We prove that every positive bounded invertible Wiener-Hopf operator admits triangular factorization. This answers the question posed by L.Sakhnovich in 1994.

2018-05-10math.FAmath.OAmath.SP

Effective construction of a class of positive operators in Hilbert space, which do not admit triangular factorization

A class of non-factorable positive operators is constructed. As a result, pure existence theorems in the well-known problems by Ringrose, Kadison and Singer are substituted by concrete examples.

2010-09-09math.CAmath.FA

Effective Construction of a Positive Operator which does not admit Triangular Factorization

We have constructed a concrete example of a non-factorable positive operator. As a result, for the well-known problems (Ringrose, Kadison and Singer problems) we replace existence theorems by concrete examples.

2015-09-04math.CAmath.FA

Sampling measures, Muckenhoupt Hamiltonians, and triangular factorization

Let $μ$ be an even measure on the real line $\mathbb{R}$ such that $$c_1 \int_{\mathbb{R}}|f|^2\,dx \le \int_{\mathbb{R}}|f|^2\,dμ\le c_2\int_{\mathbb{R}}|f|^2\,dx$$ for all functions $f$ in the Paley-Wiener space $\mathrm{PW}_{a}$. We prove that $μ$ is the spectral measure for the unique Hamiltonian $\mathcal{H}=\left(w&00&\frac{1}{w}\right)$ on $[0,a]$ generated by a weight $w$ from the Muckenhoupt class $A_2[0,a]$. As a consequence of this result, we construct Krein's orthogonal entire functions with respect to $μ$ and prove that every positive, bounded, invertible Wiener-Hopf operator on $…

2016-09-13math.FA

Reverse Cholesky factorization and tensor products of nest algebras

We prove that every positive semidefinite matrix over the natural numbers that is eventually 0 in each row and column can be factored as the product of an upper triangular matrix times a lower triangular matrix. We also extend some known results about factorization with respect to tensor products of nest algebras. Our proofs use the theory of reproducing kernel Hilbert spaces.

2017-04-14math.FA

Triangular Tetrablock-contractions, factorization of contractions, dilation and subvarieties

A commuting triple of Hilbert space operators $(A,B,P)$, for which the closed tetrablock $\bar{\mathbb E}$ is a spectral set, is called a \textit{tetrablock-contraction} or simply an $\mathbb E$-\textit{contraction}, where \[ \mathbb E=\{(a_{11},a_{22}, \det A):\, A=[a_{ij}]\in \mathcal M_2(\mathbb C), \; \|A\| <1 \} \subset \mathbb C^3 \] is a polynomially convex domain which is naturally associated with the $μ$-synthesis problem. We introduce triangular $\mathbb E$-contractions and prove that every pure triangular $\mathbb E$-contraction dilates to a pure triangular $\mathbb E$-isometry. We …

2022-05-24math.CVmath.FA

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Unitary loops and factorization

We consider a refinement of triangular factorization for unitary matrix valued loops.

2009-07-25math.FA

论正算子三角分解的稳定性

On stability of triangular factorization of positive operators

本文研究 Hilbert 空间中正算子关于套的三角分解在非正定情形下的稳定性，并给出保证因子收敛的充分条件。

2025-09-26math-phmath.FAmath.MP

Loops in SU(2) and Factorization, II

In the prequel to this paper, we proved that for a $SU(2,\mathbb C)$ valued loop having the critical degree of smoothness (one half of a derivative in the $L^2$ Sobolev sense), the following statements are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. This hinges on some Plancherel-esque formulas for determinants of Toeplitz operators. The main point of this report is is to outline a generalization of this result to loops of vanishi…

2022-03-14math.FA

Loops in SL(2,C) and Factorization

In previous work we proved that for a SU(2,C) valued loop having the critical degree of smoothness (one half of a derivative in the L^2 Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a triangular factorization, and (3) the loop has a root subgroup factorization. For a loop g satisfying these conditions, the Toeplitz determinant det(A(g)A(g^{-1})) and shifted Toeplitz determinant det(A_1(g)A_1(g^{-1})) factor as products in root subgroup coordinates. In this paper we observe that, at least in …

2017-09-18math.FA

阿戴尔环群与完美oid类比：阿戴尔射影线上的因式分解与全纯丛

Adelic Loop Groups and Perfectoid Analogies: Factorization and Holomorphic Bundles on the Adelic Projective Line

Universidad Nacional Autónoma de México（墨西哥国立自治大学）

研究在通用一维螺线管上的阿戴尔环群理论，引入阿戴尔射影线等，建立多个定理得出猜想，发展相关几何并与完美oid几何比较，得到猜想的重新表述，核心方法是构建理论体系并进行比较，主要贡献是丰富了相关理论及得出新猜想表述。

2026-07-15math.CVmath.FA