---
title: Krein Space Numerical Range Overview
url: https://www.emergentmind.com/topics/krein-space-numerical-range
type: topic
---

# Krein Space Numerical Range Overview

The Krein space numerical range is the analogue of the classical numerical range for operators acting on a vector space endowed with a nondegenerate Hermitian form of indefinite signature. In matrix form, it is built from vectors normalized by the indefinite inner product and therefore splits into positive and negative sheets; as a consequence, its geometry is generally pseudo-convex rather than classically convex, and in finite dimensions its boundary generating curve can be hyperbolic rather than elliptical. Recent results give a complete \(2\times2\) hyperbolical characterization, identify structured higher-order matrix classes in which hyperbolas persist or bifurcate, and connect the numerical range of non-negative operators with a co-numerical range that restores spectral localization [2407.03164], [2508.12039], [1208.3094].

## 1. Indefinite inner products and definitions

Let \(J\in M_n(\C)\) be Hermitian and invertible of signature \((r,n-r)\). On \(\C^n\) one defines the indefinite inner product
\[
[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.
\]
For \(A\in M_n(\C)\),
\[
W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\},
\qquad
W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},
\]
and the Krein-space numerical range is
\[
W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).
\]
When \(J=I\), this reduces to the classical numerical range \(W(A)\) [2407.03164].

A parallel operator-theoretic formulation is used for non-negative selfadjoint operators \(A\) in a Kreĭn space \((K,[\cdot,\cdot])\):
\[
W(A)\;:=\;\Bigl\{\frac{[Ax,x]}{[x,x]}:\;x\in\dom A,\;[x,x]\neq0\Bigr\}\subset\R.
\]
Equivalently, one may normalize to \([x,x]=\pm1\). This formulation is real-valued because the operator is assumed selfadjoint and non-negative [1208.3094].

For the block setting \(K=\C^{2n}\) with
\[
J = I_n \oplus (-I_n),
\]
the numerical range can be written as
\[
W_K(T)\equiv W_J(T)=W_+^J(T)\cup(-W_-^J(T)).
\]
In this setting, each of \(W_+^J(T)\) and \(W_-^J(T)\) is convex, but possibly unbounded, and \(W_J(T)\) is pseudo-convex in the sense that any two points either generate a line segment in \(W_J(T)\) or two opposing rays [2508.12039].

## 2. The \(2\times2\) hyperbolical range theorem

For the non-trivial signature
\[
J=\diag(1,-1)
\]
and \(A\in M_2(\C)\), let \(\lambda_1,\lambda_2\) be the eigenvalues of \(A\), and set
\[
\Delta \;=\;\tfrac12(\Tr(JA^*JA)) \;=\;\frac12\bigl(A^\#A\bigr),
\qquad
\Sigma \;=\;|\lambda_1|^2+|\lambda_2|^2.
\]
Then \(W^J(A)\) is bounded by a non-degenerate hyperbola with foci at \(\lambda_1,\lambda_2\) if and only if
\[
2\Re(\bar\lambda_1\lambda_2)\;<\;\Delta\;<\;\Sigma.
\]
When this holds, the transverse and non-transverse semi-axes are
\[
a_T \;=\;\sqrt{\Delta-2\Re(\bar\lambda_1\lambda_2)},
\qquad
a_N \;=\;\sqrt{\Sigma-\Delta},
\]
and after a suitable rotation and translation the boundary hyperbola has equation
\[
\frac{x^2}{a_T^2}\;-\;\frac{y^2}{a_N^2}\;=\;1.
\]
This is the finite-dimensional hyperbolical range theorem in the \(2\times2\) case [2407.03164].

The support-line mechanism is expressed through the \(J\)-Hermitian pencil
\[
H_\theta \;=\;\Re^J(A)\cos\theta + \Im^J(A)\sin\theta.
\]
For each real \(\theta\), \(H_\theta\) has two real eigenvalues \(\lambda_L(\theta)\le \lambda_R(\theta)\), and these are the support lines of \(-W^J_-\) and \(W^J_+\). One computes
\[
\det\bigl(H_\theta-zI\bigr)
=
z^2-(p+q\cos2\theta+t\sin2\theta),
\]
with real constants \(p,q,t\), so that the support-eigenvalues satisfy
\[
z^2 \;=\;p + q\cos(2\theta)+t\sin(2\theta),\qquad p^2<q^2+t^2.
\]
The condition \(p^2<q^2+t^2\) is algebraically equivalent to the double inequality above and guarantees that the two sheets
\[
z=\pm\sqrt{p+q\cos2\theta+t\sin2\theta}
\]
never meet. As \(\theta\) varies, these branches trace a nondegenerate hyperbola, and exactly one branch lies on the \(+1\)-sheet while the other lies on the \(-1\)-sheet, so \(W^J(A)\) is the pseudo-convex hull of the two branches [2407.03164].

