Takagi–Pfister Skew Eigenvectors

• Takagi–Pfister skew eigenvectors are defined as extremal solutions of an antilinear eigenvalue problem using truncated Toeplitz operators and reflection conjugation on polynomial subspaces of Hardy spaces.
• They generate rational inner approximants of Padé type for bounded holomorphic functions, linking to Montessus de Ballore-type convergence on polydisks.
• The framework connects to Agler–Herglotz–Nevanlinna interpolation, providing semialgebraic coefficient characterizations while addressing pole distribution challenges in several variables.

Searching arXiv for the specified paper and topic to ground the article in the cited source. 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" (Bhowmik et al., 19 Sep 2025), 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 (Bhowmik et al., 19 Sep 2025).

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 (Bhowmik et al., 19 Sep 2025).

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 $S$0 onto $S$1 with respect to the Hardy $S$2 inner product, the truncated Toeplitz operator

$S$3

and the conjugation

$S$4

These satisfy the symmetry relation

$S$5

The Takagi–Pfister skew-eigenvector equation is then

$S$6

With the $S$7 inner product $S$8, this is equivalent to

$S$9

The extremal value is the operator norm,

$S = S^T$0

and the paper states that $S = S^T$1 (Bhowmik et al., 19 Sep 2025). In this sense, skew eigenvectors are extremizers for a Hardy-space variational principle.

Pfister’s multivariate extension is expressed through the maximization formula

$S = S^T$2

and equivalently

$S = S^T$3

Here $S = S^T$4 denotes the trigonometric polynomials with Fourier support in $S = S^T$5. This places the skew-eigenvector problem in a Hankel/kernel formulation on $S = S^T$6 (Bhowmik et al., 19 Sep 2025).

2. Rational inner functions generated by skew eigenvectors

Given an extremal eigenpair $S = S^T$7 for $S = S^T$8, the decomposition

$S = S^T$9

holds with remainder $S = U D U^T$0 orthogonal to $S = U D U^T$1. If $S = U D U^T$2, then the Taylor expansions at $S = U D U^T$3 of $S = U D U^T$4 and of

$S = U D U^T$5

match for all monomials up to multidegree $S = U D U^T$6 (Bhowmik et al., 19 Sep 2025).

This produces a Padé-type approximation scheme. The paper identifies the matching condition as a multivariate $S = U D U^T$7-type matching of the formal series: the difference

$S = U D U^T$8

has no terms up to multidegree $S = U D U^T$9. The normality condition is supplied by choosing $U$0 from the extremal variational problem. The boundary behavior is inner because $U$1 is of polynomial/reflection form and therefore unimodular almost everywhere on $U$2 (Bhowmik et al., 19 Sep 2025).

A related notation is

$U$3

so that $U$4. In these terms,

$U$5

and

$U$6

If $U$7 and $U$8, then $U$9 is itself rational inner (Bhowmik et al., 19 Sep 2025).

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 (Bhowmik et al., 19 Sep 2025). Let $D$0 be holomorphic with $D$1, and for each multi-index $D$2 let $D$3 be an optimal solution with $D$4 attaining

$D$5

Assume that the zero function is not a weak limit point of the sequence $D$6 in $D$7 as $D$8. Then there exist an analytic hypersurface $D$9 and a subsequence $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$0, with $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$1, such that the rational functions

$S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$2

are unimodular almost everywhere on $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$3, their poles accumulate on $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$4, and $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$5 uniformly on every compact subset $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$6 (Bhowmik et al., 19 Sep 2025).

The associated remainder admits the estimate

$S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$7

On compact subsets of $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$8 with $S\,\overline{u} = \sigma u, \qquad \sigma \ge 0.$9 and $D^d$0,

$D^d$1

Because $D^d$2 and $D^d$3, this decays uniformly on compact subsets avoiding zeros of $D^d$4 (Bhowmik et al., 19 Sep 2025).

The poles of the approximants are the zeros of $D^d$5 in $D^d$6. Along the convergent subsequence, these poles accumulate on

$D^d$7

where $D^d$8 is the nonzero weak limit in $D^d$9 of $n \in \mathbb{N}_0^d$0. Outside $n \in \mathbb{N}_0^d$1, the rational approximants are holomorphic and converge uniformly to $n \in \mathbb{N}_0^d$2 (Bhowmik et al., 19 Sep 2025). 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 $n \in \mathbb{N}_0^d$3 is

$n \in \mathbb{N}_0^d$4

and the Herglotz–Nevanlinna class is

$n \in \mathbb{N}_0^d$5

For a commuting $n \in \mathbb{N}_0^d$6-tuple of strict contractions $n \in \mathbb{N}_0^d$7 on an infinite-dimensional separable Hilbert space $n \in \mathbb{N}_0^d$8, one defines

$n \in \mathbb{N}_0^d$9

The Agler–Schur class $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$0 consists of holomorphic $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$1 with $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$2, while the Agler–Herglotz–Nevanlinna class $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$3 consists of holomorphic $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$4 with

$$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$5

for all such $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$6. The paper notes that in $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$7 and $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$8 these coincide with the classical Schur and Herglotz classes, while for $$\mathbb{C}_n[z] := \{g(z)\text{ of multidegree }\le n\},$$9 they are strict subclasses (Bhowmik et al., 19 Sep 2025).

