---
title: Takagi–Pfister Skew Eigenvectors
url: https://www.emergentmind.com/topics/takagi-pfister-skew-eigenvectors
type: topic
---

# Takagi–Pfister Skew Eigenvectors

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Takagi–Pfister skew eigenvectors are extremal solutions of an antilinear eigenvalue problem associated with truncated Toeplitz operators and a reflection conjugation on polynomial subspaces of Hardy spaces over the disk or polydisk. In the framework developed in "The multivariate Herglotz-Nevanlinna class: Rational approximation" [2509.15668], they generalize the one-variable Takagi variational principle to several complex variables through Pfister’s extension, and they generate rational inner approximants of Padé type for bounded holomorphic functions on $D^d$. The same framework connects these extremals to Montessus de Ballore-type convergence on polydisks, to constructive Cayley rational inner interpolation in the Agler–Herglotz–Nevanlinna class, and to semialgebraic descriptions of finite Taylor coefficient sets. A central unresolved issue is the distribution of poles arising from the reflected denominator polynomials in several variables [2509.15668].

## 1. Variational and operator-theoretic definition

For a complex symmetric matrix $S$ with $S = S^T$, Takagi’s factorization yields a decomposition $S = U D U^T$, where $U$ is unitary and $D$ is diagonal with nonnegative entries. Equivalently, the Takagi singular values are obtained from the antilinear eigenvalue equation
$$
S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.
$$
In the analytic setting, this matrix model is replaced by a truncated Toeplitz/Hankel construction associated with a bounded holomorphic function [2509.15668].

On the polydisk $D^d$, for each multi-index $n \in \mathbb{N}_0^d$, one considers the polynomial space
$$
\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},
$$
the orthogonal projection $P_n$ onto $\mathbb{C}_n[z]$ with respect to the Hardy $L^2(\mathbb{T}^d)$ inner product, the truncated Toeplitz operator
$$
T_n(q) = P_n(fq),
$$
and the conjugation
$$
C_n(q)(z) = q^*(z) := z^n \overline{q(1/\overline{z})}.
$$
These satisfy the symmetry relation
$$
C_n T_n C_n = T_n^*.
$$
The Takagi–Pfister skew-eigenvector equation is then
$$
T_n C_n(q) = \sigma q, \qquad \sigma \ge 0,\qquad q \in \mathbb{C}_n[z]\setminus\{0\}.
$$
With the $L^2(\mathbb{T}^d)$ inner product $\langle \cdot,\cdot\rangle$, this is equivalent to
$$
\langle T_n C_n(q), q\rangle = \sigma \langle q,q\rangle.
$$

The extremal value is the operator norm,
$$
\|T_n\| = \sup\{\sigma \ge 0:\exists q\neq 0 \text{ with } T_n C_n(q)=\sigma q\}=: \sigma_n,
$$
and the paper states that $\lim_{\min n \to \infty}\sigma_n = \|f\|_\infty$ [2509.15668]. In this sense, skew eigenvectors are extremizers for a Hardy-space variational principle.

Pfister’s multivariate extension is expressed through the maximization formula
$$
\sigma_{2n} = \max\left\{\operatorname{Re}\langle T_{2n}C_{2n}(q),q\rangle : q\in \mathbb{C}_{2n}[z],\ \|q\|_2=1\right\}
$$
and equivalently
$$
\sigma_{2n} = \max\left\{\operatorname{Re}\int_{\mathbb{T}^d} f(\zeta)\,q(\zeta)^2\,dm(\zeta): q\in \mathrm{Trig}_n,\ \|q\|_2=1\right\}.
$$
Here $\mathrm{Trig}_n$ denotes the trigonometric polynomials with Fourier support in $\prod_{j=1}^d[-n_j,n_j]\cap \mathbb{Z}^d$. This places the skew-eigenvector problem in a Hankel/kernel formulation on $H^2(D^d)$ [2509.15668].

