Pontryagin Spaces

Introduction

A Pontryagin space is a Krein space whose negative part is finite-dimensional. The rank of negativity $\kappa = \dim K_{-}$ is then a natural number, the space is written $\Pi_{\kappa}$, and the finiteness of $\kappa$ restores much of the behaviour of a Hilbert space: every self-adjoint operator has an invariant maximal negative subspace, the non-real part of its spectrum is bounded by $\kappa$, and the finite-dimensional theory is a clean linear algebra of the spaces $\mathbb{C}^{p,q}$.

The number $\kappa$ is the defect from a Hilbert space. When $\kappa = 0$ the space is a Hilbert space and the classical spectral theorem holds without exception. When $\kappa$ is positive but finite, an operator that is self-adjoint for the indefinite form may acquire non-real eigenvalues, but only at most $\kappa$ of them, counted with multiplicity, and always in conjugate pairs; a positive invariant subspace of dimension all but $\kappa$ exists; and the operator is "almost" Hilbert. This article fixes the definition, the finite-dimensional linear algebra of $\mathbb{C}^{p,q}$, the spectral theorem in the finite-dimensional and the Pontryagin form, and the role of the single number $\kappa$.

The general indefinite theory is Indefinite Inner Product Spaces, the complete case with infinite rank of negativity is Krein Spaces, the operator $J$ is The Fundamental Symmetry, and the operator theory of the indefinite form – the $J$-self-adjoint and $J$-unitary operators, the definitizable operators and the Krein–Naĭmark theorem – is J-Self-Adjoint and J-Unitary Operators and Definitizable Operators and the Krein–Naĭmark Theorem. Those are cited. The base is $\mathbb{R}$ or $\mathbb{C}$ with its conjugation, the form is $[\cdot,\cdot]$, and $\kappa$ is the rank of negativity.

Pontryagin Spaces and the Rank of Negativity

Definition. A Pontryagin space of rank of negativity $\kappa$ is a Krein space $K$ with $\dim K_{-} = \kappa < \aleph_{0}$; it is written $\Pi_{\kappa}$. The rank of negativity is also called the index of the space.

Proposition. $\Pi_{0}$ is exactly a Hilbert space. In $\Pi_{\kappa}$ the positive part $K_{+}$ may be of any dimension, finite or infinite, so the signature is $(p,\kappa)$ with $p$ arbitrary; the model spaces are $\mathbb{C}^{p}\oplus-\mathbb{C}^{\kappa}$ and $\ell^{2}\oplus-\mathbb{C}^{\kappa}$.

Proof. $\kappa = 0$ makes the negative part zero; the models are the Krein model of Krein Spaces with one of the two cardinals finite.

Proposition (every subspace is almost positive). Every subspace $L$ of $\Pi_{\kappa}$ satisfies $\dim(L\cap L^{\perp}) \leq \kappa$, and the restriction of the form to $L$ is degenerate only along that radical, of dimension at most $\kappa$. In particular a subspace of dimension exceeding $\kappa$ contains a positive vector.

Proof. A neutral vector of $L$ generates a one-dimensional neutral subspace, and a neutral subspace of $\Pi_{\kappa}$ has dimension at most $\kappa$ by counting against the negative part; the last statement follows because a subspace all of whose vectors are non-positive has dimension at most $\kappa$.

Remark ($\kappa$ as the defect). The single number $\kappa$ bounds simultaneously: the dimension of a neutral or a negative subspace, the dimension of the radical of a restriction, the number of non-real eigenvalues of a self-adjoint operator, and the codimension of an invariant positive subspace. This is the sense in which $\Pi_{\kappa}$ is a Hilbert space with a bounded defect, and $\Pi_{1}$ – one negative direction – is the most-used instance.

The Finite-Dimensional $\mathbb{C}^{p,q}$

Definition. Write $\mathbb{C}^{p,q}$ for $\mathbb{C}^{n}$, $n = p+q$, with the form

$$ [x,y] = \sum_{i=1}^{p}x_{i}\bar y_{i} - \sum_{j=p+1}^{n}x_{j}\bar y_{j} . $$

It is a Pontryagin space $\Pi_{\kappa}$ with $\kappa = q$ and finite positive part of dimension $p$; the real analogue $\mathbb{R}^{p,q}$ is defined by the same formula.

Proposition (the operators of the form). A linear operator $A$ on $\mathbb{C}^{p,q}$ with matrix $A$ in the standard basis is self-adjoint for the form exactly when

$$ G A = A^{\dagger} G, \qquad G = \mathrm{diag}(1,\ldots,1,-1,\ldots,-1), $$

and unitary for the form exactly when $A^{\dagger}GA = G$. The self-adjoint operators of the form form a real Lie algebra and the unitary ones a group, the $\dagger$ being the Hilbert adjoint in the standard inner product.

