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

This work extends Favard-type spectral representations for banded matrices $T$ beyond the bounded setting. It assumes that, for every $N\in\mathbb N_0$, there exists a shift $s_N\ge 0$ such that the shifted truncation $A_N:= T^{[N]}+s_N I_{N+1}$ admits a positive bidiagonal factorization (PBF). Allowing $s_N$ to depend on $N$ leads to a natural recentering step: the discrete Gauss-type quadrature measures associated with $A_N$ are translated by $x\mapsto x-s_N$, producing a uniformly bounded family of distribution functions. Combining moment stabilization for banded truncations with Helly-type compactness theorems yields a limiting matrix-valued measure, together with a Favard-type spectral representation and the corresponding mixed-type multiple biorthogonality relations. As a consequence, the classical Favard theorem for (possibly unbounded) Jacobi matrices is recovered as a special case. Indeed, for a tridiagonal $J$ with positive sub- and superdiagonals, each truncation $J^{[N]}$ admits a shift $s_N\ge 0$ such that $J^{[N]}+s_N I_{N+1}$ is oscillatory and therefore admits a PBF. The preceding construction then produces the usual spectral measure for $J$.

💡 Research Summary

The paper tackles the problem of extending Favard‑type spectral representations, which were previously established for bounded banded matrices possessing a positive bidiagonal factorization (PBF), to the unbounded setting. The central difficulty in the unbounded case is that a semi‑infinite matrix cannot, in general, be equipped directly with a PBF because its diagonal entries may grow without bound. To overcome this, the authors introduce a shifted truncation strategy: for each truncation size (N) they select a non‑negative shift (s_N) such that the finite matrix
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