## 2. Rational inner functions generated by skew eigenvectors

Given an extremal eigenpair $(\sigma_n,q_n)$ for $T_n C_n$, the decomposition
$$
f\,C_n(q_n) = \sigma_n q_n + r_n
$$
holds with remainder $r_n$ orthogonal to $\mathbb{C}_n[z]$. If $q_n^*(0)\neq 0$, then the Taylor expansions at $0$ of $f$ and of
$$
R_n(z) = \sigma_n\,\frac{q_n(z)}{q_n^*(z)}
$$
match for all monomials up to multidegree $n$ [2509.15668].

This produces a Padé-type approximation scheme. The paper identifies the matching condition as a multivariate $[n/n]$-type matching of the formal series: the difference
$$
f-\sigma_n q_n/q_n^*
$$
has no terms up to multidegree $n$. The normality condition is supplied by choosing $q_n$ from the extremal variational problem. The boundary behavior is inner because $q_n/q_n^*$ is of polynomial/reflection form and therefore unimodular almost everywhere on $\mathbb{T}^d$ [2509.15668].

A related notation is
$$
h_n := z^n q_n,\qquad h_n^* := z^n\overline{q_n},
$$
so that $h_n^* = C_{2n}(h_n)$. In these terms,
$$
R_n(z) = \frac{h_n(z)}{h_n^*(z)} = \frac{q_n(z)}{q_n^*(z)},
$$
and
$$
f = \sigma_{2n}R_n + \frac{r_{2n}}{h_{2n}^*}.
$$
If $\sigma_{2n}=1$ and $r_{2n}=0$, then $f=q/q^*$ is itself rational inner [2509.15668].

The construction generalizes the one-variable relation between Takagi extremals and finite Blaschke products. A plausible implication is that the multivariate theory provides a structured inner rational approximation paradigm in which extremality, Taylor matching, and boundary unimodularity are encoded by the same skew-eigenvector data.

## 3. Montessus-type convergence and moving pole sets

The paper proves a Montessus de Ballore-type convergence theorem on the polydisk for the rational inner functions arising from Takagi–Pfister extremals [2509.15668]. Let $f:D^d\to \overline{D}$ be holomorphic with $\|f\|_\infty = 1$, and for each multi-index $n$ let $q_{2n}\in \mathrm{Trig}_n$ be an optimal solution with $\|q_{2n}\|_2=1$ attaining
$$
\sigma_{2n} = \max_{q\in \mathrm{Trig}_n,\ \|q\|_2=1}\operatorname{Re}\int_{\mathbb{T}^d} f(\zeta)\,q(\zeta)^2\,dm(\zeta).
$$
Assume that the zero function is not a weak limit point of the sequence $z^n\,\overline{q_{2n}}$ in $H^2(D^d)$ as $\min n\to\infty$. Then there exist an analytic hypersurface $X\subset D^d$ and a subsequence $n(k)$, with $\min n(k)\to\infty$, such that the rational functions
$$
R_k(z) = \frac{q_{2n(k)}(z)}{\overline{q_{2n(k)}(z)}}
$$
are unimodular almost everywhere on $\mathbb{T}^d$, their poles accumulate on $X$, and $R_k\to f$ uniformly on every compact subset $K\subset D^d\setminus X$ [2509.15668].

The associated remainder admits the estimate
$$
\|r_{2n}\|_2 \le (\|f\|_\infty-\sigma_{2n})\|q_{2n}\|_2 = 1-\sigma_{2n}.
$$
On compact subsets of $\delta D^d$ with $0<\delta<1$ and $m:=\min n$,
$$
|r_{2n}(z)| \le (\|f\|_\infty-\sigma_{2n})\|q_{2n}\|_2 \frac{\delta^m}{(1-\delta^2)^{d/2}}
= (1-\sigma_{2n})\frac{\delta^m}{(1-\delta^2)^{d/2}}.
$$
Because $\sigma_{2n}\to 1$ and $m\to\infty$, this decays uniformly on compact subsets avoiding zeros of $h_{2n}^*$ [2509.15668].

