Transfinite Operator Fixed Points on Hilbert Spaces: An Alpay Algebra Approach
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909726723276800 |
|---|---|
| author | Alpay, Faruk Alakkad, Hamdi Alpay, Taylan |
| author_facet | Alpay, Faruk Alakkad, Hamdi Alpay, Taylan |
| contents | This work develops a functional-analytic framework based on the transfinite iteration of a self-adjoint operator. Beginning with a densely defined self-adjoint operator $A$ on a Hilbert space $H$, a spectral-transform functor $Φ$ is applied iteratively. This process generates a transfinite sequence of operators, $\{Φ^α(A)\}_{α<Ω}$, by progressively enlarging the ambient Hilbert space at each ordinal stage. Under suitable continuity and monotonicity conditions on $Φ$, it is established via transfinite induction that the sequence converges, stabilizing at a minimal ordinal $Ω$ where $Φ^{Ω+1}(A) = Φ^Ω(A)$. The resultant limit operator, $A_{\infty} = Φ^{\infty}(A)$, is a self-adjoint fixed point of the transformation, satisfying $Φ(A_{\infty}) = A_{\infty}$. Its spectrum is characterized by the relation $$σ(A_{\infty})=\bigcap_{n<\infty}f^{\,n}\bigl(σ(A)\bigr),$$ where $f$ is the spectral map induced by $Φ$. For canonical transformations, such as $Φ(A)=A^2$ or the semigroup action $Φ_t(A)=e^{tA}$, the limit operator $A_{\infty}$ is identified as the orthogonal projection onto the iteratively invariant eigenspaces of the initial operator $A$. Principal contributions include a transfinite spectral-mapping theorem, a proof of the uniqueness of $A_{\infty}$ up to unitary equivalence, and a reinterpretation of the discrete iteration as an evolution semigroup on an $L^2$-type function space. The framework is demonstrated to subsume and generalize classical asymptotic-projection results. This study is partly motivated by the algebraic structures introduced by F. Alpay (arXiv:2505.15344). An appendix outlines a hierarchy of open problems in operator theory whose complexity is indexed by the iterative stage. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04890 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transfinite Operator Fixed Points on Hilbert Spaces: An Alpay Algebra Approach Alpay, Faruk Alakkad, Hamdi Alpay, Taylan Functional Analysis Operator Algebras Spectral Theory 47A10, 47D06, 47B25, 47A60, 46L05 G.1.0; G.1.10; F.1.1 This work develops a functional-analytic framework based on the transfinite iteration of a self-adjoint operator. Beginning with a densely defined self-adjoint operator $A$ on a Hilbert space $H$, a spectral-transform functor $Φ$ is applied iteratively. This process generates a transfinite sequence of operators, $\{Φ^α(A)\}_{α<Ω}$, by progressively enlarging the ambient Hilbert space at each ordinal stage. Under suitable continuity and monotonicity conditions on $Φ$, it is established via transfinite induction that the sequence converges, stabilizing at a minimal ordinal $Ω$ where $Φ^{Ω+1}(A) = Φ^Ω(A)$. The resultant limit operator, $A_{\infty} = Φ^{\infty}(A)$, is a self-adjoint fixed point of the transformation, satisfying $Φ(A_{\infty}) = A_{\infty}$. Its spectrum is characterized by the relation $$σ(A_{\infty})=\bigcap_{n<\infty}f^{\,n}\bigl(σ(A)\bigr),$$ where $f$ is the spectral map induced by $Φ$. For canonical transformations, such as $Φ(A)=A^2$ or the semigroup action $Φ_t(A)=e^{tA}$, the limit operator $A_{\infty}$ is identified as the orthogonal projection onto the iteratively invariant eigenspaces of the initial operator $A$. Principal contributions include a transfinite spectral-mapping theorem, a proof of the uniqueness of $A_{\infty}$ up to unitary equivalence, and a reinterpretation of the discrete iteration as an evolution semigroup on an $L^2$-type function space. The framework is demonstrated to subsume and generalize classical asymptotic-projection results. This study is partly motivated by the algebraic structures introduced by F. Alpay (arXiv:2505.15344). An appendix outlines a hierarchy of open problems in operator theory whose complexity is indexed by the iterative stage. |
| title | Transfinite Operator Fixed Points on Hilbert Spaces: An Alpay Algebra Approach |
| topic | Functional Analysis Operator Algebras Spectral Theory 47A10, 47D06, 47B25, 47A60, 46L05 G.1.0; G.1.10; F.1.1 |
| url | https://arxiv.org/abs/2508.04890 |