Krein Space Numerical Range Overview

• Krein space numerical range is the analogue of the classical numerical range for operators in a vector space with an indefinite Hermitian form, splitting into positive and negative sheets.
• It features pseudo-convex geometry where the boundary of the range can be hyperbolic rather than elliptical, as demonstrated in the complete 2×2 hyperbolical range theorem.
• Structured matrices reveal persistence and bifurcation of hyperbolic boundaries, while the co-numerical range sharpens spectral localization for non-negative selfadjoint operators.

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 (Bebiano et al., 2024, Bebiano et al., 16 Aug 2025, Philipp et al., 2012).

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 $W(A)$ (Bebiano et al., 2024).

A parallel operator-theoretic formulation is used for non-negative selfadjoint operators $J\in M_n(\C)$0 in a Kreĭn space $J\in M_n(\C)$1: $J\in M_n(\C)$2 Equivalently, one may normalize to $J\in M_n(\C)$3. This formulation is real-valued because the operator is assumed selfadjoint and non-negative (Philipp et al., 2012).

For the block setting $J\in M_n(\C)$4 with

$J\in M_n(\C)$5

the numerical range can be written as

$J\in M_n(\C)$6

In this setting, each of $J\in M_n(\C)$7 and $J\in M_n(\C)$8 is convex, but possibly unbounded, and $J\in M_n(\C)$9 is pseudo-convex in the sense that any two points either generate a line segment in $(r,n-r)$0 or two opposing rays (Bebiano et al., 16 Aug 2025).

2. The $(r,n-r)$1 hyperbolical range theorem

For the non-trivial signature

$(r,n-r)$2

and $(r,n-r)$3, let $(r,n-r)$4 be the eigenvalues of $(r,n-r)$5, and set

$(r,n-r)$6

Then $(r,n-r)$7 is bounded by a non-degenerate hyperbola with foci at $(r,n-r)$8 if and only if

$(r,n-r)$9

When this holds, the transverse and non-transverse semi-axes are

$\C^n$0

and after a suitable rotation and translation the boundary hyperbola has equation

$\C^n$1

This is the finite-dimensional hyperbolical range theorem in the $\C^n$2 case (Bebiano et al., 2024).

The support-line mechanism is expressed through the $\C^n$3-Hermitian pencil

$\C^n$4

For each real $\C^n$5, $\C^n$6 has two real eigenvalues $\C^n$7, and these are the support lines of $\C^n$8 and $\C^n$9. One computes

$[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$0

with real constants $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$1, so that the support-eigenvalues satisfy

$[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$2

The condition $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$3 is algebraically equivalent to the double inequality above and guarantees that the two sheets

$[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$4

never meet. As $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$5 varies, these branches trace a nondegenerate hyperbola, and exactly one branch lies on the $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$6-sheet while the other lies on the $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$7-sheet, so $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$8 is the pseudo-convex hull of the two branches (Bebiano et al., 2024).

3. Persistence and bifurcation in structured tridiagonal matrices

The hyperbolic boundary is not confined to the $[x,y]_J \;:=\; y^*\,J\,x,\qquad x,y\in\C^n.$9 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 (Bebiano et al., 2024).

For the $A\in M_n(\C)$0 biperiodic tridiagonal case,

$A\in M_n(\C)$1

let

$A\in M_n(\C)$2

Then $A\in M_n(\C)$3 is exactly the hyperbolic disc bounded by $A\in M_n(\C)$4 if and only if

$A\in M_n(\C)$5

In that event, the third eigenvalue $A\in M_n(\C)$6 lies in or on the boundary of the hyperbola.

For a centro-symmetric tridiagonal $A\in M_n(\C)$7 matrix with

$A\in M_n(\C)$8

a suitable $A\in M_n(\C)$9-orthogonal block-diagonalization splits the problem into a $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$0 block and a $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$1 block of the previous type. The numerical range is again a single hyperbola provided the corresponding $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$2 block satisfies the same strict double inequality. Otherwise one obtains two nested hyperbolas, and only the outer one supports $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$3.

For a centro-symmetric tridiagonal $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$4 matrix with quasi-biperiodic main diagonal and

$$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$5

the matrix is $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$6-orthogonally similar to two $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$7 Hermitian pencils. Either both yield the same single hyperbola, or they yield two distinct hyperbolas $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$8 and $$W^J_+(A)\;=\;\{[A x,x]_J : [x,x]_J=+1\}, \qquad W^J_-(A)\;=\;\{[A x,x]_J : [x,x]_J=-1\},$$9 with

$W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$0

In the latter situation, $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$1 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 $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$2 case, for example,

$W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$3

In the $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$4 case there are two such inequalities, one for each $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$5 block, and in the $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$6 case again a single double inequality for the central $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$7 block.

4. Unified block-matrix formulation

A broader unified picture is available for $W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$8 block matrices of the form

$W^J(A)\;=\;-\,W^J_-(A)\;\cup\;W^J_+(A).$9

Set

$J=I$0

For each $J=I$1, define the $J=I$2-Hermitian family

$J=I$3

A direct computation gives

$J=I$4

and after reduction to $J=I$5, the nontrivial eigenvalues occur in $J=I$6 pairs

$J=I$7

where $J=I$8 are the singular values of $J=I$9 (Bebiano et al., 16 Aug 2025).

