Spectral Theory on Krein Spaces

Introduction

A $J$-self-adjoint operator on a Krein space need not have real spectrum, and the failure is not a pathology but the general situation: the fundamental symmetry $J$ has spectrum $\{\pm1\}$, so it is itself $J$-self-adjoint with a negative spectral point, and the negativity propagates to every indefinite space of positive rank. The spectral theory of a $J$-self-adjoint operator is therefore not the spectral theorem of Hilbert space with a change of notation; it is a theory in which the real axis carries the ordinary spectral data, the upper half-plane carries at most a finite amount of extra data, and the two are related by conjugation.

What survives from the Hilbert theory is the symmetry: the spectrum of a $J$-self-adjoint operator is invariant under conjugation, because $\bar T = JTJ^{-1}$ makes $T$ similar to its complex conjugate. What changes is the semi-boundedness of the real spectrum, which is lost, and the definiteness of the eigenvectors, which is replaced by a trichotomy of positive, negative and neutral eigenvectors. What is bounded is the exceptional set: in a Pontryagin space $\Pi_{\kappa}$ the non-real spectrum consists of at most $\kappa$ conjugate pairs counting algebraic multiplicities, and in a general Krein space the non-real spectrum consists of isolated eigenvalues with finite-dimensional root subspaces that can accumulate only on the real axis. For a bounded operator the theory is self-contained; the unbounded theory, with its own difficulties, is Spectral Theory (Part III).

This article fixes the conjugate symmetry of the spectrum, the trichotomy of real eigenvalues, the Pontryagin bound on the non-real spectrum, the accumulation statement, and the pointer to definitizability.

The indefinite adjoint, $J$-self-adjointness and the $J$-positive cone are J-Self-Adjoint and J-Unitary Operators; the space, the fundamental symmetry and the ranks are Indefinite Inner Product Spaces, Krein Spaces and The Fundamental Symmetry; the ordered structure is The J-Positive Cone and the J-Order and Krein Algebras; the definitizable case is Definitizable Operators and the Krein–Naĭmark Theorem; the unbounded case is Part III. Those are cited. The operator is bounded and $J$-self-adjoint on the Krein space $K$ with $\kappa = \kappa(K)$.

The Spectrum of a J-Self-Adjoint Operator

Definition. The spectrum $\sigma(T)$ is the complement of the set of $\lambda$ with $\lambda - T$ boundedly invertible; the point spectrum $\sigma_{p}(T)$ is the set of eigenvalues, and the root subspace at $\lambda$ is $\bigcup_{n}\ker(\lambda - T)^{n}$.

Proposition (resolvent identity). If $T$ is $J$-self-adjoint then $\lambda - T$ is $J$-self-adjoint exactly for real $\lambda$, and $(\lambda - T)^{-1}$ is $J$-self-adjoint at every point of the resolvent that is real; at non-real $\lambda$ the resolvent is $J$-self-adjoint with parameter $\bar\lambda$.

Proof. $(\lambda - T)^{\dagger} = \bar\lambda - T^{\dagger} = \bar\lambda - T$, which equals $\lambda - T$ exactly for real $\lambda$; the resolvent of a $J$-self-adjoint operator at a real point is $J$-self-adjoint, and the general statement is the adjoint of $(\lambda - T)^{-1}$ computed at $\bar\lambda$.

Theorem (conjugate symmetry). The spectrum and the point spectrum of a $J$-self-adjoint operator are symmetric with respect to the real axis:

$$ \lambda\in\sigma(T) \iff \bar\lambda\in\sigma(T) , \qquad \lambda\in\sigma_{p}(T) \iff \bar\lambda\in\sigma_{p}(T) , $$

with equality of algebraic multiplicities, and the root subspace at $\bar\lambda$ is the $J$-orthogonal complement of the root subspace at $\lambda$ for the appropriate invariant pairing.

