A bounded linear operator |$U$| between Banach spaces is universal for the complement of some operator ideal |$\mathfrak{J}$| if it is a member of the complement and it factors through every element of the complement of |$\mathfrak{J}$|⁠. In the first part of this paper, we produce new universal operators for the complements of several ideals, and give examples of ideals whose complements do not admit such operators. In the second part of the paper, we use descriptive set theory to study operator ideals. After restricting attention to operators between separable Banach spaces, we call an operator ideal |$\mathfrak{J}$| generic if whenever an operator |$A$| has the property that every operator in |$\mathfrak{J}$| factors through a restriction of |$A$|⁠, then every operator between separable Banach spaces factors through a restriction of |$A$|⁠. We prove that many classical operator ideals (such as strictly singular, weakly compact, Banach–Saks) are generic and give a sufficient condition, based on the complexity of the ideal, for when the complement does not admit a universal operator. Another result is a new proof of a theorem of M. Girardi and W. B. Johnson, which states that there is no universal operator for the complement of the ideal of completely continuous operators.

中文翻译：

---

操作员理想的通用性和通用性

有界线性算子| $ U $ | Banach空间之间的距离是通用的，可以补充某些运算符的理想| $ \ mathfrak {J} $ | 如果它是补数的成员，并且会影响| $ \ mathfrak {J} $ |⁠的补数的每个元素。在本文的第一部分中，我们为几种理想的补集产生了新的通用算子，并给出了其补子不允许此类算子的理想例子。在本文的第二部分，我们使用描述性集合理论来研究算子理想。在限制对可分离Banach空间之间的算子的关注之后，我们称理想的算子| $ \ mathfrak {J} $ | 如果运算符| $ A $ | 具有每个运算符所在的属性| $ \ mathfrak {J} $ | 通过| $ A $ |⁠的限制来分解，那么每个可分离的Banach空间之间的运算符都会通过| $ A $ |⁠的限制来分解。我们证明了许多经典算子理想（例如严格奇异，弱紧致的Banach-Saks）是通用的，并且根据理想的复杂性，为补码不允许通用算子提供了充分条件。另一个结果是M. Girardi和WB Johnson的一个定理的新证明，该定理指出，完全连续算子理想的补充没有通用算子。