Let $A_{𝒩}$ be the symmetric operator given by the restriction of $A$ to $𝒩$, where $A$ is a self-adjoint operator on the Hilbert space $\mathscr{H}$ and $𝒩$ is a linear dense set which is closed with respect to the graph norm on $D\left(A\right)$, the operator domain of $A$. We show that any self-adjoint extension $A_{\Theta }$ of $A_{𝒩}$ such that $D\left(A_{\Theta }\right)\cap D\left(A\right)=𝒩$ can be additively decomposed by the sum $A_{\Theta }=\overline{A}+T_{\Theta }$, where both the operators $\overline{A}$ and $T_{\Theta }$ take values in the strong dual of $D\left(A\right)$. The operator $\overline{A}$ is the closed extension of $A$ to the whole $\mathscr{H}$ whereas $T_{\Theta }$ is explicitly written in terms...

A generalization of the Carleman criterion for selfadjointness of Jacobi matrices to the case of symmetric matrices with finite rows is established. In particular, a new proof of the Carleman criterion is found. An extension of Jørgensen's criterion for selfadjointness of symmetric operators with "almost invariant" subspaces is obtained. Some applications to hyponormal weighted shifts are given.

In this note, I wish to describe the first order semiclassical approximation to the spectrum of one frequency quasi-periodic operators. In the case of a sampling function with two critical points, the spectrum exhibits two gaps in the leading order approximation. Furthermore, I will give an example of a two frequency quasi-periodic operator, which has no gaps in the leading order of the semiclassical approximation.

This is the second instalment of my previous paper with the same title, [1]. This paper consists of two different parts. The first part is devoted to improvements of the results developed in [1]. These improvements are described in section 0.1 below and developed in sections 1 to 5, and 9 to 10; they are in fact technically distinct from [1] and rely on a systematic use of microlocalisation in the context of Hörmander-Weyl calculus. These paragraphs can therefore be read quite independently from...

Soit $X$ un espace riemannien symétrique et $C_0\left(X\right)$ l’espace des fonctions continues sur $X$ tendant vers 0 à l’infini. On démontre qu’un opérateur $(D_{A^{\text{'}}},A)$, invariant par les isométries de $X$, engendre un semi-groupe fortement continu de contractions sur $C_0\left(X\right)$ s’il est dissipatif et si son domaine contient les fonctions de classe ${𝒞}^{\infty }$ à support compact.

We study a family of commuting selfadjoint operators $={\left(A_k\right)}_{k=1}^n$, which satisfy, together with the operators of the family $={\left(B_j\right)}_{j=1}^n$, semilinear relations ${⅀}_i{f}_{ij}\left(\right)B_j{g}_{ij}\left(\right)=h\left(\right)$, ($f_{ij}$, $g_{ij}$, $h_j:{ℝ}^n\to ℂ$ are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra $U_q^{\text{'}}\left(so\left(3\right)\right)$.

Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X{\widehat{\otimes }}_{\tau }Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X{\widehat{\otimes }}_{\tau }Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck’s theorem may be applied. In this paper, we take an index $1\le p<\infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with ${\tau }_p$ the associated $L^p$-tensor product...

Sets of bounded linear operators , ⊂ ℬ(H) (ℋ is a Hilbert space) are similar if there exists an invertible (in ℬ(H)) operator G such that $G^{-1}··G=$. A bounded operator is scalar if it is similar to a normal operator. is jointly scalar if there exists a set ⊂ ℬ(H) of normal operators such that and are similar. is separately scalar if all its elements are scalar. Some necessary and sufficient conditions for joint scalarity of a separately scalar abelian set of Hilbert space operators are presented (Theorems...