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The numerical range and the Crouzeix inequality

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

For a normal matrix \(A\), the norm of a polynomial \(p(A)\) is the largest value of \(|p|\) on the spectrum. For a matrix that is not normal the spectrum gives no such control: a nonzero nilpotent matrix has spectrum \(\{0\}\), while \(p(A)\) can be large for polynomials that are small at \(0\). The numerical range \(W(A)=\{u^*Au:\|u\|=1\}\), a compact convex set containing the spectrum, does give a universal bound. Crouzeix proved in 2007 that \(\|p(A)\|\le11.08\max_{W(A)}|p|\) and conjectured that the constant \(2\) holds; Crouzeix and Palencia lowered the constant to \(1+\sqrt2\) [CP], and Ransford and Schwenninger simplified their argument [RS]. In 2026 the constant \(2\) for scalar polynomials was proved by Jin [Jin], by Lorist and Schwenninger [LoS], who also treat operators on Hilbert space and rational functions, and by Luo [Luo]; Crouzeix and Greenbaum used the theorem of Lorist and Schwenninger to lower the spectral constants of other regions, showing for example that certain annular regions are \(2\)-spectral sets [CG]. The complete form of the conjecture allows matrix coefficients and asks for the constant \(2\) uniformly in their size; Badea, Crouzeix and Delyon proved it for matrices \(A\) of order \(2\), and Åhag, Czyż and Virtanen for order at most \(3\) [ÅCV]. This course proves the complete form in all dimensions, a theorem of OpenAI [OpenAI-CC], following OpenAI's direct proof [OpenAI-DC].

The theorem

For \(B_0,\dots,B_d\in M_m(\mathbf C)\) and the polynomial \(P(z)=\sum_{k=0}^dB_kz^k\) with matrix coefficients, and for \(A\in M_n(\mathbf C)\), put \[ P[A]=\sum_{k=0}^dA^k\otimes B_k, \] an operator on \(\mathbf C^n\otimes\mathbf C^m\). All norms are operator norms for the Euclidean inner products. For \(m=1\), \(P[A]\) is the usual \(p(A)\).

Theorem A (OpenAI). For all positive integers \(n,m\), every \(A\in M_n(\mathbf C)\) and every polynomial \(P\) with coefficients in \(M_m(\mathbf C)\), \[ \|P[A]\|\le2\max_{z\in W(A)}\|P(z)\| . \] The constant \(2\) cannot be replaced by a smaller number.

The bound does not depend on \(n\), \(m\) or the degree. Its complete character is the uniformity in \(m\): the coefficients \(B_k\) need not commute with each other, and a proof for scalar polynomials need not survive this amplification. The proof below keeps the order of multiplication, using \(FG\) and \(GF\) separately. OpenAI's other manuscript [OpenAI-CC] proves the theorem by a different method: it finds a similarity \(S\) with \(\|S\|\|S^{-1}\|\le2\) that turns a conformal image of \(A\) into a contraction; it is not treated here.

The course.

  1. This lesson: the numerical range, the Toeplitz–Hausdorff theorem, matrix polynomials, sharpness, and the plan of the proof.
  2. Smooth convex neighbourhoods: every compact convex set is approximated from outside by convex domains whose boundaries have analytic polar parametrizations.
  3. An exterior conformal coordinate: the exterior of such a domain is the conformal image of \(\{|\lambda|>1\}\) under a map that extends across the unit circle.
  4. Faber polynomials and a positive kernel: the Faber polynomials of the domain, a nonnegative kernel with marginals one, and a two-sided estimate of boundary norms by coefficients.
  5. The bound on a smooth convex domain: the resolvent, its positive real part, singular vectors, and the proof of Theorem A.

What the course assumes. Linear algebra over \(\mathbf C\), including tensor products of matrices; complex analysis as in the core course Complex Analysis, whose results are cited by number from Lebl's Guide to Cultivating Complex Analysis; Fourier series of absolutely convergent trigonometric series.

