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

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

13 pages