Dilation and Functional Models for Pure $\mathbfΘ_n$-Contractions and the von Neumann Inequality on Distinguished Varieties in $\mathbfΘ_n$

Abstract: In this paper, we introduce the notion of a distinguished variety in the domain $\mathbfΘ_n$. One of the main results of the paper is a determinantal representation for every distinguished variety in $\mathbfΘ_n$. We also show that the closure of every distinguished variety is polynomially convex. Furthermore, we obtain a dilation and a functional model for a class of pure $\mathbfΘ_n$-contractions. Finally, we show that for a $\mathbfΘ_n$-contraction $\mathbf{T}=(T_1,\dots,T_n)$ such that $T_n<sup>*$ is a pure contraction, there exists an algebraic variety in $\mathbfΘ_n$ for which the von Neumann inequality holds on the intersection of the closure of the variety with the distinguished boundary of $\mathbfΘ_n$.

• The paper constructs minimal pure Θₙ-isometric dilations and Sz.-Nagy–Foiaş-type functional models using fundamental operator coefficients under symmetry and Γₙ₋₁-contraction hypotheses.
• The paper proves that distinguished varieties in Θₙ admit determinantal representations, are polynomially convex, and arise as images of distinguished varieties in the polydisc.
• The paper establishes a matricial von Neumann inequality on the distinguished boundary of a contraction-specific variety, while identifying uniqueness of the fundamental operators as a central open problem.

The domain $\mathbf{\Theta}_n$ and the operator classes it induces

The symmetrized polydisc $G_n$ and its closure $\Gamma_n$ have been the subject of an extensive operator-theoretic program, including canonical models, functional calculi, and dilation theorems for $\Gamma_n$-contractions. This paper works in the larger family of generalized symmetrized domains $\mathbf{\Theta}_n$, defined via the polynomial map $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$ with $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$ for $1\le i\le n-1$ and $\theta_n(z)=(z_1\cdots z_n)^q$, where $p\mid m$ and $G_n$0. The symmetrized polydisc is recovered at $G_n$1. Membership in the closure $G_n$2 is characterized by the condition that every zero of the polynomial

$G_n$3

lies in $G_n$4. The paper studies commuting tuples $G_n$5 for which $G_n$6 is a spectral set — the $G_n$7-contractions — together with the associated classes of $G_n$8-unitaries, isometries, co-isometries, and pure isometries (the last defined by purity of the distinguished component $G_n$9).

A recurring structural device is the system of fundamental equations

$\Gamma_n$0

for $\Gamma_n$1, generalizing the fundamental equations for $\Gamma_n$2-contractions. The paper is candid that a basic question remains unresolved here: whether the tuples $\Gamma_n$3 are uniquely determined by $\Gamma_n$4. Uniqueness is known for the $\Gamma_n$5 case, but for general $\Gamma_n$6-contractions the problem is open, and the dilation and model results below must therefore be stated relative to a chosen solution set satisfying additional compatibility conditions rather than in terms of canonical data.

Determinantal representation of distinguished varieties

The paper introduces distinguished varieties in $\Gamma_n$7: sets $\Gamma_n$8 for an algebraic variety $\Gamma_n$9 whose closure meets $\Gamma_n$0 only along the distinguished boundary $\Gamma_n$1. The technical foundation is a measure-theoretic lemma, obtained by pulling back the Agler–McCarthy construction on compact Riemann surfaces through a desingularization of the projective closure of the defining curve: for every one-dimensional distinguished variety $\Gamma_n$2 there is a finite measure $\Gamma_n$3 on $\Gamma_n$4 such that every point of $\Gamma_n$5 is a bounded point evaluation for $\Gamma_n$6 and the evaluation vectors span a dense subspace. Two consequences follow: the coordinate multiplication tuple $\Gamma_n$7 on $\Gamma_n$8 is a pure $\Gamma_n$9-isometry, and points of $\mathbf{\Theta}_n$0 are characterized as conjugates of joint eigenvalues of $\mathbf{\Theta}_n$1 — using, in the latter step, the polynomial convexity of $\mathbf{\Theta}_n$2.

The central geometric result is a determinantal representation theorem. Given square matrices $\mathbf{\Theta}_n$3 of common order satisfying the symmetry $\mathbf{\Theta}_n$4 and four compatibility conditions — pairwise commutativity of the matrix polynomials $\mathbf{\Theta}_n$5, a $\mathbf{\Theta}_n$6-interpolation condition on joint eigenvector quadratic forms, the regular sequence condition on $\mathbf{\Theta}_n$7, and irreducibility — the set

$\mathbf{\Theta}_n$8

is a distinguished variety contained in an affine algebraic curve that is a set-theoretic complete intersection. Conversely, every distinguished variety admits such a representation. The converse direction proceeds by a dimension argument showing the defining variety must be a curve (a positive-dimensional fibre would force an exit point of $\mathbf{\Theta}_n$9 through $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$0), followed by finite-dimensionality of $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$1 and the classification of pure $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$2-isometries to produce the matrices $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$3.

Two further structural results sharpen the picture. First, distinguished varieties in $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$4 are exactly images under $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$5 of distinguished varieties in $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$6; this reduces the geometry of the generalized domains to the Agler–McCarthy bidisc/polydisc picture through the covering map. Second, the closure of every distinguished variety in $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$7 is polynomially convex, proved by separating points outside $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$8 (using polynomial convexity of $\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)$9) from points inside via a determinantal polynomial vanishing identically on $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$0. Polynomial convexity is used essentially in the spectral characterization of $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$1 via joint eigenvalues, so these results are mutually reinforcing rather than merely corollaries.