1. The numerical range

For \(A\in M_n(\mathbf C)\), \[ W(A)=\{u^*Au:u\in\mathbf C^n,\ u^*u=1\}. \] It is compact, as the continuous image of the unit sphere, and it contains every eigenvalue: for a unit eigenvector \(u\) with \(Au=\lambda u\), \(u^*Au=\lambda\). It is invariant under unitary conjugation, \(W(VAV^*)=W(A)\), and \(W(aA+bI)=aW(A)+b\) for scalars \(a,b\).

Theorem 1.1 (Toeplitz–Hausdorff). \(W(A)\) is convex.

Proof. Let \(z_1=u_1^*Au_1\) and \(z_2=u_2^*Au_2\) with unit vectors \(u_1,u_2\). If they are linearly dependent, \(z_1=z_2\). Otherwise let \(e_1,e_2\) be an orthonormal basis of their span and \(A_2=(e_i^*Ae_j)_{i,j}\in M_2(\mathbf C)\); the unit vectors \(v=xe_1+ye_2\) of the span give \(v^*Av=(\bar x,\bar y)A_2(x,y)^{\mathsf T}\), so \(z_1,z_2\in W(A_2)\subseteq W(A)\). It suffices to show that \(W(A_2)\) is convex. Write \(A_2=\bigl(\begin{smallmatrix}\alpha&\beta\\\gamma&\delta\end{smallmatrix}\bigr)\) and, for \(|x|^2+|y|^2=1\), \[ w=\bigl(2\operatorname{Re}(\bar xy),\ 2\operatorname{Im}(\bar xy),\ |x|^2-|y|^2\bigr). \] Then \[ \begin{aligned} \|w\|^2&=4|\bar xy|^2+(|x|^2-|y|^2)^2\\ &=(|x|^2+|y|^2)^2=1, \end{aligned} \] and every point of the unit sphere \(S^2\subset\mathbf R^3\) arises: choose \(|x|,|y|\ge0\) with \(|x|^2-|y|^2=w_3\) and then the phases so that \(\bar xy=(w_1+iw_2)/2\), whose modulus \(\frac12\sqrt{1-w_3^2}\) equals \(|x||y|\). Expanding, \[ \begin{aligned} (\bar x,\bar y)A_2(x,y)^{\mathsf T}={}&\tfrac{\alpha+\delta}2+\tfrac{\alpha-\delta}2w_3\\ &+\tfrac{\beta+\gamma}2w_1+i\tfrac{\beta-\gamma}2w_2, \end{aligned} \] a real-affine map \(\Lambda:\mathbf R^3\to\mathbf C\cong\mathbf R^2\) evaluated at \(w\). So \(W(A_2)=\Lambda(S^2)\). The linear part of \(\Lambda\) has a nonzero kernel, and moving a point of the closed unit ball along a kernel direction until it reaches the sphere does not change its image. Hence \(\Lambda(S^2)=\Lambda(\text{ball})\), which is convex. \(\square\)

2. Matrix polynomials

If \(F(z)=\sum_iD_iz^i\) and \(G(z)=\sum_jE_jz^j\) have coefficients in \(M_m(\mathbf C)\), then \[ \begin{aligned} (FG)[A]&=\sum_{i,j}A^{i+j}\otimes D_iE_j\\ &=F[A]\,G[A], \end{aligned} \] because \((A^i\otimes D_i)(A^j\otimes E_j)=A^{i+j}\otimes D_iE_j\). Likewise \((GF)[A]=G[A]F[A]\), and \(FG\ne GF\) in general.

For a normal matrix \(A=V\operatorname{diag}(\lambda_1,\dots,\lambda_n)V^*\) with \(V\) unitary, \[ P[A]=(V\otimes I)\,D_P\,(V\otimes I)^*, \] where \(D_P\) is the block diagonal matrix with blocks \(P(\lambda_1),\dots,P(\lambda_n)\). So \(\|P[A]\|=\max_i\|P(\lambda_i)\|\), which is at most \(\max_{W(A)}\|P\|\). For normal matrices the constant is \(1\).

