This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups a...This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups are always bounded linear operators, and that generators of C0-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of C0-semigroups in quotient indecomposable spaces are not necessarily bounded.展开更多
The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimens...The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T is an element of B(Y,X such that the range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc, are also discussed.展开更多
In functional analysis, the following problem is fundamental: does every infinite-dimensional Banach space have an infinite-dimensional quotient space? It has remainedunsolved for a long time.
基金The NSF (10471025) of China and the NSF (Z0511019) of Fujian Province in China.
文摘This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups are always bounded linear operators, and that generators of C0-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of C0-semigroups in quotient indecomposable spaces are not necessarily bounded.
文摘The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T is an element of B(Y,X such that the range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc, are also discussed.
文摘In functional analysis, the following problem is fundamental: does every infinite-dimensional Banach space have an infinite-dimensional quotient space? It has remainedunsolved for a long time.
