Krein space

Krein space
Name	Krein space
Type	Topological vector space
Field	Functional analysis
Notable	Mark Grigorievich Krein, M. G. Krein

Krein space A Krein space is a topological vector space equipped with a nondegenerate indefinite inner product and a compatible Hilbert space topology that admits a decomposition into two orthogonal subspaces with positive and negative definite inner products. It generalizes Hilbert space notions while connecting to operator theory, spectral theory, and mathematical physics. Krein spaces play roles in the works of Mark Grigorievich Krein, Israel Gohberg, Gustav Herglotz, Naum Akhiezer, and in frameworks used by researchers at institutions such as Steklov Institute of Mathematics, Moscow State University, and Institute for Advanced Study.

Definition and basic properties

A Krein space is defined as a complex vector space with an indefinite inner product ⟨·,·⟩ that is nondegenerate and for which there exists a direct sum decomposition into two closed subspaces where the form is positive definite on one summand and negative definite on the other. Foundational results were developed in the context of extension theory by Mark Grigorievich Krein and later elaborated by authors affiliated with Pennsylvania State University and Technische Universität Berlin. Basic properties include completeness relative to a Hilbertian norm induced by a choice of fundamental symmetry, existence of orthogonal complements with respect to the indefinite form, and relationships to Pontryagin spaces studied by Lev Pontryagin and M. G. Krein.

Indefinite inner product and geometry

The indefinite inner product endows the space with a pseudo-Euclidean geometry akin to structures appearing in the work of Albert Einstein (relativity) and in classical studies of bilinear forms by Carl Gustav Jacob Jacobi and Hermann Grassmann. Geometric notions such as isotropic vectors, neutral subspaces, and signature are central; signature theory traces back to the Sylvester's law of inertia and the classification of quadratic forms addressed by David Hilbert and Emil Artin. Fundamental symmetries—self-adjoint involutions—connect the indefinite form to a positive definite inner product, a technique used in spectral analyses by John von Neumann and Marshall H. Stone.

Fundamental decompositions and canonical forms

Fundamental decompositions split a Krein space into maximal definite subspaces and are unique up to equivalence determined by index theory explored by Israel Gohberg and Mark Krein. Canonical forms for the indefinite metric parallel normal forms in matrix theory developed by Fritzsche, Hermann Weyl, and contributors associated with University of Cambridge and ETH Zurich. Reduction theorems and block diagonalization employ techniques from the theory of self-adjoint extensions of symmetric operators studied by M. G. Krein and by researchers at Brown University and University of California, Berkeley.

Operators on Krein spaces

Operators preserving or related to the indefinite inner product—such as J-self-adjoint, J-unitary, and J-contractive operators—are central; these classes extend notions treated by John von Neumann and Harald Bohr. Spectral properties of J-self-adjoint operators link to non-Hermitian spectral theory investigated by Eugene Wigner in quantum contexts and by analysts at Massachusetts Institute of Technology and Institute Henri Poincaré. Perturbation theory, invariant subspace problems, and dilation theory for such operators draw on work by Béla Szőkefalvi-Nagy, C. R. Putnam, and research groups at University of Oxford.

Examples and constructions

Standard examples include spaces of sequences with indefinite weight matrices studied in connection with Juliusz Schauder bases and moment problems investigated by Naum Akhiezer and M. G. Krein. Function-space constructions arise from spaces of entire functions and reproducing kernels linked to the Hardy space theory advanced by Paul Koosis and Harm Bart. Pontryagin spaces with finite negative index provide explicit finite-rank perturbation models used in interpolation problems examined by Nikolai Nikolski and teams at University of Paris. Direct sum and quotient constructions connect Krein spaces to extension theory in operator algebras studied at California Institute of Technology and Princeton University.

Applications in mathematics and physics

In mathematics, Krein space methods are applied to spectral theory of differential operators, inverse problems, and interpolation theory pursued by researchers at Rutgers University, University of Warsaw, and University of Göttingen. In physics, indefinite metric spaces appear in quantum field theory formulations considered by Paul Dirac and in scattering theory contexts related to the S-matrix studied by Werner Heisenberg and Enrico Fermi. Stability analysis of Hamiltonian systems, where Krein signatures determine Krein collisions, connects to dynamical studies by Lev Landau and modern investigations at Courant Institute of Mathematical Sciences.

Category:Functional analysis