## 3. Persistence and bifurcation in structured tridiagonal matrices

The hyperbolic boundary is not confined to the \(2\times2\) case. Bebiano–Lemos–Soares identify several structured families of larger matrices for which the boundary generating curve remains a single hyperbola or becomes the union of a small number of hyperbolas [2407.03164].

For the \(3\times3\) biperiodic tridiagonal case,
\[
A \;=\;
\begin{pmatrix}
a & b_1 & 0\\
c_1 & -a & b_2\\
0 & c_2 & a
\end{pmatrix},
\qquad
J=\diag(1,-1,1),
\]
let
\[
\Delta \;=\;a^2 + b_1c_1 + b_2c_2,
\qquad
\mathcal H\colon\;\frac{x^2}{|\Delta|} - \frac{y^2}{\tfrac12\bigl(|b_1|^2+|b_2|^2+|c_1|^2+|c_2|^2 -2a^2\bigr)}=1.
\]
Then \(W^J(A)\) is exactly the hyperbolic disc bounded by \(\mathcal H\) if and only if
\[
a^2 \;-\;2|\Delta|\;<\;(A^\#A)\;<\;a^2\,+\,2|\Delta|.
\]
In that event, the third eigenvalue \(a\) lies in or on the boundary of the hyperbola.

For a centro-symmetric tridiagonal \(5\times5\) matrix with
\[
J=\diag(1,-1,1,-1,1),
\]
a suitable \(J\)-orthogonal block-diagonalization splits the problem into a \(2\times2\) block and a \(3\times3\) block of the previous type. The numerical range is again a single hyperbola provided the corresponding \(3\times3\) block satisfies the same strict double inequality. Otherwise one obtains two nested hyperbolas, and only the outer one supports \(W^J(A)\).

For a centro-symmetric tridiagonal \(4\times4\) matrix with quasi-biperiodic main diagonal and
\[
J=\diag(1,-1,-1,1),
\]
the matrix is \(J\)-orthogonally similar to two \(2\times2\) Hermitian pencils. Either both yield the same single hyperbola, or they yield two distinct hyperbolas \(\mathcal H_+\) and \(\mathcal H_-\) with
\[
\partial W^J(A)\;=\;\mathcal H_+\;\cup\;\mathcal H_-.
\]
In the latter situation, \(W^J(A)\) is typically the pseudo-convex hull of those two branches. Flat sides or even a full-plane degeneration may occur if the support-eigenvalue-interlacing condition fails for some angle.

A notable feature of these small-order results is that the hyperbolicity criteria are entry-wise. In the \(3\times3\) case, for example,
\[
a^2 -2\bigl|a^2 + b_1c_1 + b_2c_2\bigr|
\;<\;
(A^\#A)
\;<\;
a^2 +2\bigl|a^2 + b_1c_1 + b_2c_2\bigr|.
\]
In the \(4\times4\) case there are two such inequalities, one for each \(2\times2\) block, and in the \(5\times5\) case again a single double inequality for the central \(3\times3\) block.

## 4. Unified block-matrix formulation

A broader unified picture is available for \(2n\times2n\) block matrices of the form
\[
T=
\begin{bmatrix}
a\,I_n & B\\
B^* & d\,I_n
\end{bmatrix},
\qquad a,d\in\R,\quad B\in M_n(\C).
\]
Set
\[
\omega=\frac{a-d}{2},
\qquad
c=\frac{a+d}{2}.
\]
For each \(\theta\in\R\), define the \(J\)-Hermitian family
\[
H_\theta(T):=\Re^J(e^{-i\theta}T)=\cos\theta\,\Re^J(T)+\sin\theta\,\Im^J(T).
\]
A direct computation gives
\[
H_\theta(T)=
\begin{bmatrix}
c\cos\theta\,I_n & -i\sin\theta\,B\\
i\sin\theta\,B^* & c\cos\theta\,I_n
\end{bmatrix},
\]
and after reduction to \(\omega=(a-d)/2\), the nontrivial eigenvalues occur in \(n\) pairs
\[
\lambda_{i,\pm}(\theta)
=
c\cos\theta
\pm
\sqrt{(\omega\cos\theta)^2-\sigma_i(B)^2\sin^2\theta},
\]
where \(\sigma_1(B)\ge\cdots\ge\sigma_n(B)\ge0\) are the singular values of \(B\) [2508.12039].

