Five Theorems on Splitting Subspaces and Projections in Banach Spaces and Applications to Topology and Analysis in Operators

2020/11/15 by Ma, Jipu
#58B05 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2011.07488

Abstract

Let B(E,F) denote the set of all bounded linear operators from E into F, and B⁺(E,F) the set of double splitting operators in B(E,F). When both E,F are infinite dimensional , in B(E,F) there are not more elementary transformations in matrices so that lose the way to discuss the path connectedness of such sets in B⁺(E,F) as Φ_m,n=\T∈ B(E,F): dim N(T)=m0 or n>0. Using these theorems we prove Φ is path connected.Also these theorems bear an equivalent relation in B⁺(E,F), so that the following general result follows: the equivalent class \widetildeT generated by T∈ B⁺(E,F) with either dim N(T)>0 or codim R(T)>0 is path connected. (This equivalent relation in operator topology appears for the first time.) As applications of the theorems we give that Φ is a smooth and path connected submanifold in B(E,F) with the tangent space T_XΦ=\T∈ B(E,F): TN(X)⊂ R(X)\ at any X∈ Φ, and prove that B(R^m,Rⁿ)=\bigcup^min\n,m\_k=0F_k possesses the following properties of geometric and topology : F_k ( k

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