A restriction theorem for Grushin operators

We study the Grushin operators acting on \(\mathbb{R}_x^{{d_1}} \times \mathbb{R}_t^{{d_2}}\) and defined by the formula \(L = - \sum\nolimits_{j = 1}^{{d_1}} {\partial _{{x_j}}^2} - {\sum\nolimits_{j = 1}^{{d_1}} {\left| {{x_j}} \right|} ^2}\sum\nolimits_{k = 1}^{{d_2}} {\partial _{{t_k}}^2} \). We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. M¨uller [Ann. of Math., 1990, 131: 567–587].     

A bifurcation theorem for potential operators

A general bifurcation theorem for potential operators is proved. It describes the possible behavior of the set of solutions of an operator equation as a function of the eigenvalue parameter in a neighborhood of the bifurcation point. The theorem applies in particular to buckling problems in elasticity theory as well as to other fields in which the bifurcation problems have a variational formulation.     

A mean ergodic theorem for resolvent operators

Let {R(t)} _t≥0 be a uniformly bounded strongly continuous resolvent operator for the Volterra equation of convolution typeu=g+k*Au, whereA is a closed and densely defined operator on a Banach spaceX andk is a scalar kernel. We show that whenX is reflexive and that the average given by {R(t)} _t≥0 andk converges on the closed subspace to a bounded projection. This work was partially supported by DICYT 92-33LY and FONDECYT 91-0471     

A uniqueness theorem for indefinite Sturm-Liouville operators

In this paper, we discuss the inverse problem for indefinite Sturm-Liouville operators on the finite interval [a, b]. For a fixed index n(n = 0, 1, 2, ··· ), given the weight function ω(x), we will show that the spectral sets {λ n (q, h a , h k )} +∞ k=1 and {λ-n (q, h b , h k )} +∞ k=1 for distinct h k are sufficient to determine the potential q(x) on the finite interval [a, b] and coefficients h a and h b of the boundary conditions.     

A lifting theorem for unbounded quasinormal operators

Relationships between minimal normal extensions of spectral and cyclic type of an unbounded quasinormal operator are discussed and some properties such as, for example, tightness of such extensions are established. A Yoshino type criterion on the lifting of the strong commutant of an unbounded quasinormal operator is proved.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

A local lifting theorem for subnormal operators

Criteria for the existence of lifts of operators intertwining subnormal operators are established. The main result of the paper reduces lifting questions for general subnormal operators to questions about lifts of cyclic subnormal operators. It is shown that in general the existence of local lifts (i.e. those coming from cyclic parts) for a pair of subnormal operators does not imply the existence of a global lift. However this is the case when minimal normal extensions of subnormal operators in question are star-cyclic.

     

A classification theorem for Helfrich surfaces

In this paper we study the functional $\mathcal W{} _{\lambda _1,\lambda _2}$ , which is the sum of the Willmore energy, $\lambda _1$ -weighted surface area, and $\lambda _2$ -weighted volume, for surfaces immersed in $\mathbb R ^3$ . This coincides with the Helfrich functional with zero ‘spontaneous curvature’. Our main result is a complete classification of all smooth immersed critical points of the functional with $\lambda _1\ge 0$ and small $L^2$ norm of tracefree curvature, with no assumption on the growth of the curvature in $L^2$ at infinity. This not only improves the gap lemma due to Kuwert and Schätzle for Willmore surfaces immersed in $\mathbb R ^3$ but also implies the non-existence of critical points of the functional satisfying the energy condition for which the surface area and enclosed volume are positively weighted.     

Some essentially self-adjoint one-electron dirac operators

A class of smooth functions is introduced and it is shown that, the one-electron Dirac operator corresponding to an element with atomic number less than 102, is essentially self-adjoint on this class of functions. The proof makes essential use of the fact that each of these operators admits a complete family of reducing subspaces. At the same time we use an explicit formula for the resolvent kernel of the part of these operators over such a reducing subspace. Appendix by F. H. Brownell of the University of Washington This work was supported by N.S.F. grant GP-21330.     

A classification theorem for Albert algebras

Let be a field whose characteristic is different from 2 and 3 and let be a quadratic extension. In this paper we prove that for a fixed, degree 3 central simple algebra over with an involution of the second kind over , the Jordan algebra , obtained through Tits' second construction is determined up to isomorphism by the class of in , thus settling a question raised by Petersson and Racine. As a consequence, we derive a ``Skolem Noether' type theorem for Albert algebras. We also show that the cohomological invariants determine the isomorphism class of , if is fixed.

     

A closed graph theorem for order bounded operators

The closed graph theorem is one of the cornerstones of linear functional analysis in Fréchet spaces, and the extension of this result to more general topological vector spaces is a di?cult problem comprising a great deal of technical difficulty. However, the theory of convergence vector spaces provides a natural framework for closed graph theorems. In this paper we use techniques from convergence vector space theory to prove a version of the closed graph theorem for order bounded operators on Archimedean vector lattices. This illustrates the usefulness of convergence spaces in dealing with problems in vector lattice theory, problems that may fail to be amenable to the usual Hausdorff-Kuratowski-Bourbaki concept of topology.