our recently developed assay exploiting enzyme-induced aggregation of plasmonic gold nanoparticles (AuNPs) for label-free and ultrasensitive detection of bacterial DNA. In the experiments, AuNPs are first functionalized with specific, single-stranded RNA probes so that they exhibit high stability in solution even under high electrolytic condition thus exhibiting red color. When bacterial DNA is present in a sample, a DNA-RNA heteroduplex will be formed and subsequently prone to the RNase H cleavage on the RNA probe, allowing the DNA to liberate and hybridize with another RNA strand. This continuously happens until all of the RNA strands are cleaved, leaving the nanoparticles ‘unprotected’. The addition of NaCl will cause the ‘unprotected’ nanoparticles to aggregate, initiating a colour change from red to blue. The reaction is performed in a multi-well plate format, and the distinct colour signal can be discriminated by naked eye or simple optical spectroscopy. As a result, bacterial DNA as low as pM could be unambiguously detected, suggesting that the enzyme-induced aggregation of AuNPs assay is very easy to perform and sensitive, it will significantly benefit to development of fast and ultrasensitive methods that can be used for disease detection and diagnosis.

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.

Endonuclease Controlled Aggregation of Gold Nanoparticles for the Ultrasensitive Detection of Pathogenic Bacterial DNA

The development of an ultrasensitive biosensor for the low-cost and on-site detection of pathogenic DNA could transform detection capabilities within food safety, environmental monitoring and clinical diagnosis. Herein, we present an innovative approach exploiting endonuclease-controlled aggregation of plasmonic gold nanoparticles (AuNPs) for label-free and ultrasensitive detection of bacterial DNA. The method utilizes RNA-functionalized AuNPs which form DNA-RNA heteroduplex structures through specific hybridization with target DNA. Once formed, the DNA-RNA heteroduplex is susceptible to RNAse H enzymatic cleavage of the RNA probe, allowing the target DNA to liberate and hybridize with another RNA probe. This continuously happens until all of the RNA probes are cleaved, leaving the nanoparticles unprotected and thus aggregated upon exposure to a high electrolytic medium. The assay is ultrasensitive, allowing the detection of target DNA at femtomolar level by simple spectroscopic analysis (40.7 fM and 2.45 fM as measured by UV-vis and dynamic light scattering (DLS), respectively). The target DNA spiked food matrix (chicken meat) is also successfully detected at a concentration of 1.2 pM (by UV-vis) or 18.0 fM (by DLS). In addition to the ultra-high sensitivity, the total analysis time of the assay is less than 3 hours, thus demonstrating its practicality for food analysis.