Krein

Krein refers to a family of mathematical objects and concepts, primarily originating from functional analysis. The most common interpretation relates to the Krein space, a type of vector space equipped with a non-degenerate indefinite inner product. Unlike Hilbert spaces where the inner product is positive definite, in a Krein space, certain vectors can have negative squared norms. This indefinite nature leads to interesting and sometimes counterintuitive properties.

Krein spaces are crucial in the study of linear operators. The spectral theory of self-adjoint operators on Krein spaces exhibits distinct features compared to Hilbert spaces. For instance, eigenvalues can be positive, negative, or infinite. This makes Krein spaces a valuable tool for analyzing operators that arise in various areas of physics, such as quantum mechanics and classical mechanics, where indefinite metrics appear naturally.

The term "Krein" also appears in the context of moment problems, specifically the Stieltjes moment problem and the Hausdorff moment problem. The Krein-Stieltjes string problem, for example, deals with reconstructing a string's density from its spectral properties. This connection highlights the broad applicability of Krein's work across different mathematical disciplines. In essence, Krein's contributions have provided powerful frameworks for understanding structures with indefinite metrics and their associated operators.