论文标题
关于BANACH空间中的子空间和预测的五个定理，以及对操作员的拓扑和分析的应用
Five Theorems on Splitting Subspaces and Projections in Banach Spaces and Applications to Topology and Analysis in Operators
论文作者
论文摘要
令$ b（e，f）$表示从$ e $到$ f $的所有有界线性运算符的集合，$ b^+（e，f）$ $ b（e，f）$中的双重分配运算符集。当$ e，f $都是无限的尺寸时，在$ b（e，f）$中，矩阵中没有更多的基本转换，因此失去了讨论此类集合在$ b^+（e，f）$ as $ b^+（e，f）as $ b^+（e，f）为$φ_{m，n} = \ n（e，f）in（e，f）： \ Mathrm {codim} r（t）= n <\ infty \}，$ $ f_k = \ {t \ in b（e，f）中：\ mathrm {rank} \，t = k <\ infty \} $，等等。在本文中，我们介绍了关于投影和在Banach空间中分裂子空间的五个定理，而不是基本变换。令$φ$表示$ f_k，k <\ infty $和$φ_{m，n} $，带有$ m> 0 $或$ n> 0。 $ \ widetilde {t} $由$ t \在b^+（e，f）$中生成的$ \ dim n（t）> 0 $或$ \ mathrm {codim} r（t）> 0 $是路径连接。 （操作拓扑中的这种等效关系首次出现。）作为定理的应用，我们给出的是，$φ$是$ b（e，f）$中的平稳且连接的子元素，与切线空间$t_xφ= \ {t \ { $ b（\ mathbf {r}^m，\ mathbf {r}^n）= \ bigCup^{\ min \ { $ b（e，f）$，特别是$ \ dim f_k =（m+n-k）k，k = 0,1，\ cdots，\ min \ min \ {m.n \}。特殊利益的$是$ f_k \ $ f_k \，k = 0,1，$ cdots的$ f_k \ c，\ n是$ f _k \ c，min c cytementefface的subhypersurface。代数几何形状。鉴于上述定理的证明，不能说定理$ 1.1-1.5 $提供了在$ b^+（e，f）中查找路径连接集的规则。$。
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 $Φ_{m,n}=\{T\in B(E,F): \dim N(T)=m<\infty \ \mbox{and} \ \mathrm{codim}R(T)=n<\infty\},$ $F_k=\{T\in B(E,F): \mathrm{rank}\, T =k<\infty\}$, and so forth. In this paper we present five theorems on projections and splitting subspaces in Banach spaces instead of the elementary transformation. Let $Φ$ denote any one of $F_k ,k<\infty$ and $Φ_{m,n}$ with either $m>0$ 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 $\widetilde{T}$ 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 $Φ$ is a smooth and path connected submanifold in $B(E,F)$ with the tangent space $T_XΦ=\{T\in B(E,F): TN(X)\subset R(X)\}$ at any $ X\in {Φ},$ 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 $B^+(E,F).$
