2025/08/06 by Alpay, Faruk, Alakkad, Hamdi, Alpay, Taylan
#46L05 #47A10 #47A60 #47B25 #47D06 #F.1.1 #FOS: Mathematics #Functional Analysis (math.FA) #G.1.0 #G.1.10 #Operator Algebras (math.OA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2508.04890
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, \Φα(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 ΦΩ+1(A) = ΦΩ(A). The resultant limit operator, A∞ = Φ∞(A), is a self-adjoint fixed point of the transformation, satisfying Φ(A∞) = A∞. Its spectrum is characterized by the relation σ(A∞)=\bigcapnlt;∞f n(σ(A)), where f is the spectral map induced by Φ. For canonical transformations, such as Φ(A)=A2 or the semigroup action Φt(A)=etA, the limit operator A∞ 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∞ up to unitary equivalence, and a reinterpretation of the discrete iteration as an evolution semigroup on an L2-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.