---
title: Joint Higher-Rank Numerical Ranges
url: https://www.emergentmind.com/topics/joint-higher-rank-numerical-ranges
type: topic
---

# Joint Higher-Rank Numerical Ranges

Joint higher-rank numerical ranges are operator-theoretic sets that encode summed expectation values of an \(m\)-tuple of bounded operators over orthonormal \(k\)-frames. For a complex Hilbert space \({\mathcal H}\), a positive integer \(1 \le k < \dim {\mathcal H}\), and \({\mathbf A}=(A_1,\dots,A_m)\in {\mathcal B}({\mathcal H})^m\), the joint \(k\)-numerical range is
\[
W_k({\mathbf A})=\left\{(\alpha_1,\dots,\alpha_m)\in {\mathbb C}^m:\ \alpha_i=\sum_{j=1}^k \langle A_i x_j,x_j\rangle \text{ for an orthonormal set } \{x_1,\dots,x_k\}\subset {\mathcal H}\right\}.
\]
Its study links the geometry of \(W_k({\mathbf A})\) to algebraic features such as reducing subspaces, diagonal compressions, commutativity, normality, and essential spectral data; in infinite dimensions, closure and polyhedrality are controlled in part by the joint essential numerical range [2105.04621].

## 1. Definitions and equivalent formulations

The basic definition of \(W_k({\mathbf A})\) admits several equivalent formulations. In projection form,
\[
W_k({\mathbf A})=\{(\operatorname{Tr}(PA_1),\dots,\operatorname{Tr}(PA_m)):\ P=P^*=P^2,\ \operatorname{rank}P=k\}.
\]
In isometric form,
\[
W_k({\mathbf A})=\{(\operatorname{Tr}(X^*A_1X),\dots,\operatorname{Tr}(X^*A_mX)):\ X^*X=I_k\}.
\]
There is also a compression characterization: \((\alpha_1,\dots,\alpha_m)\in W_k({\mathbf A})\) if and only if there exists a unitary \(U\in {\mathcal B}({\mathcal H})\) and a \(k\times k\) matrix tuple \(\widetilde A\) with \(\operatorname{Tr}(\widetilde A_i)=\alpha_i\) such that
\[
U^*A_iU=
\begin{bmatrix}
\widetilde A_i & 0\\
0 & B_i
\end{bmatrix}.
\]
These formulations make explicit that \(W_k\) is determined by rank-\(k\) compressions rather than by scalar restrictions [2105.04621].

In finite dimension \(n=\dim {\mathcal H}\), the endpoint cases are fixed by convention or trace identities:
\[
W_0({\mathbf A})=\{(0,\dots,0)\},\qquad
W_n({\mathbf A})=\{(\operatorname{Tr}A_1,\dots,\operatorname{Tr}A_m)\},
\]
and
\[
W_k({\mathbf A})=(\operatorname{Tr}A_1,\dots,\operatorname{Tr}A_m)-W_{n-k}({\mathbf A}).
\]
The range is translation-covariant and linear-covariant: for \((u_1,\dots,u_m)\in {\mathbb C}^m\),
\[
W_k(A_1+u_1I,\dots,A_m+u_mI)=W_k({\mathbf A})+k(u_1,\dots,u_m),
\]
and under linear recombination by a matrix \(T=(T_{ij})\), one has
\[
W_k(B_1,\dots,B_p)=\{(a_1,\dots,a_m)T:\ (a_1,\dots,a_m)\in W_k({\mathbf A})\}.
\]
If \({\mathcal H}= {\mathcal H}_1\oplus \cdots \oplus {\mathcal H}_r\) and each \(A_i\) decomposes accordingly, then \(W_k({\mathbf A})\) is constrained by convex combinations of sums of lower-rank joint numerical ranges on the summands; equality holds when \(W_k({\mathbf A})\) is convex [2105.04621].

A persistent source of terminological ambiguity is the distinction between \(W_k\) and the higher-rank numerical range \(\Lambda_k\). For a single operator \(A\),
\[
\Lambda_k(A)=\{\lambda\in {\mathbb C}:\ \exists V\subset {\mathcal H},\ \dim V=k,\ A|_V=\lambda I\},
\]
and for tuples the analogous notion requires simultaneous scalar compressions. By contrast, \(W_k({\mathbf A})\) records summed expectations over orthonormal \(k\)-tuples and generally differs from \(\Lambda_k\) unless strong structural conditions are present. In the matricial-range literature, the joint higher-rank numerical range \(\Lambda_k(A_1,\dots,A_m)\) appears as the scalar slice of the joint higher-rank matricial range \(W^{(k)}(A_1,\dots,A_m)\), and the more general joint \((k:p)\)-matricial range \(\Lambda_{(k:p)}({\mathbf A})\) compresses each \(A_j\) to \(D_j\otimes I_k\) with diagonal \(D_j\in M_p\) [1911.12744].

