Many Parameter Lipschitz Perturbation of Unbounded Operators

If u 7→ A(u) is a C1,α-mapping having as values unbounded selfadjoint operators with compact resolvents and common domain of definition, parametrized by u in an (even infinite dimensional) space then any continuous arrangement of the eigenvalues u 7→ λi(u) is C0,1 in u. If u 7→ A(u) is C0,1, then the eigenvalues may be chosen C0,1/N (even C0,1 if N = 2), locally in u, where N is locally the maximal multiplicity of the eigenvalues. Theorem. Let U ⊆ E be a c∞-open subset in a convenient vector space E. Let u 7→ A(u), for u ∈ U , be a mapping with values unbounded self-adjoint operators in a Hilbert space H with common domain of definition and with compact resolvent. (A) If u 7→ A(u) is C, for some 0 < α ≤ 1, then any continuous arrangement of the eigenvalues of A(u) (e.g., ordered by size), is C. (B) If u 7→ A(u) is C, then the increasingly ordered continuous eigenvalues of A(u) are C locally in u ∈ U , where N is locally the maximal multiplicity of the eigenvalues. If N = 2, the increasingly ordered eigenvalues are even locally C. Remarks and definitions. This paper is a complement to [10] and builds upon it. A function f : R → R is called C if it is k times differentiable and for the k-th derivative the expression f (t)−f(s) |t−s|α is locally bounded in t 6= s. For k = 0 and 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. [3] has shown that this holds for even more general concepts of Hölder differentiable maps. A convenient vector space (see [9]) 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∞ if and only if ` ◦ c is C∞ for all continuous linear functionals `. The c∞-topology on E is the final topology with respect to all smooth curves. Mappings f defined on open (or even c∞-open) subsets of convenient vector spaces E are called C if f ◦c is C for every smooth curve c. If E is a Banach space then a C-mapping is k-times differentiable and the k-th derivative is locally Hölder-continuous of order α in the usual sense. This has been proved in [4], which is not easily accessible, thus we include a proof of the simplest case k = 0 in the lemma below. That a mapping t 7→ A(t) defined on a c∞-open subset U of a convenient vector space E is real analytic, C∞, or 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, Date: May 25, 2007. 2000 Mathematics Subject Classification. Primary 47A55, 47A56, 47B25.