Dining room

In functional analysis , a Krein space (after Mark Krein ) is a Hilbert space with a weakened structure: an i. A. indefinite inner product instead of the usual inner product. A precise definition can be found below. In many applications, the theory of free spaces is a very useful tool, for example with operator matrices or with certain differential operators .

Inner product

Let it be a complex vector space with an indefinite inner product . We use it to define the subsets ${\displaystyle K}$ ${\displaystyle [\cdot ,\cdot ]:K\times K\rightarrow \mathbb {C} }$

${\displaystyle P_{++}:=\{x\in K:[x,x]>0\},}$

${\displaystyle P_{0}:=\{x\in K:[x,x]=0\},}$

${\displaystyle P_{--}:=\{x\in K:[x,x]<0\},}$

${\displaystyle P_{+}:=P_{++}\cup P_{0},}$

${\displaystyle P_{-}:=P_{--}\cup P_{0}.}$

The vectors in these quantities are called positive, neutral, negative, non-negative or non-positive . A subspace with , , , and are called positive, neutral, negative, not negative and not positive . In all of these cases it is said to be semidefinite . A subspace that is not semidefinite is called indefinite . ${\displaystyle L\subset K}$ ${\displaystyle L\subset P_{++}\cup \{0\}}$ ${\displaystyle L\subset P_{0}}$ ${\displaystyle L\subset P_{--}\cup \{0\}}$ ${\displaystyle L\subset P_{+}}$ ${\displaystyle L\subset P_{-}}$ ${\displaystyle L}$

Definition of the dining area

Let there be a complex vector space and an inner product . Then a space is called , if there is a decomposition ${\displaystyle K}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle K}$ ${\displaystyle (K,[\cdot ,\cdot ])}$

${\displaystyle K=K_{+}[\dotplus ]K_{-}}$

exists such that and are Hilbert dreams. here denotes the orthogonal direct sum (that is, 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 shape is called a fundamental decomposition . ${\displaystyle (K_{+},[\cdot ,\cdot ])}$ ${\displaystyle (K_{-},-[\cdot ,\cdot ])}$ ${\displaystyle [\dotplus ]}$ ${\displaystyle K_{+}}$ ${\displaystyle K_{-}}$ ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle K}$

Fundamental symmetry

In the following is a dining room. With the help of the above fundamental decomposition, a scalar product can be defined ${\displaystyle (K,[\cdot ,\cdot ]\,)}$ ${\displaystyle K}$

${\displaystyle (x,y):=[x_{+},y_{+}]-[x_{-},y_{-}],}$ With ${\displaystyle x=x_{+}+x_{-},\;y=y_{+}+y_{-},\;x_{\pm },y_{\pm }\in K_{\pm }.}$

This is a Hilbert space (see e.g. in the book by T.Ya. Azizov and IS Iokhvidov). is the orthogonal sum of the Hilbert spaces and . Now we are introducing the following projectors : ${\displaystyle (K,(\cdot ,\cdot ))}$ ${\displaystyle (K,(\cdot ,\cdot ))}$ ${\displaystyle (K_{+},[\cdot ,\cdot ]\,)}$ ${\displaystyle (K_{-},-[\cdot ,\cdot ]\,)}$ ${\displaystyle P_{\pm }}$

${\displaystyle P_{\pm }x:=x_{\pm },{\mbox{ falls }}x\in K\,,\,x=x_{+}+x_{-}\,,\,x_{\pm }\in K_{\pm }.}$

The operator is called the fundamental symmetry of . Now applies and , where the adjoint operator is denoted with respect to the Hilbert space scalar product . Furthermore is ${\displaystyle J:=P_{+}-P_{-}}$ ${\displaystyle K}$ ${\displaystyle J^{2}=I}$ ${\displaystyle J=J^{*}=J^{-1}}$ ${\displaystyle *}$ ${\displaystyle (\cdot ,\cdot )}$

${\displaystyle [x,y]=(Jx,y)\;,\;(x,y)=[Jx,y]}$ For ${\displaystyle x,y\in K.}$

The Hilbert space scalar product depends on the chosen fundamental decomposition, which, with the exception of the case that the whole space is positive or negative, is not uniquely determined. But it can be shown (see e.g. Proposition 1.1 and 1.2 in the work of H. Langer in the literature list below) that for two fundamental decompositions ${\displaystyle (\cdot ,\cdot )}$

${\displaystyle K=K_{+}[\dotplus ]\,K_{-}}$ and ${\displaystyle K=K_{+}^{\prime }[\dotplus ]\,K_{-}^{\prime }}$

of the dimensions of the corresponding subspaces match, ${\displaystyle K}$

${\displaystyle {\rm {dim}}\;K_{\pm }={\rm {dim}}\;K_{\pm }^{\prime },}$

and generate the associated Hilbert space scalar products and equivalent norms . All terms in a space that refer to a topology , such as continuity , closure , spectrum of an operator in , etc., refer to this Hilbert space topology . ${\displaystyle (\cdot ,\cdot )}$ ${\displaystyle (\cdot ,\cdot )^{\prime }}$ ${\displaystyle K}$

Pontryagin room

If it is, the chalk room is called a Pontryagin room or also a room (named after Lev Pontryagin ). In this case, (or ) the number of positive (negative) squares of the inner product mentioned. ${\displaystyle \kappa :=\min\{\dim K_{+},\dim K_{-}\}<\infty }$ ${\displaystyle (K,[\cdot ,\cdot ])}$ ${\displaystyle \Pi _{\kappa }}$ ${\displaystyle \dim K_{+}}$ ${\displaystyle \dim K_{-}}$ ${\displaystyle [\cdot ,\cdot ]}$

Web link

• Harald Woracek: Operator theory in the Krein space (Chapter 3, page 49; PDF; 1.7 MB). TU Vienna, lecture notes.

literature

• T. Ya. Azizov, IS Iokhvidov: Linear operators in spaces with an indefinite metric , John Wiley & Sons, Chichester, 1989, ISBN 0-471-92129-7 .
• J. Bognár: Indefinite inner product spaces , Springer-Verlag, Berlin-Heidelberg-New York, 1974, ISBN 3-540-06202-5 .
• H. Langer: Spectral functions of definable operators in Krein spaces , Functional Analysis. Proceedings of a conference held at Dubrovnik, Yugoslavia, November 2-14, 1981, Lecture Notes in Mathematics, 948, Springer-Verlag Berlin-Heidelberg-New York, 1982, 1-46, ISSN  0075-8434 .