## 2. Geometry and convexity

A central fact is that the joint setting departs sharply from the classical Toeplitz–Hausdorff picture. For a single operator, higher-rank numerical ranges are convex, but for \(m\ge 2\) operators the joint \(k\)-numerical range need not be convex. Chan, Li, and Poon formulate convexity criteria in terms of the affine span of \(\{I,A_1,\dots,A_m\}\). If \(\dim {\mathcal H}=2\), then \(W_k({\mathbf A})\) is convex if and only if \(\operatorname{span}\{I,A_1,\dots,A_m\}\) has dimension at most \(3\). If \(\dim {\mathcal H}\ge 3\) and \(\operatorname{span}\{I,A_1,\dots,A_m\}\) has dimension at most \(4\), then \(W_k({\mathbf A})\) is convex; in particular, \(W_k(A_1,A_2,A_3)\) is always convex. Conversely, if \(\dim {\mathcal H}\ge 3\) and the span has dimension at least \(4\), there exists \(A_0\) such that \(W_k(A_0,A_1,\dots,A_3)\) is not convex [2105.04621].

The Pauli matrices supply the canonical low-dimensional examples. With \(X,Y,Z\), one has \(W(X,Y)\) equal to the unit disk, while \(W(X,Y,Z)\) is the unit sphere in \({\mathbb R}^3\), hence nonconvex. This contrast encapsulates the fact that adding one more coordinate can destroy convexity even in dimension \(2\) [2105.04621].

Certain block forms force convexity. If a unitary \(U\) brings each \(A_j\) into one of the block forms
\[
\begin{bmatrix}
a_j I_k & R_j^*\\
R_j & 0
\end{bmatrix}
\qquad\text{or}\qquad
\begin{bmatrix}
0 & P_j\\
P_j^* & 0
\end{bmatrix},
\]
with \(\dim {\mathcal H}\ge 2k\) in the second case, then \(W_k({\mathbf A})\) is convex. These criteria are structural rather than purely dimensional: they arise from convexity of the admissible projection blocks in the compression model [2105.04621].

Convexity does not propagate monotonically in \(k\). Example 3.3 shows a tuple built from Pauli blocks for which \(W_{k+1}({\mathbf A})\) is not convex while \(W_r({\mathbf A})\) is convex for broad ranges of \(r\). Nonetheless, monotonicity survives at the level of convex hulls:
\[
W_{k+1}({\mathbf A})\subseteq \frac{k+1}{k}\,\operatorname{conv} W_k({\mathbf A}),
\qquad
\operatorname{conv} W_{k+1}({\mathbf A})\subseteq \frac{k+1}{k}\,\operatorname{conv} W_k({\mathbf A}).
\]
The geometry of \(\operatorname{cl}(\operatorname{conv}W_k({\mathbf A}))\) is governed by support half-spaces. For a real unit vector \(u=(u_1,\dots,u_m)\), define \(A_u=\sum_{j=1}^m u_jA_j\). Then
\[
\Pi_u(k;{\mathbf A})=
\left\{(s_1,\dots,s_m):\ \sum_{j=1}^m u_js_j\le \sum_{i=1}^k \lambda_i(A_u)\right\}
\]
contains \(W_k({\mathbf A})\), its boundary is a support hyperplane, and
\[
\operatorname{cl}(\operatorname{conv}W_k({\mathbf A}))=\bigcap_{u\ \text{unit}}\Pi_u(k;{\mathbf A}).
\]
Thus the support function in direction \(u\) is the sum of the top \(k\) eigenvalues of the self-adjoint linear combination \(A_u\) [2105.04621].

## 3. Closure, essential numerical range, and infinite-dimensional phenomena

The finite-dimensional and infinite-dimensional theories diverge most clearly at the level of topology. In finite dimension, \(W_k({\mathbf A})\) is closed. In infinite dimension, closedness can fail even for diagonal operators and even when neighboring ranks behave differently; the paper records examples where \(W_k({\mathbf A})\) is closed while \(W_{k+1}({\mathbf A})\) is not [2105.04621].

