We discuss several natural metrics on spaces of unbounded self– adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of paths which is pa...

We introduce the notion of formally self-adjoint conformally covariant polydifferential operators and give some constructions families such operators. In one direction, we show that any homogeneous variational scalar Riemannian invariant (CVI) induces these another use ambient metric to alternative certain produced this way, which is a self-adjoint, fourth-order, tridifferential operator should...

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let A and V be bounded self-adjoint operators. Assume that the spectrum of A consists of two disjoint parts σ and Σ such that d = dist(σ,Σ) > 0. We show that the norm of the difference of the spectral projections EA(σ) and EA+V ( {λ | dist(λ, σ) < d/2} ) for A and A+ V is...

The polarized Gowdy T 3 vacuum spacetimes are characterized, modulo gauge, by a " point particle " degree of freedom and a function ϕ that satisfies a linear field equation and a non-linear constraint. The quantum Gowdy model has been defined by using a representation for ϕ on a Fock space F. Using this quantum model, it has recently been shown that the dynamical evolution determined by the lin...

Spectra of observables in the q-oscillator and q-analogue of the Fourier transform Abstract. Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + − qa + a = 1) are studied when q > 1. These operators are symmetric but not self-adjoint. They have one-parameter family of self-adjoint extensions. These extensions are derived explicit...

in this paper, we find explicit solution to the operator equation$txs^* -sx^*t^*=a$ in the general setting of the adjointable operators between hilbert $c^*$-modules, when$t,s$ have closed ranges and $s$ is a self adjoint operator.

A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space X is constructed. This model is analogous to the Nagy– Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-plane. The transformation map from X to the mode...

Associated with every commuting m-tuples of operators on a complex Hilbert space X is its Aluthge transform. In this paper we show that and transform have the same joint essential spectrum. Further, it shown spectrum contained in numerical range