The paper is a short survey around the author’s recent works on topics related to complex Monge-Ampère equations and strictly pseudoconvex pseudo-Hermitian manifolds. 1. Invariant differential operators In complex analysis of one variable, the fact that the invariant property for Laplace operator under holomorphic change of coordinates plays an important role. Namely, Let φ : D1 → D2 be a holom...

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the x and p appearing in the Hamiltonian, but quantities X and P constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find t...

We show that the R/Z part of the analytically defined η invariant of Atiyah-PatodiSinger for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, we discuss the relation with the Atiyah-PatodiSinger R/Z index theorem for unitary flat vector bundles, an...

We present the detailed account of the quantum(-like) viewpoint to common knowledge. The Binmore-Brandenburger operator approach to the notion of common knowledge is extended to the quantum case. We develop a special quantum(-like) model of common knowledge based on information representations of agents which can be operationally represented by Hermitian operators. For simplicity, we assume tha...

In harmonic analysis on classical domains of matrices, the differential operator whose symbol is the determinant polynomial plays important roles. Particularly, the operator is substantial in study of invariant Hilbert spaces of holomorphic functions on the domain [1, 2, 7, 14, 15, 21]. Considering the Siegel domain realization of a certain symmetric domain with Fourier-analytic methods, Jakobs...

This paper offers the functional model of a class of non-selfadjoint extensions of a Hermitian operator with equal deficiency indices. The explicit form of dilation of a dissipative extension is offered and the symmetric form of Sz.Nagy-Foiaş model as developed by B. Pavlov is constructed. A variant of functional model for a general non-selfadjoint non-dissipative extension is formulated. We il...