Transfinite Operator Fixed Points on Hilbert Spaces: An Alpay Algebra Approach

TL;DR

This paper introduces a transfinite iterative framework for self-adjoint operators on Hilbert spaces, establishing convergence to fixed points and characterizing their spectra, generalizing classical asymptotic projection results.

Contribution

It develops a novel transfinite spectral-iteration method for self-adjoint operators, including a spectral-mapping theorem and a fixed point characterization, extending classical operator theory.

Findings

Convergence of transfinite operator sequences to fixed points.

Spectral characterization of limit operators via iterative spectral maps.

Identification of limit operators as orthogonal projections onto invariant eigenspaces.

Abstract

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 $\Phi$ is applied iteratively. This process generates a transfinite sequence of operators, $\{\Phi^{\alpha}(A)\}_{\alpha<\Omega}$, by progressively enlarging the ambient Hilbert space at each ordinal stage. Under suitable continuity and monotonicity conditions on $\Phi$, it is established via transfinite induction that the sequence converges, stabilizing at a minimal ordinal $\Omega$ where $\Phi^{\Omega+1}(A) = \Phi^{\Omega}(A)$. The resultant limit operator, $A_{\infty} = \Phi^{\infty}(A)$, is a self-adjoint fixed point of the transformation, satisfying $\Phi(A_{\infty}) = A_{\infty}$. Its spectrum is characterized by the relation $$\sigma(A_{\infty})=\bigcap_{n<\infty}f^{\,n}\bigl(\sigma(A)\bigr),$$ where $f$ is the spectral map induced by $\Phi$. For canonical transformations, such as $\Phi(A)=A^2$ or the semigroup action $\Phi_t(A)=e^{tA}$, 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.