An operator T on a separable, infinite dimensional, complex Hilbert space \({\mathcal {H}}\) is called complex symmetric if T has a symmetric matrix representation relative to some orthonormal basis for \({\mathcal {H}}\). This paper aims to describe reducing subspaces of complex symmetric operators from the view point of approximation. In particular, given a complex symmetric operator T, \(1\le n\le \aleph _0\) and \(\varepsilon >0\), it is proved that there exists a compact operator K with \(\Vert K\Vert <\varepsilon \) such that \(T+K\) is complex symmetric and has exactly n minimal reducing subspaces.

中文翻译：

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简化复对称算子的子空间

如果T具有相对于\（{\ mathcal {H}}的正交正交性的对称矩阵表示形式，则可分离的，无限维的复杂希尔伯特空间\（{\ mathcal {H}} \）上的算子T被称为复对称。\）。本文旨在从逼近的角度描述复杂对称算子的约化子空间。特别地，给定一个复杂的对称算Ť，\（1 \了N \文件\阿列夫_0 \）和\（\ varepsilon> 0 \） ，证明存在一个紧算ķ与\（\ Vert的ķ\垂直<\ varepsilon \）使得\（T + K \）是复对称的，并且具有n个最小化还原子空间。