1 · The essentials
The question is how large a polynomial expression in a matrix or operator can become. The numerical range is a region of the complex plane associated with that matrix or operator. The result compares the size of the expression with the largest value of the same polynomial on this region. It says that the expression is at most twice as large, even when the polynomial’s coefficients are themselves finite matrices. The factor two cannot be reduced for the result as a whole, and it does not grow with the size of those coefficient matrices. The main manuscript treats finite matrices, while its companion extends the inequality to bounded operators on complex Hilbert spaces, including infinite-dimensional ones. Both manuscripts are marked as Lean-formalized in the supplied source.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine checking a recipe’s output across a whole region of test conditions and then using it in a more complicated machine. The inequality says that the machine’s output stays within twice the largest test output; the image does not explain why the numerical range is the right testing region.
Why it matters
The source presents this family as resolving the complete Crouzeix conjecture, with a sharp factor of two. The same factor works for every finite coefficient-matrix size and, in the companion manuscript, in infinite-dimensional spaces without a separability assumption. The companion also connects this estimate with a change of coordinates and a positive boundary representation under additional geometric hypotheses.
2 · Going further
The precise inequality is , where is a bounded operator on a complex Hilbert space, is its numerical range, and is a finite matrix-valued polynomial. Here denotes evaluation at the operator rather than at a complex number. The constant is sharp and independent of the coefficient-matrix size. The source identifies this as the complete Crouzeix conjecture.
The main manuscript, “A direct proof of the complete Crouzeix inequality,” gives a direct treatment in all finite base and coefficient dimensions. Its stated scope includes matrices whose numerical ranges are points or line segments. The abstract does not describe the steps of the direct argument, so no more specific mechanism can be extracted from the supplied text.
The companion manuscript extends the setting to every bounded operator on a complex Hilbert space, with no separability assumption. It states that is a complete -spectral set and extends the bound to finite matrix-valued functions holomorphic near that closure. This complements the main manuscript’s finite-dimensional formulation rather than merely changing the constant.
The companion also supplies a structural perspective under extra assumptions: for a finite matrix and a bounded convex domain containing its numerical range, with regular real-analytic Jordan boundary, an optimal similarity making the conformal disk image contractive is attained with condition number at most two. For the resulting similar matrix, a single continuous positive boundary density, whose total mass is the identity, represents evaluation of every matrix-valued function holomorphic near the closed domain. The abstract states these connections but does not specify how they enter the argument for the general inequality.
Useful notions
- Complex numbers and the complex plane
- Matrices and polynomial evaluation
- Norms as measures of size
- Hilbert spaces and bounded operators
- Holomorphic functions
What was known before / what is new
Before
The source identifies the target statement as the complete Crouzeix conjecture. It supplies no earlier bounds, chronology, or account of previous approaches, so no more detailed historical comparison is available here.
New
The main manuscript establishes the sharp constant-two inequality for matrix-valued polynomials in all finite base and coefficient dimensions, including point and line-segment numerical ranges. The companion states the bounded-operator version without separability and the extension to functions holomorphic near the closed numerical range. It also gives the optimal-similarity and common-positive-boundary-representation results under its stated finite-dimensional and domain hypotheses.
Limits and assumptions
The general inequality concerns bounded operators on complex Hilbert spaces and finite matrix-valued polynomials; the holomorphic extension requires functions defined near the closure of the numerical range. The similarity and boundary-density statements require a finite matrix and a bounded convex containing domain with regular real-analytic Jordan boundary. Sharpness means that two cannot be reduced uniformly, not that every individual case reaches it. Both manuscripts are labeled Lean-formalized, with the main manuscript’s stated scope being finite base and coefficient dimensions; the supplied text does not give a claim-by-claim inventory of the companion’s formalization. No remaining open problem is specified in the source.
Mini glossary
- Numerical range —
- The region associated with an operator on which the theorem measures polynomial values.
- Matrix-valued polynomial —
- A polynomial whose coefficients are finite matrices, so its value at a complex number is a matrix.
- Sharp constant —
- A constant that cannot be replaced by a smaller one while keeping the stated inequality valid in every allowed case.
- Complete 2-spectral set —
- In the companion’s formulation, a set controlling operator evaluation by a factor of two uniformly across all finite coefficient-matrix sizes.
- Common positive boundary representation —
- A representation of function evaluation using one continuous positive boundary density of total mass the identity for all the functions specified in the companion theorem.
Keywords : Complete Crouzeix conjecture · Numerical range · Matrix-valued polynomials
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has , where is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A direct proof of the complete Crouzeix inequality
26/09/2026We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.
The complete Crouzeix theorem: optimal similarity and a common positive boundary representation
23/09/2026We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete 2-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure. For a finite matrix and a bounded convex domain containing its numerical range with regular real-analytic Jordan boundary, the optimal similarity making its conformal disk image contractive is attained with condition number at most two. For the similar matrix, one continuous positive boundary density of mass the identity represents the evaluation of every matrix-valued function holomorphic near the closed domain.