The relevant asymptotic object is the joint essential numerical range
\[
W_e({\mathbf A})=\bigcap_{K\in K({\mathcal H})}\operatorname{cl}\big(W_1({\mathbf A}+K)\big),
\]
where \(K({\mathcal H})\) is the ideal of compact operators. An equivalent characterization is that \(u\in W_e({\mathbf A})\) if and only if there exists a weakly null sequence of unit vectors, equivalently an orthonormal sequence, \(\{v_n\}\) such that
\[
\langle A_jv_n,v_n\rangle \to u_j,\qquad j=1,\dots,m.
\]
The set \(W_e({\mathbf A})\) is always convex and closed [2105.04621].

Closure of \(W_k\) is described by an essential-diagonal augmentation. For self-adjoint \({\mathbf A}\), one forms diagonal operators \(D_j\) on a Hilbert space with basis indexed by \(W_e({\mathbf A})\), then sets
\[
\widetilde A_j=A_j\oplus (I_k\otimes D_j).
\]
The resulting identities are
\[
W_\ell(I_k\otimes D_1,\dots,I_k\otimes D_m)=\ell\cdot W_e({\mathbf A}),\qquad \ell=1,\dots,k,
\]
and
\[
\operatorname{cl}W_k({\mathbf A})=W_k(\widetilde{\mathbf A}).
\]
Moreover, \(W_k({\mathbf A})\) is closed if and only if \(W_k({\mathbf A})=W_k(\widetilde{\mathbf A})\), equivalently every point in \(W_k({\mathbf A})\) has the form
\[
(\operatorname{tr}(A_1P),\dots,\operatorname{tr}(A_mP))+(k-\operatorname{tr}P)\cdot u
\]
for some positive semidefinite contraction \(P\) with rank at most \(k\) and some \(u\in W_e({\mathbf A})\) [2105.04621].

The convex closure satisfies the exact formula
\[
\operatorname{cl}(\operatorname{conv}W_k({\mathbf A}))=\operatorname{conv}(\operatorname{cl}W_k({\mathbf A}))
=\operatorname{conv}S,
\]
where
\[
S=\bigcup_{\ell=0}^k \big(W_\ell({\mathbf A})+(k-\ell)\cdot W_e({\mathbf A})\big).
\]
Consequently, \(\operatorname{conv}W_k({\mathbf A})\) is closed if and only if
\[
W_\ell({\mathbf A})+(k-\ell)\cdot W_e({\mathbf A})\subseteq \operatorname{conv}W_k({\mathbf A})
\]
for each \(\ell=0,\dots,k-1\) [2105.04621].

Closedness has a partial inheritance property. If \(m\le 3\) and \(W_{k+1}({\mathbf A})\) is closed, then \(W_k({\mathbf A})\) is closed. If \(m\ge 4\) and \(\operatorname{conv}W_{k+1}({\mathbf A})\) is closed, then \(\operatorname{conv}W_k({\mathbf A})\) is closed. The general question whether \(W_{k+1}({\mathbf A})\) closed implies \(W_k({\mathbf A})\) closed for \(m>3\) remains open [2105.04621].

## 4. Polyhedrality and structural decomposition

Polyhedrality provides the most rigid geometric regime. In this context, a polyhedral set is the convex hull of finitely many points. The central equivalence for operator tuples states that for \({\mathbf A}\in {\mathcal B}({\mathcal H})^m\) and fixed \(k\), the following are equivalent: \(W_\ell({\mathbf A})\) is polyhedral for all \(\ell=1,\dots,k\); \(\operatorname{conv}W_k({\mathbf A})\) is polyhedral; and there exists \(r\ge 2k\) and a unitary \(U\) such that
\[
U^*A_jU=D_j\oplus B_j,\qquad D_j\in M_r \text{ diagonal},
\]
with
\[
W_k({\mathbf A})=W_k(D_1,\dots,D_m).
\]
Thus polyhedrality is equivalent to reduction to a finite-dimensional common reducing subspace carrying diagonal compressions that already generate the full joint \(k\)-numerical range [2105.04621].

A parallel result characterizes polyhedrality of closures. The following are equivalent: \(\operatorname{cl}W_\ell({\mathbf A})\) is polyhedral for all \(\ell=1,\dots,k\); \(\operatorname{cl}(\operatorname{conv}W_k({\mathbf A}))\) is polyhedral; and there exist \(r\ge 2k\) and isometries \(X_n:\mathbb C^r\to {\mathcal H}\) such that
\[
D^{(n)}=(X_n^*A_1X_n,\dots,X_n^*A_mX_n)
\]
is a sequence of diagonal \(m\)-tuples converging to \(D\), with
\[
W_k(D^{(n)})\to W_k(D)=\operatorname{cl}W_k({\mathbf A})
\]
in the Hausdorff metric. This realizes polyhedral closure as a Hausdorff limit of polyhedra produced by finite-dimensional diagonal compressions [2105.04621].