The indefinite Kippenhahn curve \(C^J(T)\) is the envelope of the support lines
\[
x\cos\theta+y\sin\theta=\lambda_{i,\pm}(\theta),
\]
with \(\theta\) varying over an open interval where the radicand is positive. For each \(i\) such that \(\omega^2>\sigma_i^2\), the envelope is a nondegenerate hyperbola centered at \((c,0)\), with transverse semi-axis \(|\omega|\) and nontransverse semi-axis \(\sigma_i\). In Cartesian form,
\[
\frac{(x-c)^2}{\omega^2}-\frac{y^2}{\sigma_i^2}=1,
\]
and the foci are
\[
c \pm \sqrt{\omega^2+\sigma_i^2}
\;=\;
\frac{a+d}{2}\pm\frac12\sqrt{(a-d)^2+4\sigma_i^2}.
\]

The full numerical range \(W^J(T)\) is the pseudo-convex hull of the collection of hyperbolas corresponding to indices \(i\) with \(\omega^2>\sigma_i^2\), together with degenerate points when \(\omega^2=\sigma_i^2\), and possibly the singleton \(c\) when \(B\) has smaller rank. The outer hyperbola, corresponding to \(i=1\), bounds \(W^J(T)\) as a hyperbolic disc exactly when
\[
\omega^2>\sigma_1(B)^2
\quad\Longleftrightarrow\quad
\frac{|a-d|}{2}>\sigma_1(B)
\quad\Longleftrightarrow\quad
2\sigma_1(B)<|a-d|.
\]
This criterion unifies and extends earlier hyperbolical range theorems for \(2\times2\) and block-Toeplitz cases, and it produces new families with multiple nested hyperbolas when \(\operatorname{rank} B>1\) [2508.12039].

## 5. Spectral localization and the co-numerical range

For non-negative selfadjoint operators in a Kreĭn space, the numerical range behaves differently from the bounded Hilbert-space numerical range. Let
\[
u_+:=\sup\bigl(\sigma(A)\cap\R_+\bigr)\in[0,+\infty],
\qquad
u_-:=\inf\bigl(\sigma(A)\cap\R_-\bigr)\in[-\infty,0].
\]
Then \(W(A)\) is always unbounded above and below unless \(A=0\). If \(\ker A\) is indefinite, then
\[
W(A)=\R.
\]
Otherwise,
\[
W(A)\cup\{u_-,u_+\}
=
(-\infty,u_-]\cup[u_+,\infty),
\]
with the conventions \(u_-=-\infty\) if \(\sigma(A)\cap\R_-=\emptyset\) and \(u_+=+\infty\) if \(\sigma(A)\cap\R_+=\emptyset\). In particular, \(W(A)\) is pseudo-convex in the sense that whenever \(a,b\in W(A)\), either the closed segment \([a,b]\subset W(A)\) or the complement in the line of its open interior is contained in \(W(A)\) [1208.3094].

To recover spectral information more sharply, Philipp–Trunk define the Kreĭn-space co-numerical range
\[
W_{\rm co}(A)
:=
\Bigl\{
\frac{[Ax,Ax]}{[Ax,x]}
:\;x\in\dom A,\;Ax\neq0
\Bigr\}\subset\R.
\]
If \(A\) is boundedly invertible, then \((K,[A\cdot,\cdot])\) is a Hilbert space and \(W_{\rm co}(A)\) is the usual Hilbert-space numerical range of the selfadjoint operator \(A\) there, hence a closed real interval. More generally,
\[
\overline{W_{\rm co}(A)}
=
\bigl[\inf(\sigma(A)\setminus\{0\}),\;\sup(\sigma(A)\setminus\{0\})\bigr].
\]

The main spectral inclusion states that if \(A\neq0\) is non-negative in an indefinite Kreĭn space, then, except in the single borderline case where \(0\) is an isolated eigenvalue with \(\ker A=\ker A^2\) and one of \(\sigma(A)\cap\R_\pm\) is empty,
\[
\sigma(A)\subset\overline{W(A)\cap W_{\rm co}(A)}.
\]
In the exceptional case,
\[
\sigma(A)\setminus\{0\}\subset\overline{W(A)\cap W_{\rm co}(A)}.
\]
This shows that \(W(A)\) by itself is generally too large to localize the spectrum pointwise, while the intersection \(W(A)\cap W_{\rm co}(A)\) does capture all nonzero spectral values [1208.3094].