3. Sharpness

Let \(A=\bigl(\begin{smallmatrix}0&2\\0&0\end{smallmatrix}\bigr)\), \(m=1\) and \(P(z)=z\). Then \(\|P[A]\|=\|A\|=2\). For a unit vector \(u\), \(u^*Au=2\bar u_1u_2\) and \(|2\bar u_1u_2|\le|u_1|^2+|u_2|^2=1\), with equality when \(|u_1|=|u_2|=1/\sqrt2\). Varying the phases and the moduli, \(W(A)\) is the closed unit disc, and \(\max_{W(A)}|z|=1\). So the ratio \(2\) is attained, and no smaller constant works in Theorem A.

4. The plan of the proof

The proof replaces \(W(A)\) by a slightly larger convex domain \(\Omega\) with a smooth boundary and proves \(\|F[A]\|\le2\) whenever \(\|F(z)\|\le1\) on \(\partial\Omega\). For such domains there is a conformal map \(h\) from \(\{|\lambda|>1\}\) onto the exterior of \(\Omega\), extending across the unit circle. The expansion \[ \frac{\lambda h'(\lambda)}{h(\lambda)-z}=\sum_{k\ge0}b_k(z)\lambda^{-k} \] defines polynomials \(b_k\) of degree \(k\), the Faber polynomials, whose values on \(\partial\Omega=h(\{|\mu|=1\})\) are \(\mu^k\) plus negative powers of \(\mu\). Convexity makes a certain kernel on the torus nonnegative, and this gives a two-sided estimate of the boundary norm of \(G=\sum_kb_kC_k\) by the coefficients \(C_k\). Substituting \(A\) gives \(\lambda h'(\lambda)(h(\lambda)I-A)^{-1}=\sum_kb_k(A)\lambda^{-k}\), whose Hermitian part is positive semidefinite on the circle because \(W(A)\subset\Omega\). Testing with the singular vectors of \(F[A]\), with products \(FG\) and \(GF\) in both orders, and comparing a positive \(2\times2\) block matrix gives the constant \(2\). A limit over shrinking domains reaches \(W(A)\) itself.

5. Exercises

1.1 (easy). Show that the numerical range of \(\bigl(\begin{smallmatrix}a&b\\0&a\end{smallmatrix}\bigr)\) is the closed disc with centre \(a\) and radius \(|b|/2\).

1.2 (easy). Give \(F\), \(G\) with coefficients in \(M_2(\mathbf C)\) and \(A=I_1\) (so \(n=1\)) with \(F[A]G[A]\ne G[A]F[A]\).

1.3 (medium). Show that for a normal matrix \(W(A)\) is the convex hull of the eigenvalues.

1.4 (medium). Show that \(\|A\|\le2\max_{z\in W(A)}|z|\) for every \(A\in M_n(\mathbf C)\), using \(A=H+iJ\) with Hermitian \(H=(A+A^*)/2\) and \(J=(A-A^*)/2i\). This is Theorem A for \(P(z)=z\).

6. Solutions

1.1. Subtracting \(aI\) reduces to \(a=0\). For a unit vector \(u\), \(u^*Au=b\,\bar u_1u_2\), and \(|\bar u_1u_2|\) takes every value in \([0,\frac12]\) with every phase.

1.2. For \(n=1\) and \(A=1\), \(F[A]=\sum_iD_i\) and \(G[A]=\sum_jE_j\). Take \(F(z)=\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)\) and \(G(z)=\bigl(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\bigr)\), constant polynomials whose products in the two orders differ.

1.3. With \(A=V\operatorname{diag}(\lambda_i)V^*\) and \(v=V^*u\), \(u^*Au=\sum_i\lambda_i|v_i|^2\), a convex combination with arbitrary weights \(|v_i|^2\).

1.4. \(u^*Hu=\operatorname{Re}(u^*Au)\) and \(u^*Ju=\operatorname{Im}(u^*Au)\), so \(\|H\|,\|J\|\le\max_{W(A)}|z|\), and \(\|A\|\le\|H\|+\|J\|\).

References