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

TL;DR

This paper establishes five theorems on projections and splitting subspaces in Banach spaces, demonstrating path connectedness of certain operator sets and their applications to topology and geometry of operator spaces.

Contribution

It introduces five new theorems on projections and splitting subspaces, providing a framework for analyzing path connectedness in operator sets in Banach spaces.

Findings

Sets of finite rank operators are path connected.

Theorems reveal smooth manifold structures in operator spaces.

Dimensional formulas for algebraic geometric properties of operator subsets.

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 $\Phi_{m,n}=\{T\in B(E,F): \dim N(T)=m<\infty \ \mbox{and} \ \mathrm{codim}R(T)=n<\infty\},$ $F_k=\{T\in B(E,F): \mathrm{rank}\, T =k<\infty\}$, 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 $B^+(E,F).$