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In this paper we generalize the concept of Davis-Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator A is considered. Moreover, we investigate the parallelism of A-bounded operators with respect to the seminorm and the numerical radius induced by A. Mainly, we characterize A-normaloid operators in terms of their A-Davis-Wielandt radii. In addition, a connection between A-seminorm-parallelism to the identity operator and an equality condition for the A-Davis-Wielandt radius is proved. This generalizes the well-known results in [23,10]. Some other related results are also discussed.

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This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L 1 -space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C 0 -semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.