Unbounded banded matrices, shifted positive bidiagonal factorizations, and mixed-type multiple orthogonality

This paper extends Favard-type spectral representations to unbounded banded matrices by utilizing $N$-dependent shifts to ensure positive bidiagonal factorizations of truncated operators, thereby establishing a limiting matrix-valued measure and mixed-type multiple biorthogonality relations that recover the classical spectral theory for Jacobi matrices as a special case.

The Gist

Imagine you are trying to understand the "sound" or the "soul" of a giant, infinite machine. In mathematics, this machine is represented by a banded matrix—a grid of numbers that is mostly empty, with the action happening only in a few diagonal stripes.

For a long time, mathematicians could only analyze these machines if they were "bounded," meaning their numbers didn't get infinitely large. It was like studying a piano where the keys were all within a comfortable reach. A famous rule, called Favard's Theorem, told them exactly how to translate the machine's structure into a set of musical notes (a spectral measure) that explained how it worked.

However, the real world often deals with "unbounded" machines—systems where the numbers can grow as large as you want, like a piano with keys stretching out to infinity. The old rules broke down because the machine was too wild to analyze directly.

The Problem: The Infinite Machine is Too Wild

The authors of this paper wanted to extend that famous rule to these wild, infinite machines. But there was a catch: you can't just look at the whole infinite machine at once; it's too messy. You have to look at it in chunks (truncations), like listening to a song one minute at a time.

The problem was that as you listened to longer and longer chunks, the "volume" of the music would get louder and louder, eventually drowning out the signal. In math terms, the numbers in these chunks were getting so big that the standard way of analyzing them failed.

The Solution: The "Shift" Trick

The authors' brilliant idea was to use a shift.

Imagine you are trying to photograph a runner who is sprinting away from you. If you try to keep the camera fixed, the runner eventually disappears into the distance. But, if you move the camera to keep up with the runner, you can keep them in the frame.

In this paper, the "camera" is a mathematical adjustment. For every chunk of the machine they analyzed, they added a specific number (a "shift") to the diagonal of that chunk.

• Why? This shift acts like a counterweight. It pushes the numbers back down to a manageable size, ensuring that every chunk of the machine has a special, orderly structure called a Positive Bidiagonal Factorization (PBF).
• The Metaphor: Think of PBF as a "perfectly stacked tower of blocks." If the blocks are messy, the tower falls. The shift ensures that no matter how big the chunk is, the blocks can always be stacked perfectly.

The Process: From Chunks to a Whole Picture

Once they had these "shifted" chunks, they followed a three-step process:

1. Analyze the Chunks: Because each shifted chunk was now a "perfect tower" (had PBF), they could easily calculate a set of "weights" and "positions" (like the notes on a piano) for that specific chunk.
2. Re-center the View: Since they had added a shift to make the math work, they had to subtract it back out. They took the results from the shifted chunks and "translated" them back to their original position. This is like taking the photo of the runner and moving the camera back to its original spot to see where the runner actually is.
3. The Helly Selection Principle (The Magic Filter): Now they had a sequence of these translated results. Some might wobble, some might jump. But the authors proved that these results were "uniformly bounded"—meaning they didn't run off to infinity.
   • They used a mathematical tool called Helly's Selection Principle. Imagine you have a bag of wobbly jellybeans. Even if they wiggle, if you keep them in a box that doesn't expand, you can eventually find a subset of jellybeans that settles down into a stable shape.
   • By applying this, they found a "limiting" shape. This stable shape is the Spectral Measure for the original, wild, infinite machine.

The Result: A New Rule for Infinite Machines

The paper proves that even for these unbounded, infinite machines, you can still find that "musical score" (the spectral measure) that explains how they work.

• The "Mixed-Type" Twist: The authors also deal with a specific type of math problem where you have two different sets of rules interacting (left and right sides). They show that their method works for this complex interaction too, ensuring that the "notes" (polynomials) they find are perfectly balanced and don't get lost.
• The Jacobi Case: They specifically show how this works for a very common type of machine called a Jacobi matrix (which looks like a tridiagonal band). They prove that for these, you can always find the right "shift" to make the math work, recovering the classic results as a special case.

In Summary

The authors took a rule that only worked for "tame" mathematical machines and extended it to "wild" ones. They did this by:

1. Shifting the view to tame the wild numbers.
2. Analyzing the tame chunks to find their structure.
3. Re-centering the view to see the original machine.
4. Using a filter (Helly's principle) to smooth out the wobbles and reveal the true, infinite pattern underneath.

They didn't invent a new machine; they just built a better pair of glasses to see how the existing, infinite ones behave.