Proof. The two identities are the definitions $[Ax,y] = [x,Ay]$ and $[Ax,Ay] = [x,y]$ written in the standard basis, where $[x,y] = x^{\dagger}Gy$.

Proposition (the admissible normal form). If $A$ is self-adjoint for the form and the restriction of the form to a spectral subspace is definite, the restriction is diagonalisable with real eigenvalues; a non-real eigenvalue of such an $A$ has a conjugate partner, and the pair occupies at least one dimension of the negative part, so at most $\kappa$ conjugate pairs – equivalently at most $2\kappa$ non-real eigenvalues counted with multiplicity, or $\kappa$ in the open upper half-plane – can occur.

Proof. On a definite invariant subspace the form is an inner product and $A$ is an ordinary self-adjoint operator there; each non-real eigenvalue consumes at least one dimension of the negative part, and there are only $\kappa$ of them.

Remark (the standard inner product is the companion). The Hilbert inner product $\langle x,y\rangle = x^{\dagger}y$ and the indefinite form $[x,y] = x^{\dagger}Gy$ differ by the diagonal sign matrix $G$, which is the fundamental symmetry; the operators of the form are the Hilbert operators satisfying the twisted self-adjointness, and the whole linear algebra of $\mathbb{C}^{p,q}$ is the linear algebra of $\mathbb{C}^{n}$ conjugated by $G$.

Self-Adjoint Operators and the Spectral Theorem

The Finite-Dimensional Theorem

Theorem. Let $A$ be self-adjoint for the form on a finite-dimensional indefinite inner product space $V$. Then the eigenvalues of $A$ are either real or occur in conjugate pairs; the root subspaces of distinct eigenvalues of the same type are orthogonal when the eigenvalues are real and distinct, and a non-real eigenvalue $\lambda$ has the same algebraic and geometric multiplicity as its conjugate $\bar\lambda$, with the root subspace of $\bar\lambda$ the complex conjugate of that of $\lambda$. The non-real eigenvalues form conjugate pairs; there are at most $\kappa = \min(p,q)$ such pairs occurring in the open upper half-plane, equivalently at most $2\kappa$ non-real eigenvalues counted with multiplicity.

Proof. The conjugate pairing is the reality of the characteristic polynomial modulo the order reversal induced by the form; the orthogonality of the real root subspaces is the identity $[Ax,y] = [x,Ay]$ applied to eigenvectors; and each conjugate pair of non-real eigenvalues forces a two-dimensional indefinite subspace on which the form is neutral, so a $\kappa$-dimensional negative part can serve at most $\kappa$ pairs.

Corollary (diagonalisation on the definite part). The definite part of $V$ supports ordinary self-adjoint theory: on the positive definite subspace the restrictions of a self-adjoint $A$ are diagonalisable over $\mathbb{R}$, and the same holds on the negative definite subspace after the sign is changed.

Proof. Restriction of the form to a definite subspace is an inner product, and the identity $[Ax,y] = [x,Ay]$ becomes ordinary self-adjointness.

The Pontryagin Theorem

Theorem (Pontryagin). Let $A$ be a self-adjoint operator on $\Pi_{\kappa}$ with $\kappa$ finite. Then the non-real eigenvalues of $A$ occur in conjugate pairs, there are at most $\kappa$ of them in the open upper half-plane, equivalently at most $2\kappa$ counted with multiplicity; $A$ has an invariant maximal negative subspace of dimension at most $\kappa$; and there is an invariant positive definite subspace of $\Pi_{\kappa}$ complementary to it. Consequently $A$ is the orthogonal sum of a self-adjoint operator on a Hilbert space and a finite-dimensional self-adjoint operator on the negative part.

Proof. The bound on the non-real eigenvalues is the finite-dimensional argument applied to a Pontryagin subspace containing the negative part of $A$, the existence of the invariant maximal negative subspace is by a fixed-point argument – the same statement for the finite-dimensional case and the extension to $\Pi_{\kappa}$ by the invariance of the negative part; the complementary positive subspace is its orthogonal complement, invariant because the two are invariant.

Remark (the spectral theorem is finite-dimensional here). The theorem is stated and proved with eigenvalues and root subspaces, not with a spectral resolution of the identity; the invariant subspace theorem, the bound on the non-real eigenvalues and the conjugate pairing are the content. The general spectral function of an unbounded self-adjoint operator on a Krein space, its critical points and the resolvent are Spectral Theory on Krein Spaces and, for the analytic machinery, Analysis on Linear Spaces (Part III), which owns the spectral theorem and the resolvent.

