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We consider the spectrum of the Stokes operator with and without rotation effect for the whole space and exterior domains in $L^q$-spaces. Based on similar results for the Dirichlet-Laplacian on $mathbf{R}^n$, $n \geq 2$, we prove in the whole space case that the spectrum as a set in $\mathbf{c}$ does not change with $q \in (1,\infty)$, but it changes its type from the residual to the continuous or to the point spectrum with $q$. The results for exterior domains are less complete, but the spectrum of the Stokes operator with rotation mainly is an essential one, consisting of infinitely many equidistant parallel half-lines in the left complex half-plane. The tools are strongly based on Fourier analysis in the whole space case and on stability properties of the essential spectrum for exterior domains.

We give a complete analysis of the spectrum of the unique self-adjoint extension of the sub-Laplacian on the one-dimensional Heisenberg group.

Let be the free product of q + 1 copies of Zn+1 and let denote its Cayley graph (with respect to aj, 1 ≤ j ≤ q + 1). We may think of G as a group acting on the “homogeneous space” , This point of view is inspired by the case of SL2(R) acting on the hyperbolic disk and is developed in [FT-P] [I-P] [FT-S] [S] (but see also [C]).Since G is a group we may investigate some classical topics: the full (reductive) C* algebra, its dual space, the regular Von Neumann algebra and so on. See [B] [P] [L] [V] and also [H]. These approaches give results pointing up the analogy between harmonic analysis on these groups and harmonic analysis on more classical objects.

In this paper we investigate perturbations of the left essential spectrum, right essential spectrum, essential spectrum and the regular spectrum of an upper triangular operator matrix such as M sub C - [A sub 1 sub 1 C sub 1 sub 2 O sub 2 sub 1 B sub 2 sub 2] acting on a Hilbert space H + K.

Let $T(\cdot)$ be a bounded $C_0$-semigroup on the Banach space $X$, with generator $A$. It is shown that the denseness of range $A$ is necessary and sufficient for the semigroup's mean stability with respect to suitable weights. Analogous results are valid for power bounded operators, tensor product semigroups, and cosine operator functions.

Let be a connected bounded open set in R N , N 2, with lipschitzian boundary. The best constant in the Poincar e type inequality: j uj 2 C()k grad(u)k 2 1 ;8u2 L 2 ()=R is the inverse of the smallest spectral value of the the bounded self-adjoint linear operator T = div( ) 1 grad in L 2 ()=R ([4]). In this paper we show that, in the case of an elliptical domain of R 2 , the point spectrum of this operator is the set p(T )= fn; e n; 1; n2 N*g, where 1 is an eigenvalue

We will investigate the point spectrum on the imaginary axis of the linearized Boltzmann operator with an external-force potential in an unbounded domain. The boundary condition considered is the perfectly reflective boundary condition. We suppose that the boundary surface is sufficiently smooth, but is not cylindrical. In this case, the point spectrum on the imaginary axis is only equal to f0g.