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

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 $\Phi_{m,n}=\{T\in B(E,F): \dim N(T)=m<\infty \ \mbox{and} \ \mathrm{codim}R(T)=n<\infty\},$ $F_k=\{T\in B(E,F): \mathrm{rank}\, T =k<\infty\}$, and so forth. In this paper we present five theorems on projections and splitting subspaces in Banach spaces instead of the elementary transformation. Let $\Phi$ denote any one of $F_k ,k<\infty$ and $\Phi_{m,n}$ with either $m>0$ or $n>0.$ Using these theorems we prove $\Phi$ is path connected.Also these theorems bear an equivalent relation in $B^+(E,F)$, so that the following general result follows: the equivalent class $\widetilde{T}$ generated by $T\in B^+(E,F)$ with either $\dim N(T)>0$ or $\mathrm{codim} R(T)>0$ is path connected. (This equivalent relation in operator topology appears for the first time.) As applications of the theorems we give that $\Phi$ is a smooth and path connected submanifold in $B(E,F)$ with the tangent space $T_X\Phi =\{T\in B(E,F): TN(X)\subset R(X)\}$ at any $X\in {\Phi },$ and prove that $B(\mathbf{R}^m,\mathbf{R}^n)=\bigcup^{\min\{n,m\}}\limits_{k=0}F_k$ possesses the following properties of geometric and topology : $F_k ( k <\min\{ m,n\})$ is a smooth and path connected subhypersurface in $B(E,F)$, and specially, $\dim F_k=(m+n-k)k, k=0,1, \cdots , \min\{m.n\}.$ Of special interest is the dimensional formula of $F_k \, \, k=0,1, \cdots , \min\{m.n\},$ which is a new result in algebraic geometry. In view of the proofs of the above theorems it can not be too much to say that Theorems $1.1-1.5$ provide the rules of finding path connected sets in $B^+(E,F).$