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

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.