The indefinite Kippenhahn curve $W(A)$0 is the envelope of the support lines

$W(A)$1

with $W(A)$2 varying over an open interval where the radicand is positive. For each $W(A)$3 such that $W(A)$4, the envelope is a nondegenerate hyperbola centered at $W(A)$5, with transverse semi-axis $W(A)$6 and nontransverse semi-axis $W(A)$7. In Cartesian form,

$W(A)$8

and the foci are

$W(A)$9

The full numerical range $J\in M_n(\C)$00 is the pseudo-convex hull of the collection of hyperbolas corresponding to indices $J\in M_n(\C)$01 with $J\in M_n(\C)$02, together with degenerate points when $J\in M_n(\C)$03, and possibly the singleton $J\in M_n(\C)$04 when $J\in M_n(\C)$05 has smaller rank. The outer hyperbola, corresponding to $J\in M_n(\C)$06, bounds $J\in M_n(\C)$07 as a hyperbolic disc exactly when

$J\in M_n(\C)$08

This criterion unifies and extends earlier hyperbolical range theorems for $J\in M_n(\C)$09 and block-Toeplitz cases, and it produces new families with multiple nested hyperbolas when $J\in M_n(\C)$10 (Bebiano et al., 16 Aug 2025).

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

$J\in M_n(\C)$11

Then $J\in M_n(\C)$12 is always unbounded above and below unless $J\in M_n(\C)$13. If $J\in M_n(\C)$14 is indefinite, then

$J\in M_n(\C)$15

Otherwise,

$J\in M_n(\C)$16

with the conventions $J\in M_n(\C)$17 if $J\in M_n(\C)$18 and $J\in M_n(\C)$19 if $J\in M_n(\C)$20. In particular, $J\in M_n(\C)$21 is pseudo-convex in the sense that whenever $J\in M_n(\C)$22, either the closed segment $J\in M_n(\C)$23 or the complement in the line of its open interior is contained in $J\in M_n(\C)$24 (Philipp et al., 2012).

To recover spectral information more sharply, Philipp–Trunk define the Kreĭn-space co-numerical range

$J\in M_n(\C)$25

If $J\in M_n(\C)$26 is boundedly invertible, then $J\in M_n(\C)$27 is a Hilbert space and $J\in M_n(\C)$28 is the usual Hilbert-space numerical range of the selfadjoint operator $J\in M_n(\C)$29 there, hence a closed real interval. More generally,

$J\in M_n(\C)$30

The main spectral inclusion states that if $J\in M_n(\C)$31 is non-negative in an indefinite Kreĭn space, then, except in the single borderline case where $J\in M_n(\C)$32 is an isolated eigenvalue with $J\in M_n(\C)$33 and one of $J\in M_n(\C)$34 is empty,

$J\in M_n(\C)$35

In the exceptional case,

$J\in M_n(\C)$36

This shows that $J\in M_n(\C)$37 by itself is generally too large to localize the spectrum pointwise, while the intersection $J\in M_n(\C)$38 does capture all nonzero spectral values (Philipp et al., 2012).

6. Representative geometries and degeneracies

Several explicit examples illustrate the range of possible geometries. In the $J\in M_n(\C)$39 theorem of Bebiano–Lemos–Soares,

$J\in M_n(\C)$40

Here $J\in M_n(\C)$41, $J\in M_n(\C)$42, and $J\in M_n(\C)$43 satisfy the strict inequalities. The boundary generating curve is a hyperbola whose foci sit at the two nonreal eigenvalues $J\in M_n(\C)$44, and the third eigenvalue $J\in M_n(\C)$45 lies inside that hyperbolic arc, so $J\in M_n(\C)$46 is exactly that hyperbolic disc. For a suitable centro-symmetric $J\in M_n(\C)$47 example with $J\in M_n(\C)$48, the $J\in M_n(\C)$49-orthogonal reduction yields a small hyperbola $J\in M_n(\C)$50 from the $J\in M_n(\C)$51 block and a larger hyperbola $J\in M_n(\C)$52 from the $J\in M_n(\C)$53 block; only the larger one supports $J\in M_n(\C)$54. In the $J\in M_n(\C)$55 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 $J\in M_n(\C)$56 and $J\in M_n(\C)$57, with possibilities including nesting, crossing, disjointness, hyperbolic arcs with flat sides, or full-plane degeneration (Bebiano et al., 2024).

The unified block formalism yields equally explicit models. When

$J\in M_n(\C)$58

one has $J\in M_n(\C)$59, $J\in M_n(\C)$60, $J\in M_n(\C)$61, $J\in M_n(\C)$62, and $J\in M_n(\C)$63. Thus $J\in M_n(\C)$64 is bounded by the single nondegenerate hyperbola

$J\in M_n(\C)$65

with foci at $J\in M_n(\C)$66. By contrast, when

$J\in M_n(\C)$67

one has $J\in M_n(\C)$68, $J\in M_n(\C)$69, $J\in M_n(\C)$70, $J\in M_n(\C)$71, and $J\in M_n(\C)$72 consists of two nested hyperbolas centered at $J\in M_n(\C)$73: $J\in M_n(\C)$74 The outer one bounds $J\in M_n(\C)$75, while the inner one remains part of the boundary-generating set (Bebiano et al., 16 Aug 2025).

At the operator-theoretic end, the example

$J\in M_n(\C)$76

shows the need for the co-numerical range. In this Kreĭn space, $J\in M_n(\C)$77 is non-negative, and one obtains

$J\in M_n(\C)$78

The example illustrates that the numerical range alone can be very large, while the co-numerical range restores a sharp spectral constraint (Philipp et al., 2012).

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.