Technical Summary

Technical Summary: Unbounded Banded Matrices, Shifted Positive Bidiagonal Factorizations, and Mixed-Type Multiple Orthogonality

Problem Statement
The paper addresses the extension of the spectral Favard theorem for banded matrices from the bounded setting to the unbounded setting. Previous work (specifically [4] and [3]) established that for bounded banded matrices admitting a positive bidiagonal factorization (PBF), one can construct a matrix-valued spectral measure and derive mixed-type multiple biorthogonality relations for associated polynomials. However, in the unbounded regime, the positive bidiagonal factorization cannot generally be imposed directly on the semi-infinite matrix. The central challenge is to recover a spectral representation and a limiting matrix-valued measure for unbounded banded matrices $T$ without assuming global boundedness, while preserving the structural properties (positivity, total positivity) that ensure the existence of the factorization and the normality of the associated polynomials.

Methodology
The authors employ a constructive "discrete-to-limit" philosophy, adapting the framework established in [4] for the bounded case. The methodology proceeds through the following key steps:

1. Shifted Truncations: Instead of assuming the semi-infinite matrix $T$ admits a PBF, the authors assume that for every finite truncation size $N$, there exists a non-negative shift $s_N \geq 0$ such that the shifted truncation $A_N \coloneqq T[N] + s_N I_{N+1}$ admits a positive bidiagonal factorization. This shift ensures the truncated matrix is oscillatory (possessing simple positive eigenvalues and specific sign-variation structures in eigenvectors), a property linked to total positivity [9, 7].
2. Recentering of Measures: Since the shift $s_N$ depends on $N$, the discrete Gauss-type quadrature measures associated with $A_N$ are not uniformly bounded in their original position. The authors introduce a recentering step, translating the discrete measures by $x \mapsto x - s_N$. This produces a family of distribution functions $\psi^{[N]}_{b,a}$ that are uniformly bounded in total mass.
3. Moment Stabilization: Using the band structure of $T$, the authors prove that matrix elements $(T^n)_{k,l}$ stabilize for sufficiently large truncation sizes $N$. Specifically, for a given power $n$, the moments computed via the truncated shifted matrix $(T[N] + s_N I)^n$ (after recentering) coincide exactly with the moments of the full unbounded operator $T^n$ once $N$ exceeds a threshold determined by the bandwidth and the indices.
4. Compactness and Convergence: The authors apply Helly's selection principle [6] to the uniformly bounded family of recentered distribution functions. This guarantees the existence of a convergent subsequence of measures. By combining this with Helly's second theorem and tail estimates (ensuring no mass is lost at infinity), they identify the limit as a matrix-valued measure that reproduces the moments of the unbounded operator.

Key Contributions and Results

• Theorem 2.2 (Favard Spectral Representation for Unbounded Matrices): The primary result establishes that if a semi-infinite banded matrix $T$ admits a sequence of shifts $s_N$ such that every shifted truncation $T[N] + s_N I_{N+1}$ admits a PBF, then there exists a matrix-valued measure $d\Psi$ (composed of $p \times q$ non-decreasing positive functions) satisfying mixed-type multiple biorthogonality relations.
  • The measure satisfies the spectral representation:
    $$(T^n)_{k,l} = \sum_{a=1}^p \sum_{b=1}^q \int_{\mathbb{R}} B^{(b)}_l(x) x^n d\psi_{b,a}(x) A^{(a)}_k(x)$$
  • This generalizes the classical Favard theorem to unbounded operators, recovering the standard result for Jacobi matrices as a special case.
• Degree Patterns and Normality: The paper analyzes the degrees of the associated mixed-type multiple orthogonal polynomials.
  • If the initial conditions are determined by positive triangular matrices, the degrees satisfy specific upper bounds.
  • If the initial conditions are determined by totally positive triangular matrices, the polynomials achieve maximal degrees (normality), mirroring the results for the bounded case found in [3].
• Application to Unbounded Jacobi Matrices: The authors provide a concrete application to unbounded Jacobi matrices $J$ with positive off-diagonals. They explicitly construct the shift $s_N = \|J[N]\|_\infty + 1$. They prove that for this choice, $J[N] + s_N I$ is an oscillatory matrix (all leading principal minors are strictly positive), thereby satisfying the PBF hypothesis. This demonstrates that the spectral Favard theorem applies to a broad class of unbounded Jacobi operators.

Significance and Claims
The paper claims to successfully remove the boundedness assumption from the spectral Favard theorem for banded matrices admitting a PBF. The significance lies in:

1. Generalization: It extends the existence of matrix-valued spectral measures and mixed-type multiple orthogonality to unbounded operators, a setting where direct factorization is often impossible.
2. Methodological Rigor: It resolves the analytic difficulty of $N$-dependent shifts by introducing a recentering mechanism and proving uniform tightness of the resulting measures, ensuring the limit is well-defined and reproduces the operator's moments.
3. Recovery of Classical Results: The construction naturally recovers the classical spectral measure for unbounded Jacobi matrices with positive off-diagonals, validating the approach against known theory.

The authors note that while the method relies on the existence of shifts $s_N$ that yield PBFs, this condition is satisfied by a wide class of operators, including unbounded Jacobi matrices. The work maintains the constructive spirit of previous bounded analyses while addressing the specific analytic challenges (tail control, mass preservation) inherent to the unbounded regime.