For $S$00, the Korányi–Pukanszky representation gives a unique positive regular Borel measure $S$01 on $S$02 satisfying the moment vanishing constraints

$S$03

unless either all $S$04 or all $S$05, such that

$S$06

This supplies the kernel representation linking positive-real-part functions to Hardy-space techniques (Bhowmik et al., 19 Sep 2025).

The Cayley transform implements a bijection between Schur and Herglotz classes:

$S$07

The same formulas connect the corresponding Agler subclasses (Bhowmik et al., 19 Sep 2025). 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

$S$08

is solvable by some $S$09, then there exists a Cayley rational inner solution of multidegree at most

$S$10

matching $S$11 for all $S$12. The solution is rational, has poles off $S$13, and satisfies $S$14 almost everywhere on $S$15 (Bhowmik et al., 19 Sep 2025).

The construction uses Woerdeman’s Agler matrix identity. One finds positive matrices $S$16 on $S$17 satisfying

$S$18

where $S$19 encodes the shift in polynomial coordinates, $S$20 stacks the data $S$21, and $S$22 stacks the basepoint. Extending an induced isometry to a unitary

$S$23

and setting

$S$24

one obtains

$S$25

with

$S$26

The paper states that this $S$27 belongs to $S$28, matches the truncated Taylor data, has poles off $S$29 since $S$30 on $S$31, and satisfies

$S$32

whose radial limit on $S$33 is zero almost everywhere (Bhowmik et al., 19 Sep 2025).

The same Hilbert-space proof yields a structural result for finite Taylor sections. For each $S$34,

$S$35

The paper proves that $S$36 is semialgebraic (Bhowmik et al., 19 Sep 2025). The argument expresses positivity constraints using factorizations $S$37 and then applies Tarski–Seidenberg after projection onto coefficient coordinates. The same source states that $S$38 is closed, convex, has nonempty interior, and is compact, with boundedness derived from the Korányi–Pukanszky integral representation giving $S$39.

In the bidisk with $S$40, writing the data as $S$41, the paper gives the explicit characterization

$S$42

if and only if

$S$43

and

$S$44

It further notes that these are polynomial inequalities in the real and imaginary parts of $S$45 (Bhowmik et al., 19 Sep 2025).

6. Pole distribution, examples, and computational issues

The principal open problem identified in the paper is the zero set problem for extremal skew eigenvectors (Bhowmik et al., 19 Sep 2025). In several variables, an extremal eigenfunction $S$46 may have its reflection $S$47 vanish inside $S$48, so that the rational inner function $S$49 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 $S$50, equivalently the zero set of $S$51, and identify conditions guaranteeing that poles avoid $S$52 (Bhowmik et al., 19 Sep 2025).

A concrete bidisk example illustrates the difficulty. For

$S$53

the maximal eigenvalue is

$S$54

with eigenfunction

$S$55

and reflected polynomial

$S$56

The zero set of $S$57 intersects $S$58, and $S$59 and $S$60 are relatively prime. The corresponding rational inner approximant

$S$61

matches Taylor data up to degree $S$62 but has poles in $S$63 (Bhowmik et al., 19 Sep 2025).

By contrast, for tensor products $S$64, 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 (Bhowmik et al., 19 Sep 2025). 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 $S$65 and $S$66, writing $S$67, one computes

$S$68

so

$S$69

The skew-eigenvector equation

$S$70

forces

$S$71

The paper states that for $S$72 or larger, nontrivial eigenpairs appear and the approximant $S$73 matches Taylor coefficients up to degree $S$74. In one variable, if $S$75 at some $S$76, then $S$77 is a finite Blaschke product; otherwise $S$78 and $S$79 uniformly on compact sets (Bhowmik et al., 19 Sep 2025).

The computational procedure described in the paper is finite-dimensional. One inputs $S$80 and a multidegree $S$81, forms the truncated Toeplitz matrix of $S$82 in the monomial basis, defines $S$83 by reflection, and constructs the antilinear operator

$S$84

One then computes its Takagi singular values and skew eigenvectors by solving

$S$85

selects the maximal $S$86 and its corresponding $S$87, and forms

$S$88

Pole verification is reduced to locating the zeros of $S$89 in $S$90; in the Agler/Herglotz setting, one instead solves the semidefinite feasibility problem for the matrices $S$91 and constructs the realization

$S$92

(Bhowmik et al., 19 Sep 2025).

7. Conceptual placement within interpolation and Padé theory

The article situates the skew-eigenvector method at the intersection of several classical theories (Bhowmik et al., 19 Sep 2025). 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

$S$93

is the Hardy-space analogue of the matrix equation

$S$94

from Takagi factorization. The conjugation $S$95 plays the role of the symmetric reflection, and the extremal eigenpairs maximize the corresponding Hankel form (Bhowmik et al., 19 Sep 2025).

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 $S$96 inner rational approximants, with a Montessus-type convergence theorem valid away from a moving pole set (Bhowmik et al., 19 Sep 2025).

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

$S$97

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 (Bhowmik et al., 19 Sep 2025).