Denseness for norm attaining operator-valued functions
Abstract In this note we offer a short, constructive proof for Hilbert spaces of Lindenstrauss' famous result on the denseness of norm attaining operators. Specifically, we show given any A∈ L ( H ) there is a sequence of rank-1 operators K n such that A+K n is norm attaining for each n and K n converges in norm to zero. We then apply our construction to establish denseness results for norm attaining operator-valued functions.
Finite rank harmonic operator-valued functions
Abstract The purpose of this paper is to characterize when a harmonic function with values in the finite rank operators on a Hilbert space is expressible as a harmonic matrix-valued function. We show that harmonic function with values in the rank 1 normal operators is expressible as a harmonic matrix-valued function. We also prove that for any natural number, n , a harmonic function with values in the rank n non-negative operators is expressible as a matrix-valued function and we give examples showing that these decomposition theorems fail when various hypotheses are relaxed.
Main tendencies of religious consciousness development in modern society
The article is devoted to the investigation of the tendencies of religious consciousness development in modern society. At the present stage of development religious consciousness, like any other social and psychological phenomenon, is undergoing significant changes. The development of religious consciousness at the present stage of society evolution leads to the development of common moral norms and requirements. There is a tendency to reduce human ties with the religious community as a translator of religious consciousness. It is possible to speak about the transformation of religious consciousness but not about its disappearance.