Minimal isometric dilation and functional model

The dilation theory is developed for pure $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$2-contractions satisfying the compatibility condition

$\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$3

together with the symmetry $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$4. The unitary $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$5 given by $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$6 is an isometry precisely because $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$7 is pure; this is the only point where purity enters the construction, and the results do not extend to non-pure tuples without modification.

Under the additional hypothesis that $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$8 is a $\theta_i(z)=s_i(z_1^m,\dots,z_n^m)$9-contraction for every $1\le i\le n-1$0 — where $1\le i\le n-1$1 — the tuple

$1\le i\le n-1$2

is a $1\le i\le n-1$3-isometry on $1\le i\le n-1$4, and the intertwining relations $1\le i\le n-1$5 hold for all $1\le i\le n-1$6. Minimality of the dilation space follows from density of $1\le i\le n-1$7. The proof that $1\le i\le n-1$8 is a $1\le i\le n-1$9-isometry invokes a structure theorem for $\theta_n(z)=(z_1\cdots z_n)^q$0-isometries from prior work on these domains, so the dilation theorem is conditional on both the fundamental-equation solution and the $\theta_n(z)=(z_1\cdots z_n)^q$1-contraction hypothesis on the coefficient polynomials.

The functional model follows by combining the dilation with the Sz.-Nagy–Foiaş identity $\theta_n(z)=(z_1\cdots z_n)^q$2. Since purity of $\theta_n(z)=(z_1\cdots z_n)^q$3 makes $\theta_n(z)=(z_1\cdots z_n)^q$4 inner, $\theta_n(z)=(z_1\cdots z_n)^q$5 coincides with the model space $\theta_n(z)=(z_1\cdots z_n)^q$6, and $\theta_n(z)=(z_1\cdots z_n)^q$7 becomes a unitary onto it. Consequently $\theta_n(z)=(z_1\cdots z_n)^q$8 is unitarily equivalent to the compression of $\theta_n(z)=(z_1\cdots z_n)^q$9 (and $p\mid m$0 to the compression of $p\mid m$1) to $p\mid m$2 — a direct analogue of the classical model for pure contractions, with the fundamental operator data replacing the single characteristic function.

Matricial von Neumann inequality on distinguished varieties

The final main theorem combines the preceding strands. Let $p\mid m$3 be a $p\mid m$4-contraction with $p\mid m$5 pure, whose coefficient operators $p\mid m$6 on the finite-dimensional space $p\mid m$7 satisfy the commutator conditions, the fundamental-equation condition, the symmetry, the $p\mid m$8-contraction hypothesis on $p\mid m$9, and determine a distinguished variety $G_n$00 via the determinantal representation. Then for every matrix-valued polynomial $G_n$01,

$G_n$02

and the same bound holds for $G_n$03. The proof shows that $G_n$04 is a $G_n$05-unitary on $G_n$06 whose boundary spectrum is contained in $G_n$07; the adjoint tuple then serves as a co-isometric dilation of $G_n$08, and the inequality reduces to the spectral bound for the normal tuple via the spectral mapping theorem, with the conjugation identity between $G_n$09 and its adjoint variety $G_n$10 handling the adjoint case.

Two implications deserve emphasis. First, the inequality is a boundary-only estimate: the supremum is taken over the intersection of the variety's closure with the distinguished boundary, not over all of $G_n$11, so it is strictly sharper than the defining spectral-set inequality for $G_n$12-contractions and constitutes a genuine von Neumann-type inequality on a subvariety. Second, the proof establishes as a by-product that both $G_n$13 and $G_n$14 admit normal boundary dilations with joint spectra contained in $G_n$15 — a boundary dilation theorem in the spirit of Agler–McCarthy's work on the bidisc, here relativized to the distinguished variety determined by the contraction's own model data.

Limitations and open questions

The main results carry substantive hypotheses that delimit their scope. The dilation and model theorems apply only to pure $G_n$16-contractions satisfying the fundamental equations with a coefficient family that additionally satisfies the symmetry and the pointwise $G_n$17-contraction condition on $G_n$18; the paper does not establish that every $G_n$19-contraction admits such a family, nor that the dilation is unique up to unitary equivalence. The uniqueness question for the fundamental operator tuples $G_n$20 is explicitly stated as open. The von Neumann inequality requires finite-dimensionality of $G_n$21 (so that the coefficients are matrices) and that the resulting determinantal data satisfy the regular sequence, irreducibility, and interpolation conditions of the representation theorem; whether these conditions are automatic for the fundamental tuples of an arbitrary pure $G_n$22-contraction is not addressed. Finally, a canonical functional model in the sense of Sz.-Nagy–Foiaş — one built from intrinsic invariants of $G_n$23 alone rather than from a chosen solution of the fundamental equations — remains outside the scope of the paper.

Conclusion

This paper extends the distinguished-variety program of Agler, McCarthy, and Pal to the generalized symmetrized domains $G_n$24, proving a determinantal representation for such varieties, their polynomial convexity, and their equivalence to images of distinguished varieties in $G_n$25 under the covering map. On the operator-theoretic side it constructs minimal pure $G_n$26-isometric dilations and Sz.-Nagy–Foiaş-type functional models for a class of pure $G_n$27-contractions, and derives a matricial von Neumann inequality on distinguished varieties together with normal boundary dilations for the tuple and its adjoint. The theory is conditional on compatibility hypotheses on the fundamental operator tuples, and the uniqueness of those tuples stands as the principal open problem gating a fully canonical theory.