Many Parameter Hölder Perturbation of Unbounded Operators

If u 7→ A(u) is a C0,α-mapping, for 0 < α ≤ 1, having as values unbounded self-adjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space, then any continuous (in u) arrangement of the eigenvalues of A(u) is indeed C0,α in u. Theorem. Let U ⊆ E be a c∞-open subset in a convenient vector space E, and 0 < α ≤ 1. Let u 7→ A(u), for u ∈ U , be a C-mapping with values unbounded self-adjoint operators in a Hilbert space H with common domain of definition and with compact resolvent. Then any (in u) continuous eigenvalue λ(u) of A(u) is C in u. Remarks and definitions. This paper is a complement to [9] and builds upon it. A function f : R→ R is called C if f(t)−f(s) |t−s|α is locally bounded in t 6= s. For α = 1 this is Lipschitz. Due to [2] a mapping f : R → R is C if and only if f ◦ c is C for each smooth (i.e. C∞) curve c. [4] has shown that this holds for even more general concepts of Hölder differentiable maps. A convenient vector space (see [8]) is a locally convex vector space E satisfying the following equivalent conditions: Mackey Cauchy sequences converge; C∞-curves in E are locally integrable in E; a curve c : R → E is C∞ (Lipschitz) if and only if ` ◦ c is C∞ (Lipschitz) for all continuous linear functionals `. The c∞-topology on E is the final topology with respect to all smooth curves (Lipschitz curves). Mappings f defined on open (or even c∞-open) subsets of convenient vector spaces E are called C (Lipschitz) if f ◦ c is C (Lipschitz) for every smooth curve c. If E is a Banach space then a C-mapping is locally Hölder-continuous of order α in the usual sense. This has been proved in [5], which is not easily accessible, thus we include a proof in the lemma below. For the Lipschitz case see [7] and [8, 12.7]. That a mapping t 7→ A(t) defined on a c∞-open subset U of a convenient vector space E is C with values in unbounded operators means the following: There is a dense subspace V of the Hilbert space H such that V is the domain of definition of each A(t), and such that A(t)∗ = A(t). And furthermore, t 7→ 〈A(t)u, v〉 is C for each u ∈ V and v ∈ H in the sense of the definition given above. This implies that t 7→ A(t)u is of the same class U → H for each u ∈ V by [8, 2.3], [7, 2.6.2], or[5, 4.1.14]. This is true because C can be described by boundedness conditions only; and for these the uniform boundedness principle is valid. Date: June 8, 2009. 2000 Mathematics Subject Classification. Primary 47A55, 47A56, 47B25.