Spectral properties of truncated Toeplitz operators by equivalence after extension

We study truncated Toeplitz operators in model spaces View the MathML source for 1<p<∞, with essentially bounded symbols in a class including the algebra View the MathML source, as well as sums of analytic and anti-analytic functions satisfying a θ -separation condition, using their equivalence after extension to Toeplitz operators with 2×2 matrix symbols. We establish Fredholmness and invertibility criteria for truncated Toeplitz operators with θ -separated symbols and, in particular, we identify a class of operators for which semi-Fredholmness is equivalent to invertibility. For symbols in View the MathML source, we extend to all p∈(1,∞) the spectral mapping theorem for the essential spectrum. Stronger results are obtained in the case of operators with rational symbols, or if the underlying model space is finite-dimensional

Paired kernels and their applications

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly $S^*$-invariant subspace of $H^2$, is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators.Comment: 26 page

Model spaces and Toeplitz kernels in reflexive Hardy spaces

This paper considers model spaces in an Hp setting. The existence of unbounded functions and the characterisation of maximal functions in a model space are studied, and decomposition results for Toeplitz kernels, in terms of model spaces, are establishedFCT/Portuga

Dark linear seesaw mechanism

We propose a minimal model where a dark sector, odd under a $\mathcal{Z}_2$ discrete symmetry, is the seed of lepton number violation in the neutrino sector at the loop level, in the context of the linear seesaw mechanism. Neutrino mass suppression stems from a naturally small scalar potential coupling which breaks the lepton number symmetry softly. The fact that we consider (dark) Dirac vector-like neutral leptons, prevents the appearance of other mass terms that could contribute to neutrino masses via alternative mechanisms. We study the dark-matter phenomenology of the model, focusing on the case in which the stable particle is the lightest neutral scalar arising from the dark scalar sector. Prospects for testing our framework with the results of current and future lepton flavour violation searches are also discussed.Comment: 13 LaTeX pages, 3 figure

Kernels of unbounded Toeplitz operators and factorization of symbols

We consider kernels of unbounded Toeplitz operators in Hp(C+) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in Hp(C+), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.This work was partially supported by FCT/Portugal through UID/MAT/04459/2020. The research of M. T. Malheiro was partially supported by Portuguese Funds through FCT/Portugal within the Projects UIDB/00013/2020 and UIDP/00013/2020

Hybrid scoto/seesaw: flavour and dark matter

We propose a model based on the interplay between the type-II seesaw and scotogenic neutrino mass generation mechanisms. The setup features a $\mathbb{Z}_8$ discrete flavour symmetry which is broken down to a residual $\mathbb{Z}_2$ responsible for stabilising dark matter. A singlet scalar field is introduced to implement spontaneous CP violation. The effective neutrino mass matrix two-texture zero structure leads to sharp neutrino sector predictions. We analyse the constraints imposed on the model by current and future charged lepton flavour violation experiments. This framework provides two viable dark matter candidates, scalar or fermion. We investigate the scalar dark matter scenario considering relic density, direct-detection and collider constraints.Comment: 8 LaTeX pages; 4 figures. Contribution to the Proceedings of the 8th Symposium on Prospects in the Physics of Discrete Symmetries (DISCRETE 2022), 7-11 November 2022, Baden-Baden, German