In this work we prove that the eigenvalues of $n$-dimensional massive Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly non-Hermitian potential $V$, are localized in union two disjoint disks complex plane, provided $V$ is sufficiently small with respect to mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$. massless case, instead discrete spectrum empty...

The notion of automatic selfadjointness all ideals in a multiplicative semigroup the bounded linear operators on separable Hilbert space B(H) arose 2015 discussion with Heydar Radjavi who pointed out that and finite rank F(H) possessed this unitary invariant property which category we named SI semigroups (for selfadjoint ideal semigroups). Equivalent to is solvability, for each A semigroup, bil...

The indefinite Sturm-Liouville operator A = (sgn x)(−d2/dx2 + q(x)) is studied. It is proved that similarity of A to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of Weyl functions are given. For operators with a finite-zone potential, the components Aess and Adisc of A corresponding to essential and discrete spectrums, respe...

We consider elliptic operators A on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of A through an observation, with an exponential cost. Following the strategy of...

We study low lying eigenvalues for non-selfadjoint semiclassical differential operators, where symmetries play an important role. In the case of the Kramers-Fokker-Planck operator, we show how the presence of certain supersymmetric and PT -symmetric structures leads to precise results concerning the reality and the size of the exponentially small eigenvalues in the semiclassical (here the low t...