Finite-rank kernel realization and completion

James Tian

doi:10.33205/cma.1942507

Finite-rank kernel realization and completion

Abstract

We consider when local finite-rank positive definite kernels come from a single global finite-rank kernel. The first part treats the case where the global kernel is fixed and shows that the missing mixed blocks control when a global rank-$r$ realization exists and how rank grows when it does not. The second part treats the completion problem, where only local kernels are given. In that setting, rank-preserving completion becomes a unitary patching problem on overlap generated subspaces. Forests always patch, full overlaps are governed by cycle conditions for the induced unitaries, and partial overlaps lead to finite-dimensional subspace compatibility conditions. We also give a rank bound for finite completions obtained by joining two completed pieces along their common feature subspace.

Keywords

References

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Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

James Tian ^*
0000-0003-4996-1227
United States

Early Pub Date

June 30, 2026

Publication Date

September 15, 2026

Submission Date

May 2, 2026

Acceptance Date

June 27, 2026

Published in Issue

Year 2026 Volume: 9 Number: 3

DOI

https://doi.org/10.33205/cma.1942507

IZ

https://izlik.org/JA78HT33UZ

Cite

RIS / Bibtex

APA

Tian, J. (2026). Finite-rank kernel realization and completion. Constructive Mathematical Analysis, 9(3), 119-137. https://doi.org/10.33205/cma.1942507

AMA

1.Tian J. Finite-rank kernel realization and completion. CMA. 2026;9(3):119-137. doi:10.33205/cma.1942507

Chicago

Tian, James. 2026. “Finite-Rank Kernel Realization and Completion”. Constructive Mathematical Analysis 9 (3): 119-37. https://doi.org/10.33205/cma.1942507.

EndNote

Tian J (September 1, 2026) Finite-rank kernel realization and completion. Constructive Mathematical Analysis 9 3 119–137.

IEEE

[1]J. Tian, “Finite-rank kernel realization and completion”, CMA, vol. 9, no. 3, pp. 119–137, Sept. 2026, doi: 10.33205/cma.1942507.

ISNAD

Tian, James. “Finite-Rank Kernel Realization and Completion”. Constructive Mathematical Analysis 9/3 (September 1, 2026): 119-137. https://doi.org/10.33205/cma.1942507.

JAMA

1.Tian J. Finite-rank kernel realization and completion. CMA. 2026;9:119–137.

MLA

Tian, James. “Finite-Rank Kernel Realization and Completion”. Constructive Mathematical Analysis, vol. 9, no. 3, Sept. 2026, pp. 119-37, doi:10.33205/cma.1942507.

Vancouver

1.James Tian. Finite-rank kernel realization and completion. CMA. 2026 Sep. 1;9(3):119-37. doi:10.33205/cma.1942507