LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ...LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ∥ T0 # (T - T0 ∥ < 1, being a generalized inverse ofT, is presented, and hence, a rank theorem of operators onE intoF is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.展开更多
Let X be a topological space and A_x with closed range R(A_x) be a continuous mapping from X into B(H). It is well known that even if dim(H)<∞, the Moore-Penrose inverses A_x^+ may fail to be continuous. So it is ...Let X be a topological space and A_x with closed range R(A_x) be a continuous mapping from X into B(H). It is well known that even if dim(H)<∞, the Moore-Penrose inverses A_x^+ may fail to be continuous. So it is necessary to find sufficient and necessary conditions for A_x^+ being continuous. This will be important for the advanced analysis and its application. This paper gives these conditions in the cases that X is a topological space or X is locally compact, and A_x is a Fredholm family, respectively.展开更多
文摘LetE andF be Banach spaces, andB(E,F) all of bounded linear operators onE intoF. LetT 0 ∈B( E,F) with an outer inverseT 0 # ∈B( F,E). Then a characteristic condition ofS=(I + T0 # (T-T0)-1 T0 # with T∈B(E, F) and ∥ T0 # (T - T0 ∥ < 1, being a generalized inverse ofT, is presented, and hence, a rank theorem of operators onE intoF is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.
基金Project supported by the National Natural Science Foundation of China.
文摘Let X be a topological space and A_x with closed range R(A_x) be a continuous mapping from X into B(H). It is well known that even if dim(H)<∞, the Moore-Penrose inverses A_x^+ may fail to be continuous. So it is necessary to find sufficient and necessary conditions for A_x^+ being continuous. This will be important for the advanced analysis and its application. This paper gives these conditions in the cases that X is a topological space or X is locally compact, and A_x is a Fredholm family, respectively.