Conical boundary points expose the same structure from the boundary inward. If
\[
p=(\operatorname{Tr}(X^*A_1X),\dots,\operatorname{Tr}(X^*A_mX))
\]
is a conical point of \(W_k({\mathbf A})\), where \(X:\mathbb C^k\to {\mathcal H}\) is an isometry, then the range of \(X\) is a reducing subspace for each \(A_j\). If \(p\) is a conical point of \(\operatorname{cl}W_k({\mathbf A})\) but \(p\notin W_k({\mathbf A})\), then \(p\) is approximated by compressions coming from a sequence of isometries \(X_\ell\) [2105.04621].

In the matrix case, polyhedrality also detects commutativity near half-rank. Li, Poon, and Wang proved that a family of \(n\times n\) matrices is a family of mutually commuting normal matrices if and only if \(W_k(A_1,\dots,A_m)\) is polyhedral for some \(k\) satisfying \(|n/2-k|\le 1\); equivalently, for a generating family it suffices that \(W_k(X,Y)\) be polyhedral for any two matrices \(X,Y\) in the family. More generally, they characterized when the joint \(c\)-numerical range \(W_c(A_1,\dots,A_m)\) is polyhedral [2002.02768].

## 5. Commuting normal families and explicit spectral descriptions

For commuting normal operators, joint higher-rank numerical ranges admit explicit spectral models. If \(A_1,\dots,A_m\) are commuting normal compact operators, then there is a joint diagonalization
\[
U^*A_jU=\operatorname{diag}(d_j(1),d_j(2),\dots),\qquad j=1,\dots,m,
\]
and one writes
\[
v(\ell)=(d_1(\ell),\dots,d_m(\ell))\in {\mathbb C}^m.
\]
In finite dimension \(n\), if \(D_k\) denotes the set of diagonal \(k\)-projections, then
\[
W_k({\mathbf A})=
\left\{
\left(\sum_{j=1}^n u_j d_1(j),\dots,\sum_{j=1}^n u_j d_m(j)\right):\ u\in D_k
\right\}.
\]
Since
\[
D_k=\operatorname{conv}\{Pv:\ P \text{ permutation},\ v=(1,\dots,1,0,\dots,0)\},
\]
one obtains
\[
W_k({\mathbf A})=\operatorname{conv}\Sigma_k({\mathbf A}),
\]
where \(\Sigma_k({\mathbf A})\) is the set of \(k\)-sums of the joint eigenvalue vectors \(v(j)\). In particular, in the commuting normal finite-dimensional case, \(W_k({\mathbf A})\) is a polytope whose vertices are the \(k\)-sums of joint eigenvalue vectors [2105.04621].

For compact operators, commuting normality is characterized by these spectral sums. If \({\mathbf A}\) is compact, then \(\{A_1,\dots,A_m\}\) is a commuting family of normal operators if and only if
\[
W_k({\mathbf A})=\operatorname{conv}E_k({\mathbf A})
\]
for every \(k\ge 1\), where \(E_k({\mathbf A})\) is the set of sums of \(k\) joint eigenvalues corresponding to \(k\) linearly independent common eigenvectors. Yet commuting normality does not force polyhedrality in infinite dimension: there exist commuting compact self-adjoint operators \(H,G\) with \(W_k(H,G)\) not polyhedral and with smooth extreme points for all \(k\) [2105.04621].

For compact tuples, closure is particularly simple:
\[
\operatorname{cl}W_k({\mathbf A})=W_k({\mathbf A}\oplus 0_k)=W_k({\mathbf A}\oplus 0_\infty),
\]
and
\[
\operatorname{cl}(\operatorname{conv}W_k({\mathbf A}))=\operatorname{conv}W_k({\mathbf A}\oplus 0_k)=\operatorname{conv}W_k({\mathbf A}\oplus 0_\infty).
\]
Consequently, for compact operators the following are equivalent: \(\operatorname{cl}W_k({\mathbf A})\) is polyhedral for every \(k\); \(\operatorname{cl}(\operatorname{conv}W_k({\mathbf A}))\) is polyhedral for every \(k\); and \(\{A_1,\dots,A_m\}\) is a commuting family of normal operators with \(\operatorname{conv}E_k({\mathbf A}\oplus 0_\infty)\) polyhedral for every \(k\). In the finite-rank case, commuting normality is equivalent to polyhedrality of \(W_k({\mathbf A})\) for all \(k\), and also equivalent to polyhedrality of \(\operatorname{conv}W_k({\mathbf A})\) for some \(k\ge r\), where \(r\) is the dimension of the sum of the ranges of the \(A_j\) [2105.04621].