Proof. $T$ is similar to its complex conjugate, $\bar T = JTJ^{-1}$, so the spectrum is conjugation-invariant; multiplicities agree because the similarity is explicit; the pairing statement is the $J$-orthogonality of root subspaces at distinct conjugate eigenvalues.

Real Eigenvalues and Their Types

Definition. A real eigenvalue $\lambda$ of $T$ is of positive type when $[x,x] > 0$ for every eigenvector $x$ at $\lambda$, of negative type when $[x,x] < 0$, and of neutral type when the eigenvectors at $\lambda$ form an isotropic set; the type is a property of the eigenvalue and the operator.

Proposition (the type is a spectral invariant). The set of eigenvalues of positive type and of negative type are invariant under $J$-unitary equivalence, and the algebraic multiplicity of an eigenvalue of definite type is bounded by the rank of the space in the corresponding sign.

Proof. $J$-unitary operators preserve the form and hence the signs of vectors; the multiplicity bound is the dimensionality of a positive or negative definite subspace of $K$.

Remark (the real spectrum need not be semi-bounded). For a Hilbert-self-adjoint operator the spectrum lies in an interval $[m,M]$ of the line. For a $J$-self-adjoint operator the real spectrum may be bounded below but not above, or exhibit both signs, because the operator $J$ itself has real spectrum $\{-1,+1\}$ with the two points of opposite type. The real spectrum is a union of sets of positive, negative and neutral type, and only its restriction to the definite parts behaves as in the Hilbert theory.

The Pontryagin Case

Theorem (finiteness of the non-real spectrum). Let $K = \Pi_{\kappa}$ be a Pontryagin space and $T$ bounded and $J$-self-adjoint. Then the non-real spectrum of $T$ consists of finitely many conjugate pairs, and the total algebraic multiplicity of the non-real eigenvalues is at most $2\kappa$; every non-real eigenvalue has a root subspace that is a nondegenerate invariant subspace.

Proof. The invariant subspaces of a $J$-self-adjoint operator at non-real eigenvalues are of uniform type, and the geometry of $\Pi_{\kappa}$ bounds their total dimension by $2\kappa$; the conjugate pairing follows from the symmetry theorem.

Corollary (the non-real part is spectrally isolated). In $\Pi_{\kappa}$ the non-real spectrum is finite, its points are poles of the resolvent, and the restriction of $T$ to the direct sum of the root subspaces at the non-real eigenvalues is similar to a normal operator on a finite-dimensional Hilbert space.

Proof. The direct sum of the finitely many root subspaces is finite dimensional, invariant and nondegenerate, and on it the operator is $J$-self-adjoint with no real spectrum; a finite-dimensional $J$-self-adjoint operator with no real spectrum is similar to a normal operator.

Proposition (the Pontryagin bound is sharp). For every $\kappa$ there is a $J$-self-adjoint operator on $\Pi_{\kappa}$ with $\kappa$ conjugate pairs of non-real eigenvalues, namely the direct sum of $\kappa$ copies of the two-dimensional example $\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ on a neutral pair of vectors.

Proof. The two-dimensional example is $J$-self-adjoint and non-real on $\mathbb{C}^{1,1}$; taking the direct sum of $\kappa$ copies saturates the bound of the theorem.

Accumulation of the Non-Real Spectrum

Theorem (the general Krein case). For a bounded $J$-self-adjoint operator on an arbitrary Krein space, the non-real spectrum consists of eigenvalues whose root subspaces are finite dimensional, and every point of the non-real spectrum is isolated in the non-real part of the spectrum: the non-real spectrum can accumulate only on the real axis.

Proof (sketch). The root subspace at a non-real eigenvalue $\lambda$ is nondegenerate, finite dimensional and invariant, and the root subspaces at distinct non-real eigenvalues are $J$-orthogonal; the indefinite form therefore controls the total dimension of the non-real part, and a non-real eigenvalue has a neighbourhood meeting the non-real spectrum only finitely. The full statement is the theorem of Kreĭn and Langer on the spectral function of a $J$-self-adjoint operator.

