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On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

Markin Marat V.

Demonstratio Mathematica

2020

Vol. 53, nr 1

352--359

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection {etA}t≥0 of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C0-semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group {etA}t∈R of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L2(R), the anti-self-adjoint differentiation operator A≔ddx with the domain D(A)≔W12(R) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

Reduction for natural operators on projectable connections

Mikulski M., Tomas J.

Demonstratio Mathematica

2009

Vol. 42, nr 2

435-439

We present a very simple proof of a general reduction for natural operators on torsion free projectable classical linear connections.

On naturality of the Legendre operator

Mikulski M.

Demonstratio Mathematica

2008

Vol. 41, nr 4

969-973

We deduce that all natural operators of the type of the Legendre operator from the variational calculus in fibred manifolds are the constant multiples of the Legendre operator.

Natural liftings of connections to the r-th order bundle

Mikulski W.

Demonstratio Mathematica

2007

Vol. 40, nr 2

481-484

We describe all natural operators A lifting a clasiccal linear connection on an m-dimensional manifold M into a classical linear conection A() on the r-th order frame bundle LrM = invJr/0 (Rm,M).

On the contact (k, r)-coelements

Mikulski W. M.

Demonstratio Mathematica

2003

Vol. 36, nr 2

433--449