Stanislav Shkarin

Time operator for diffusion

We extend the concept of time operator for general semigroups and construct a non-self-adjoint time operator for the diffusion equation which is intertwined with the unilateral shift. We obtain the spectral resolution, the age eigenstates and a new shift representation of the solution of the diffusion equation. Based on previous work we obtain similarly a self-adjoint time operator for Relativistic Diffusion.

Time operators and shift representation of dynamical systems

We introduce and characterise time operators for unilateral shifts and exact endomorphisms. The associated shift representation of evolution is related to the spectral representation by a generalized Fourier transform. We illustrate the results for a simple exact system, namely the Renyi map.

The spectrum of the Liouville-von Neumann operator in the Hilbert-Schmidt space

The singular continuous spectrum of the Liouville operator of quantum statistical physics is, in general, properly included in the difference of the spectral values of the singular continuous spectrum of the associated Hamiltonian. The absolutely continuous spectrum of the Liouvillian may arise from a purely singular continuous Hamiltonian. We provide the correct formulas for the spectrum of the Liouville operator and show that the decaying states of the singular continuous subspace of the Hamiltonian do not necessarily contribute to the absolutely continuous subspace of the Liouvillian.

Antisupercyclic operators and orbits of the Volterra operator

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any

SOME RESULTS ON SOLVABILITY OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS IN LOCALLY CONVEX SPACES

x\in B

Existence theorems in linear chaos

either

Hypercyclic tuples of operators on ℂ n and ℝ n

T^nx=0

The Golod-Shafarevich inequality for Hilbert series of quadratic algebras and the Anick conjecture

for some positive integer n or the sequence

New extended spectral decompositions of the Renyi map

\{T^nx/\|T^nx\|\}

Hypercyclic and mixing operator semigroups

weakly converges to zero in B . Antisupercyclicity of T means that the angle criterion of supercyclicity is not satisfied for T in the strongest possible way. Normal antisupercyclic operators and antisupercyclic bilateral weighted shifts are characterized. As for the Volterra operator V , it is proved that if