Random Operator-Valued Frames in Hilbert Spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917085228040192 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | We study strongly measurable random bounded operators on separable Hilbert spaces and analyze two simple iterations driven by independent random positive contractions. The first, a Kaczmarz-like iteration, converges in mean square and almost surely and produces a random operator-valued frame. In the projection case it yields a Parseval identity. The second, a residual-weighted iteration, enjoys an exact step-by-step identity: the accumulated analysis terms plus a residual equal the identity operator. Under a mild mean-coercivity condition, the residual shrinks at a geometric rate in expectation, vanishes almost surely, and admits nonasymptotic tail bounds. As a result, the construction delivers an almost-sure Parseval frame for any independent sequence of positive contractions, not only projections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01914 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Operator-Valued Frames in Hilbert Spaces Tian, James Functional Analysis Primary 42C15. Secondary 47B80, 47B90, 94A12 We study strongly measurable random bounded operators on separable Hilbert spaces and analyze two simple iterations driven by independent random positive contractions. The first, a Kaczmarz-like iteration, converges in mean square and almost surely and produces a random operator-valued frame. In the projection case it yields a Parseval identity. The second, a residual-weighted iteration, enjoys an exact step-by-step identity: the accumulated analysis terms plus a residual equal the identity operator. Under a mild mean-coercivity condition, the residual shrinks at a geometric rate in expectation, vanishes almost surely, and admits nonasymptotic tail bounds. As a result, the construction delivers an almost-sure Parseval frame for any independent sequence of positive contractions, not only projections. |
| title | Random Operator-Valued Frames in Hilbert Spaces |
| topic | Functional Analysis Primary 42C15. Secondary 47B80, 47B90, 94A12 |
| url | https://arxiv.org/abs/2508.01914 |