The matrix-theoretic results of Li, Poon, and Wang sharpen this picture in finite dimensions. For Hermitian tuples they describe \(\operatorname{conv}W_c(A_1,\dots,A_m)\) as the intersection of half-spaces determined by eigenvalues of linear combinations \(v_1A_1+\cdots+v_mA_m\), and show that conical points of \(W_c(A_1,\dots,A_m)\) force common direct-sum decompositions. When the weight matrix \(C\) has \(n\) distinct eigenvalues, the existence of a conical point implies that \(\{A_1,\dots,A_m\}\) is a commuting family of normal matrices [2002.02768].

## 6. Related generalizations, applications, and open problems

Joint higher-rank numerical ranges sit alongside several related constructions. One direction replaces scalar sums over orthonormal \(k\)-frames by compressions to scalar matrices or block-diagonal matricial forms. In the notation of higher-rank matricial ranges,
\[
W^{(k)}(A_1,\dots,A_m)=\{(V^*A_1V,\dots,V^*A_mV):\ V^*V=I_k\},
\]
and the joint higher-rank numerical range \(\Lambda_k(A_1,\dots,A_m)\) is its scalar slice:
\[
(\lambda_1,\dots,\lambda_m)\in \Lambda_k(A_1,\dots,A_m)
\Longleftrightarrow
(\lambda_1I_k,\dots,\lambda_mI_k)\in W^{(k)}(A_1,\dots,A_m).
\]
The joint rank \((k:p)\)-matricial range \(\Lambda_{(k:p)}({\mathbf A})\) further requires
\[
V^*A_jV=D_j\otimes I_k,\qquad D_j\in \mathcal D_p.
\]
This framework is tied directly to hybrid quantum error correction: for a noisy quantum channel with Kraus operators \(\{E_\alpha\}\), a hybrid \((k:p)\) code exists if and only if
\[
\Lambda_{(k:p)}(E_1^*E_1,\ E_1^*E_2,\ \dots,\ E_c^*E_c)\neq \emptyset
\]
[1911.12744].

Non-emptiness and geometry of these matricial ranges are controlled by dimension bounds. If \({\mathbf A}=(A_1,\ldots,A_m)\in H_n^m\) and \(k>1\), then
\[
n \ge (m+1)\Big((m+1)(k-1)+k(p-1)\Big)\quad\Longrightarrow\quad \Lambda_{(k:p)}({\mathbf A})\ne \emptyset.
\]
Under the stronger bound
\[
n \ge (kp(m+2)-1)(m+1)^2,
\]
\(\Lambda_{(k:p)}({\mathbf A})\) is star-shaped with star center \((a_1I_p,\dots,a_mI_p)\) for any \((a_1,\dots,a_m)\in \Lambda_{(kp(m+2):1)}({\mathbf A})\). These results extend the higher-rank geometry from scalar compressions to block-diagonal compressions motivated by operator-algebraic quantum error correction [1911.12744].

A different extension appears in max algebra. There the paper on generalized numerical ranges in max algebra introduces a rank-\(k\) numerical range, a max joint \(k\)-numerical range, and a max joint \(C\)-numerical range for entry-wise nonnegative matrices. In that setting,
\[
W_{\max}^k(\mathbb A)
=
\{(tr_{\otimes}(X^t\otimes A_1\otimes X),\dots,tr_{\otimes}(X^t\otimes A_m\otimes X)):\ X\in \mathcal X_{n\times k}\},
\]
and the resulting theory replaces linear algebra over \(\mathbb C\) by max-times semiring structure, permutation invariance, and interval-type geometry [2412.10375]. This suggests that the higher-rank viewpoint is robust under substantial changes in ambient algebraic structure.

Several open problems remain explicit in the operator-theoretic theory. Chan, Li, and Poon list the following: prove or disprove
\[
W_{k+1}({\mathbf A})/(k+1)\subseteq W_k({\mathbf A})/k;
\]
prove or disprove that \(W_k({\mathbf A})\) is closed whenever \(W_{k+1}({\mathbf A})\) is closed for \(m>3\); construct \({\mathbf A}\) with \(\operatorname{conv}W_k({\mathbf A})\) closed but \(W_k({\mathbf A})\) not closed for \(k>1\); and extend the theory to joint \(c\)-numerical ranges \(W_c({\mathbf A})\) [2105.04621]. These questions mark the boundary between currently understood spectral-compression phenomena and the unresolved geometry of multivariable higher-rank ranges.

Source: https://www.emergentmind.com/topics/joint-higher-rank-numerical-ranges