### A counterexample to a compact embedding theorem for functions with values in a Hilbert space.

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This paper is devoted to the spectral analysis of a non elliptic operator A , deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator A has been derived, we determine its continuous spectrum. Then, we show that A is unbounded from below and that it has a sequence of negative eigenvalues tending to -∞. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some...

This paper is devoted to the spectral analysis of a non elliptic operator $A$, deriving from the study of superconducting micro-strip lines. Once a sufficient condition for the self-adjointness of operator $A$ has been derived, we determine its continuous spectrum. Then, we show that $A$ is unbounded from below and that it has a sequence of negative eigenvalues tending to $-\infty $. Using the Min-Max principle, a characterization of its positive eigenvalues is given. Thanks to this characterization, some conditions...

We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators An, $n\in \mathbb{N}$, in a suitable Hilbert space. We show that the essential spectrum of An is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators ${A}_{n}$, $n\in \mathbb{N}$, in a suitable Hilbert space. We show that the essential spectrum of ${A}_{n}$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

The Fourier expansion in eigenfunctions of a positive operator is studied with the help of abstract functions of this operator. The rate of convergence is estimated in terms of its eigenvalues, especially for uniform and absolute convergence. Some particular results are obtained for elliptic operators and hyperbolic equations.

The paper concerns an approximation of an eigenvalue problem for two forms on a Hilbert space $X$. We investigate some approximation methods generated by sequences of forms ${a}_{n}$ and ${b}_{n}$ defined on a dense subspace of $X$. The proof of convergence of the methods is based on the theory of the external approximation of eigenvalue problems. The general results are applied to Aronszajn’s method.

Sufficient spectral conditions for the existence of a spectral decomposition of an operator T defined on a Banach space X, with countable spectrum, are given. We apply the results to obtain the West decomposition of certain Riesz operators.