The poles of the approximants are the zeros of $h_{2n(k)}^*$ in $D^d$. Along the convergent subsequence, these poles accumulate on
$$
X = \{g=0\},
$$
where $g$ is the nonzero weak limit in $H^2(D^d)$ of $h_{2n(k)}^*$. Outside $X$, the rational approximants are holomorphic and converge uniformly to $f$ [2509.15668]. This identifies the singular geometry of the denominator sequence as the main obstruction to full-domain uniform convergence in several variables.

## 4. Relation to Schur, Herglotz–Nevanlinna, and Agler classes

The skew-eigenvector framework is embedded in a broader function-theoretic setting. The Schur class on $D^d$ is
$$
\mathcal{S}(D^d)=\{f\text{ holomorphic on }D^d: |f(z)|\le 1\},
$$
and the Herglotz–Nevanlinna class is
$$
\mathcal{H}(D^d)=\{\phi\text{ holomorphic on }D^d: \operatorname{Re}\phi(z)\ge 0\}.
$$
For a commuting $d$-tuple of strict contractions $T=(T_1,\dots,T_d)$ on an infinite-dimensional separable Hilbert space $\mathcal{E}$, one defines
$$
\phi(T)=\sum_{\alpha\in \mathbb{N}_0^d} c_\alpha(\phi) T^\alpha,\qquad T^\alpha=T_1^{\alpha_1}\cdots T_d^{\alpha_d}.
$$
The Agler–Schur class $\mathcal{AS}(D^d)$ consists of holomorphic $\phi$ with $\sup_T \|\phi(T)\|<\infty$, while the Agler–Herglotz–Nevanlinna class $\mathcal{AH}(D^d)$ consists of holomorphic $\phi$ with
$$
\operatorname{Re}\phi(T)=\frac{\phi(T)+\phi(T)^*}{2}\ge 0
$$
for all such $T$. The paper notes that in $d=1$ and $d=2$ these coincide with the classical Schur and Herglotz classes, while for $d\ge 3$ they are strict subclasses [2509.15668].

For $\phi\in \mathcal{H}(D^d)$, the Korányi–Pukanszky representation gives a unique positive regular Borel measure $\nu$ on $\mathbb{T}^d$ satisfying the moment vanishing constraints
$$
\widehat{\nu}(n_1,\dots,n_d)=0
$$
unless either all $n_j\ge 0$ or all $n_j\le 0$, such that
$$
\phi(z)= i\,\operatorname{Im}\phi(0) + \int_{\mathbb{T}^d}\left[\frac{2}{\prod_{j=1}^d(1-z_j\overline{\xi_j})}-1\right]\,d\nu(\xi).
$$
This supplies the kernel representation linking positive-real-part functions to Hardy-space techniques [2509.15668].

The Cayley transform implements a bijection between Schur and Herglotz classes:
$$
F=\frac{1+B}{1-B},\qquad B=\frac{F-1}{F+1}.
$$
The same formulas connect the corresponding Agler subclasses [2509.15668]. Within the article’s framework, this transform is the mechanism through which rational inner objects on the Schur side correspond to Cayley rational inner objects on the Herglotz side.

## 5. Cayley rational inner interpolation and semialgebraic coefficient sets

A constructive theorem in the Agler–Herglotz–Nevanlinna setting states that if truncated Taylor data
$$
\{c_\beta:\beta\in \Lambda_n,\ c_0=1\}
$$
is solvable by some $\phi\in \mathcal{AH}(D^d)$, then there exists a Cayley rational inner solution of multidegree at most
$$
(|\Lambda_n|d,\dots,|\Lambda_n|d),
$$
matching $c_\beta$ for all $\beta\le n$. The solution is rational, has poles off $D^d$, and satisfies $\operatorname{Re}\phi=0$ almost everywhere on $\mathbb{T}^d$ [2509.15668].

