Given two different complex numbers a and b, a bounded linear operator T acting on an infinite-dimensional complex Hilbert space H is said to be {a, b}-quadratic if (T − a)(T − b) = 0. We provide in this paper a complete description of all surjective maps Φ (not necessarily additive) on the algebra \({\cal B}(H)\) of all bounded linear operators on H that satisfy S − λT is {a, b}-quadratic if and only if Φ(S) − λΦ(T) is {a, b}-quadratic for every \(S,T \in {\cal B}(H)\) and λ ∈ ℂ.

中文翻译：

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涉及二次运算符的非线性保护器

给定两个不同的复数a和b，如果 ( T - a ) ( T - b ) = 0 ，则作用于无限维复数希尔伯特空间H的有界线性算子T被称为 { a, b } -二次型。在本文中提供所有满足S - λT 的H上所有有界线性算子的代数\ ({\ cal B} (H) \)上的所有满射映射 Φ（不一定是可加的）的完整描述是 { a, b } -二次当且仅当 Φ ( S ) - λΦ (T )对于每个\ (S, T \ in {\ cal B} (H) \)和λ ∈ ℂ是 { a, b } -二次型。