## 6. Representative geometries and degeneracies

Several explicit examples illustrate the range of possible geometries. In the \(3\times3\) theorem of Bebiano–Lemos–Soares,
\[
A=
\begin{pmatrix}
4 & 3+i & 0\\
-4+i & -4 & -i\\
0 & -i & 4
\end{pmatrix},
\qquad
J=\diag(1,-1,1).
\]
Here \(a=4\), \(\Delta=3-2i\), and \((A^\#A)\approx 50\) satisfy the strict inequalities. The boundary generating curve is a hyperbola whose foci sit at the two nonreal eigenvalues \(\pm\sqrt{3-2i}\), and the third eigenvalue \(4\) lies inside that hyperbolic arc, so \(W^J(A)\) is exactly that hyperbolic disc. For a suitable centro-symmetric \(5\times5\) example with \(\mathbf a=(6,-6,6,-6,6)\), the \(J\)-orthogonal reduction yields a small hyperbola \(\mathcal H_1\) from the \(2\times2\) block and a larger hyperbola \(\mathcal H_2\) from the \(3\times3\) block; only the larger one supports \(W^J(A)\). In the \(4\times4\) case, if the middle subdiagonal vanishes the two hyperbolas coincide and the result is a single nondegenerate hyperbolic disc, whereas otherwise one may obtain two separated hyperbolas \(\mathcal H_+\) and \(\mathcal H_-\), with possibilities including nesting, crossing, disjointness, hyperbolic arcs with flat sides, or full-plane degeneration [2407.03164].

The unified block formalism yields equally explicit models. When
\[
n=2,\qquad a=5,\qquad d=-3,\qquad
B=
\begin{bmatrix}
1&1\\
1&1
\end{bmatrix},
\]
one has \(\sigma_1(B)=2\), \(\sigma_2(B)=0\), \(\omega=4\), \(c=1\), and \(16>4\). Thus \(W^J(T)\) is bounded by the single nondegenerate hyperbola
\[
\frac{(x-1)^2}{4^2}-\frac{y^2}{2^2}=1,
\]
with foci at \(1\pm\sqrt{20}\). By contrast, when
\[
n=2,\qquad a=3,\qquad d=-1,\qquad B=\diag(2,1),
\]
one has \(\sigma_1=2\), \(\sigma_2=1\), \(\omega=2\), \(c=1\), and \(C^J(T)\) consists of two nested hyperbolas centered at \((1,0)\):
\[
H_1:\ \frac{(x-1)^2}{2^2}-\frac{y^2}{2^2}=1,
\qquad
H_2:\ \frac{(x-1)^2}{2^2}-\frac{y^2}{1^2}=1.
\]
The outer one bounds \(W^J(T)\), while the inner one remains part of the boundary-generating set [2508.12039].

At the operator-theoretic end, the example
\[
K=\C^2,\qquad J=\diag(1,-1),\qquad
A=
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
\]
shows the need for the co-numerical range. In this Kreĭn space, \(A\) is non-negative, and one obtains
\[
W(A)=\R\setminus\{0\},
\qquad
W_{\rm co}(A)=\{1\}.
\]
The example illustrates that the numerical range alone can be very large, while the co-numerical range restores a sharp spectral constraint [1208.3094].

Taken together, these results show that the Krein space numerical range is governed by the interaction between indefinite normalization, support-line pencils, and block or tridiagonal structure. In low dimensions the geometry can be completely hyperbolical; in higher-order structured settings it can persist as a single hyperbola, split into nested or distinct hyperbolas, or degenerate when separation and interlacing conditions fail. In the non-negative selfadjoint setting, the same pseudo-convex character persists, but spectral information requires the additional co-numerical range.

Source: https://www.emergentmind.com/topics/krein-space-numerical-range