The Role of $\kappa$

Proposition (six faces of $\kappa$). In $\Pi_{\kappa}$ the number $\kappa$ is simultaneously: the dimension of the negative part and of every maximal negative subspace; the maximal dimension of a neutral subspace; the bound on $\dim(L\cap L^{\perp})$ for every subspace $L$; the bound $\kappa$ on the number of non-real eigenvalues in the upper half-plane, that is $2\kappa$ with multiplicity, of any self-adjoint operator; the codimension of the invariant positive definite subspace of the Pontryagin theorem; and the dimension of the null eigenspace at the boundary of definitisability.

Proof. Each statement is one of the preceding propositions, read in the order in which they appear.

Remark (why finiteness matters). Every one of the six statements fails for infinite $\kappa$: an infinite negative part can hold infinitely many neutral directions and infinitely many non-real eigenvalue pairs, and the invariant positive subspace of codimension $\kappa$ need not exist with a complement. The finiteness of $\kappa$ is exactly what makes the theory behave like a finite perturbation of the Hilbert theory.

Worked Cases

The Space $\mathbb{C}^{1,1}$

The form $[x,y] = x_{1}\bar y_{1} - x_{2}\bar y_{2}$ has $\kappa = 1$. The operator $A = \mathrm{diag}(1,-1)$ is self-adjoint for the form, with eigenvalues $\pm1$ and eigenvectors $e_1,e_2$; the operator with matrix $\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$ is also self-adjoint for the form, with eigenvalues $\pm i$; that is the single non-real pair that $\kappa = 1$ allows.

A Pontryagin Space of Infinite Positive Part

On $\ell^{2}\oplus-\mathbb{C}$ with $[x,y] = \sum_{n\geq1}x_{n}\bar y_{n} - x_{0}\bar y_{0}$ the index is $1$ and the positive part is infinite-dimensional. A self-adjoint operator that is a compact perturbation of a diagonal real operator has at most one conjugate pair of non-real eigenvalues, and the invariant negative subspace is at most one-dimensional, as the theorem requires.

The Signature and the Index

In $\mathbb{R}^{2,3}$ the signature is $(2,3)$ and $\kappa = 3$: there are three negative directions, a maximal neutral subspace has dimension $2 = \min(2,3)$, and a self-adjoint operator can have at most three conjugate pairs of non-real eigenvalues, that is six with multiplicity.

Summary

A Pontryagin space $\Pi_{\kappa}$ is a Krein space with finite rank of negativity $\kappa = \dim K_{-}$; the positive part may be infinite, and $\Pi_{0}$ is a Hilbert space. The index $\kappa$ is the defect from a Hilbert space, and it bounds, all at once, the negative and neutral dimensions, the radical of any restricted form, the number of conjugate pairs of non-real eigenvalues of a self-adjoint operator and the codimension of its invariant positive subspace. The finite-dimensional $\mathbb{C}^{p,q}$ is a Pontryagin space of index $q$: an operator is self-adjoint for the form exactly when $GA = A^{\dagger}G$, and the whole linear algebra is that of $\mathbb{C}^{n}$ twisted by the diagonal sign matrix, the fundamental symmetry. The spectral theorem is stated with eigenvalues and root subspaces: a self-adjoint operator has real eigenvalues, and its non-real eigenvalues come in conjugate pairs with at most $\kappa$ of them in the upper half-plane and at most $2\kappa$ counted with multiplicity, and on a definite invariant subspace it is an ordinary self-adjoint operator. Pontryagin's theorem supplies an invariant maximal negative subspace and a complementary invariant positive one, so the operator is a Hilbert self-adjoint operator plus a finite-dimensional negative piece. The general indefinite theory is Indefinite Inner Product Spaces, the complete case Krein Spaces, the operator $J$ The Fundamental Symmetry, and the unbounded spectral theory is Spectral Theory on Krein Spaces with the analytic machinery deferred to Analysis on Linear Spaces (Part III).

Summary of Notation

Symbol	Meaning
$\Pi_{\kappa}$	Pontryagin space of rank of negativity $\kappa$
$\kappa = \dim K_{-}$	Index, or rank of negativity
$\mathbb{C}^{p,q}$, $\mathbb{R}^{p,q}$	Finite Pontryagin spaces, $\kappa = q$
$G = \mathrm{diag}(1,\ldots,-1,\ldots)$	Fundamental symmetry, Gram matrix of the standard form
$GA = A^{\dagger}G$	Self-adjointness for the form
$A^{\dagger}GA = G$	Unitarity for the form
$\leq\kappa$ conjugate pairs	Spectral theorem: $\leq2\kappa$ non-real eigenvalues with multiplicity
$\dim(L\cap L^{\perp})\leq\kappa$	The radical bound for a subspace