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We study integral operators on the space of square-integrable functions from a compact set, X, to a separableHilbert space,H. The kernel of such an operator takes values in the ideal of Hilbert–Schmidt operators on H.We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer’s theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on L2(X; H) under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is Hölder continuous with Hölder exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on H. Finally, when dim H < ∞, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds. 

Abstract This work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity. 

Abstract We study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.