Corollary (the real axis carries the continuous spectrum). The continuous spectrum, if any, lies on the real axis, and the non-real part of the spectrum is a purely point spectrum with finite-dimensional root subspaces.

Proof. The complement statement from the theorem, and the Hilbert-space fact that an isolated point of the spectrum of a bounded operator is an eigenvalue.

Remark (the shape of the spectral theory). The spectral theory of a $J$-self-adjoint operator is therefore layered: on the real axis the spectral data may be as complicated as in the Hilbert theory and in general is not semi-bounded; off the real axis the data is a finite (in $\Pi_{\kappa}$) or thin (in general) set of invariant subspaces with a conjugate pairing; and the two layers are coupled by the form. The tool that controls the coupling is definitizability, taken up in Definitizable Operators and the Krein–Naĭmark Theorem, where the spectral function of the operator is built.

Worked Cases

The Fundamental Symmetry

For $T = J$ the spectrum is $\{-1,+1\}$; the eigenvalue $+1$ is of positive type with multiplicity $\kappa_{+}$ and the eigenvalue $-1$ is of negative type with multiplicity $\kappa_{-}$; the non-real spectrum is empty and the structure is purely real.

The Two-Dimensional Example

For $T = \left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ on $\mathbb{C}^{1,1}$ the spectrum is $\{i,-i\}$, a conjugate pair, and the root subspaces are one dimensional; this saturates the Pontryagin bound for $\kappa = 1$ and shows that a $J$-self-adjoint operator with real rank one can have all of its spectrum non-real.

A Hilbert-Self-Adjoint Operator

If $T$ commutes with $J$ and is Hilbert-self-adjoint, then $T$ is $J$-self-adjoint with real spectrum, all eigenvalues of definite type, and the spectral theory reduces to the Hilbert one; the indefinite theory is nontrivial exactly when $T$ fails to commute with $J$.

Summary

The spectrum of a bounded $J$-self-adjoint operator on a Krein space is symmetric with respect to the real axis, $\lambda\in\sigma(T)\iff\bar\lambda\in\sigma(T)$ with equal multiplicities, because $T$ is similar to its complex conjugate, $\bar T = JTJ^{-1}$. Real eigenvalues carry a type — positive, negative or neutral according to the sign of $[x,x]$ on the eigenvectors — and the real spectrum need not be semi-bounded, since $J$ itself has the two real points of opposite type. The non-real spectrum is bounded by the indefiniteness: in a Pontryagin space $\Pi_{\kappa}$ the non-real spectrum consists of at most $\kappa$ conjugate pairs counting algebraic multiplicities, and the bound is sharp; in a general Krein space the non-real spectrum consists of isolated eigenvalues with finite-dimensional root subspaces, accumulating only on the real axis, so the continuous spectrum lies on the real line. The controlling tool for the coupling of the two layers is definitizability, which is Definitizable Operators and the Krein–Naĭmark Theorem; the indefinite adjoint and the $J$-self-adjoint operators are J-Self-Adjoint and J-Unitary Operators, the space and its ranks are Krein Spaces and Indefinite Inner Product Spaces, and the unbounded theory is Part III.

Summary of Notation

Symbol	Meaning
$\sigma(T)$, $\sigma_{p}(T)$	Spectrum and point spectrum
$\bar T = JTJ^{-1}$	Similarity to the complex conjugate
$\lambda\in\sigma(T)\iff\bar\lambda\in\sigma(T)$	Conjugate symmetry
Positive, negative, neutral type	Sign of $[x,x]$ on eigenvectors
$\Pi_{\kappa}$	Pontryagin space of rank $\kappa$
$\leq\kappa$ conjugate pairs	Bound on the non-real spectrum
Isolated root subspaces	Accumulation only on the real axis
$\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$	Sharp example on $\mathbb{C}^{1,1}$