The construction uses Woerdeman’s Agler matrix identity. One finds positive matrices $\Gamma_1,\dots,\Gamma_d$ on $\mathbb{C}^{|\Lambda_n|}$ satisfying
$$
2(EC^*+CE^*) = \sum_{j=1}^d \Gamma_j - \sum_{j=1}^d T_j \Gamma_j T_j^*,
$$
where $T_j$ encodes the shift in polynomial coordinates, $C$ stacks the data $c_\beta$, and $E$ stacks the basepoint. Extending an induced isometry to a unitary
$$
\mathcal{U}=\begin{bmatrix}U_{11}&U_{12}\\ U_{21}&U_{22}\end{bmatrix},
$$
and setting
$$
U:=U_{11}-U_{12}U_{21},\qquad V:=-U_{12},
$$
one obtains
$$
\phi(z)=1+2\,V^*U(I-\Delta(z)U)^{-1}\Delta(z)V,
$$
with
$$
\Delta(z)=\operatorname{diag}(z_1 I,\dots,z_d I).
$$
The paper states that this $\phi$ belongs to $\mathcal{AH}(D^d)$, matches the truncated Taylor data, has poles off $D^d$ since $\det(I-\Delta(z)U)\neq 0$ on $D^d$, and satisfies
$$
2\operatorname{Re}\phi(z) =
2\,V^*(I-\Delta(z)U)^{-1}[I-U\Delta(z)\Delta(z)^*U^*](I-\Delta(z)^*U^*)^{-1}V,
$$
whose radial limit on $\mathbb{T}^d$ is zero almost everywhere [2509.15668].

The same Hilbert-space proof yields a structural result for finite Taylor sections. For each $n\in \mathbb{N}_0^d$,
$$
K_n=\{(c_\alpha(\phi))_{\alpha\in \Lambda_n}: \phi\in \mathcal{AH}(D^d),\ c_0(\phi)=1\}\subset \mathbb{R}^{2|\Lambda_n|}.
$$
The paper proves that $K_n$ is semialgebraic [2509.15668]. The argument expresses positivity constraints using factorizations $\Gamma_j = X_j^*X_j$ and then applies Tarski–Seidenberg after projection onto coefficient coordinates. The same source states that $K_n$ is closed, convex, has nonempty interior, and is compact, with boundedness derived from the Korányi–Pukanszky integral representation giving $|c_\alpha(\phi)|\le 2$.

In the bidisk with $n=(1,1)$, writing the data as $(1,c_{01},c_{10},c_{11})$, the paper gives the explicit characterization
$$
(1,c_{01},c_{10},c_{11})\in K_{11}
$$
if and only if
$$
2|c_{11}-c_{10}c_{01}| + |c_{10}|^2 + |c_{01}|^2 \le 4,
$$
and
$$
|c_{10}|+|c_{01}| \le 2.
$$
It further notes that these are polynomial inequalities in the real and imaginary parts of $c_{01},c_{10},c_{11}$ [2509.15668].

## 6. Pole distribution, examples, and computational issues

The principal open problem identified in the paper is the zero set problem for extremal skew eigenvectors [2509.15668]. In several variables, an extremal eigenfunction $q$ may have its reflection $q^*$ vanish inside $D^d$, so that the rational inner function $q/q^*$ develops poles in the domain. This prevents full-domain uniform convergence even when Taylor matching and boundary unimodularity hold. The paper formulates the open question as follows: describe asymptotically, along increasing multidegrees, the pole distribution of the Takagi–Pfister rational inner interpolants $\sigma_n q_n/q_n^*$, equivalently the zero set of $h_n^*=z^n\overline{q_n}$, and identify conditions guaranteeing that poles avoid $D^d$ [2509.15668].

A concrete bidisk example illustrates the difficulty. For
$$
f(z,w)=\frac{z+w}{2},\qquad n=(1,1),
$$
the maximal eigenvalue is
$$
\sigma_{(1,1)}=\frac{1}{\sqrt{2}},
$$
with eigenfunction
$$
q(z,w)=\frac{z+w}{2}+\frac{1}{\sqrt{2}}zw,
$$
and reflected polynomial
$$
q^*(z,w)=\frac{z+w}{2}+\frac{1}{\sqrt{2}}.
$$
The zero set of $q^*$ intersects $D^2$, and $q$ and $q^*$ are relatively prime. The corresponding rational inner approximant
$$
R(z,w)=\sigma_{(1,1)}\,\frac{q(z,w)}{q^*(z,w)}
$$
matches Taylor data up to degree $(1,1)$ but has poles in $D^2$ [2509.15668].

