We provide the proof of a previously announced result that resolves the following problem posed by A. A. Kirillov. Let T be a presentation of a group \(\mathcal {G}\) by bounded linear operators in a Banach space G and \(E\subset G\) be a closed invariant subspace. Then T generates in the natural way presentations \(T_1\) in E and \(T_2\) in \(F:=G/E\). What additional information is required besides \(T_1, T_2\) to recover the presentation T? In finite-dimensional (and even in infinite dimensional Hilbert) case the solution is well known: one needs to supply a group cohomology class \(h\in H^1(\mathcal {G},Hom(F,E))\). The same holds in the Banach case, if the subspace E is complemented in G. However, every Banach space that is not isomorphic to a Hilbert one has non-complemented subspaces, which aggravates the problem significantly and makes it non-trivial even in the case of a trivial group action, where it boils down to what is known as the three-space problem. This explains the title we have chosen. A solution of the problem stated above has been announced by the author in 1976, but the complete proof, for non-mathematical reasons, has not been made available. This article contains the proof, as well as some related considerations of the functor \(Ext^1\) in the category Ban of Banach spaces.

我们提供了先前宣布的结果的证明，该结果解决了A. A. Kirillov提出的以下问题。令T为Banach空间G中有界线性算子对\（\ mathcal {G} \）的表示，而\（E \ subset G \）为封闭不变子空间。然后Ť生成自然的方式演示\（T_1 \）在ë和\（T_2 \）在\（F：= G / E \） 。除\（T_1，T_2 \）外，还需要什么其他信息来恢复演示文稿T？在有限维（甚至是无限维希尔伯特）的情况下，该解决方案是众所周知的：需要提供一组同调类\（h \ in H ^ 1（\ mathcal {G}，Hom（F，E））\ ）。如果子空间E补充G，则在Banach情况下也是如此。但是，每个不与希尔伯特同构的Banach空间都具有不互补的子空间，这使问题变得更加严重，甚至在琐碎的集体动作的情况下也变得不平凡，归结为所谓的三空间问题。这解释了我们选择的标题。作者于1976年宣布了上述问题的解决方案，但由于非数学原因，尚未提供完整的证明。本文包含证明和Banach空间类别Banc中的仿函数\（Ext ^ 1 \）的一些相关注意事项。