Indefinite inner product space

In functional analysis a Krein space ( according to Mark Krein ) is a Hilbert space with a weakened structure: a iA indefinite inner product instead of the usual scalar product. A precise definition is provided below. In many applications, the theory of Krein spaces is a very useful tool, such as operator matrices or for certain differential operators.

Inner Product

It is a complex vector space with an indefinite inner product. We thus define the subsets

The vectors lying within these sets are called positive, neutral, negative, not negative or not positive. A subspace with, or is called positive, neutral, negative, non-negative or non- positive. In all these cases, it says, semidefinite. A sub-space that is not semi-definite, called indefinite.

Definition of the Krein space

Let a complex vector space and an inner product. Then is called a Krein space if a decomposition

Exists such that and Hilbert spaces. denotes the orthogonal direct sum (ie, the sum is direct, and and are perpendicular to each other with respect to the inner product ). A decomposition of the space of the above form is called the fundamental decomposition.

Fundamental symmetry

The following is a Krein space. Using the above fundamental decomposition can be defined in a scalar

This is a Hilbert space ( see, eg, in the book of T.Ya. Azizov and IS Iokhvidov ). is the orthogonal sum of Hilbert spaces and. Now we introduce the following projectors:

The operator is called the fundamental symmetry of. Now applies and, being designated by the adjoint operator with respect to the Hilbertraumskalarproduktes. Furthermore, it is

Hilbertraumskalarprodukt the fundamental depends on the chosen separation, which is not uniquely determined, except for the case that the whole area is positive or negative. But it can be shown ( see, eg, Proposition 1.1 and 1.2 in the work of H. Langer in the below list of references ) that for two fundamental decompositions

Match the dimensions of the corresponding subspaces

Generate and the associated Hilbertraumskalarprodukte and equivalent standards. All terms in a Krein space, which make reference to a topology such as continuity, seclusion, spectrum of an operator in etc. refer to this Hilbert space topology.

Pontrjaginraum

If it is, then the Krein space a Pontrjaginraum or space is called (named after Lev Pontryagin ). In this case, (or ) the number of positive (negative) square of the inner product is called.

Weblink

• Harald Woracek: Operator theory in Krein space (Chapter 3, page 49, PDF, 1.7 MB). Vienna University of Technology, lecture notes.