By contrast, for tensor products $f(z,w)=g(z)h(w)$, the paper states that products of one-variable Takagi extremals yield poles outside the closed bidisk by the one-variable theorem, giving inner approximants with good pole control [2509.15668]. This suggests that separable structure may ameliorate the denominator-zero problem, although no general theorem of that form is stated beyond the given setting.

The paper also includes a one-variable example. For $f(z)=z/2 \in \mathcal{S}(\mathbb{D})$ and $n=1$, writing $q(z)=a+bz$, one computes
$$
T_1(q)=\frac{a}{2}z,\qquad C_1(q)(z)=\overline{a}z+\overline{b},
$$
so
$$
T_1C_1(q)=\frac{\overline{a}}{2}z.
$$
The skew-eigenvector equation
$$
T_1C_1(q)=\sigma q
$$
forces
$$
\sigma a=0,\qquad \sigma b=\frac{\overline{a}}{2}.
$$
The paper states that for $n=2$ or larger, nontrivial eigenpairs appear and the approximant $R_n=\sigma_n q/q^*$ matches Taylor coefficients up to degree $n$. In one variable, if $\sigma_n=1$ at some $n$, then $f$ is a finite Blaschke product; otherwise $\sigma_n\uparrow 1$ and $R_n\to f$ uniformly on compact sets [2509.15668].

The computational procedure described in the paper is finite-dimensional. One inputs $f$ and a multidegree $n$, forms the truncated Toeplitz matrix of $T_n$ in the monomial basis, defines $C_n$ by reflection, and constructs the antilinear operator
$$
A_n := T_n C_n.
$$
One then computes its Takagi singular values and skew eigenvectors by solving
$$
A_n\,\overline{v} = \sigma v,
$$
selects the maximal $\sigma_n$ and its corresponding $q_n$, and forms
$$
R_n=\sigma_n\,q_n/q_n^*.
$$
Pole verification is reduced to locating the zeros of $q_n^*$ in $D^d$; in the Agler/Herglotz setting, one instead solves the semidefinite feasibility problem for the matrices $\Gamma_j$ and constructs the realization
$$
\phi(z)=1+2V^*U(I-\Delta(z)U)^{-1}\Delta(z)V
$$
[2509.15668].

## 7. Conceptual placement within interpolation and Padé theory

The article situates the skew-eigenvector method at the intersection of several classical theories [2509.15668]. First, it is described as a Hilbert-space realization of Carathéodory–Fejér interpolation and Nevanlinna–Pick theory. In one variable, solvability is encoded by contractivity or positivity of Toeplitz and Hankel matrices formed from the data; in several variables, the Agler framework provides positivity certificates and realizations through operator-theoretic decompositions.

Second, the Takagi–Pfister eigenproblem
$$
T_n C_n(q)=\sigma q
$$
is the Hardy-space analogue of the matrix equation
$$
S\overline{u}=\sigma u
$$
from Takagi factorization. The conjugation $C_n$ plays the role of the symmetric reflection, and the extremal eigenpairs maximize the corresponding Hankel form [2509.15668].

Third, the resulting rational functions are explicitly Padé-type. They match finite Taylor sections while controlling denominator degree and preserving an inner boundary condition. The paper characterizes them as multivariate $[n/n]$ inner rational approximants, with a Montessus-type convergence theorem valid away from a moving pole set [2509.15668].

Finally, the Agler-side realization theory complements the skew-eigenvector construction by giving a separate but connected route to rational interpolation with positivity constraints. The Cayley transform
$$
F=\frac{1+B}{1-B},\qquad B=\frac{F-1}{F+1}
$$
converts between Schur and Herglotz–Nevanlinna formulations, while Woerdeman’s positive matrix identity yields realizable finite sections and semialgebraic coefficient bodies. Taken together, these ingredients form what the source presents as a cohesive framework for rational approximation on polydisks, centered on Takagi–Pfister extremals and their denominator geometry [2509.15668].

Source: https://www.emergentmind.com/topics/takagi-pfister-skew-eigenvectors