Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

On quasi-affine transforms of Read's operator

We show that C. J. Read's example of an operator on which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of . Furthermore, we present a bounded diagonal operator such that even though is unbounded, the operator is a bounded operator on with invariant subspaces, and is adjoint to an operator on .

     

A new invariant of stable equivalences of Morita type

It was proved in an earlier paper by the author that the Hochschild cohomology algebras of self-injective algebras are invariant under stable equivalences of Morita type. In this note we show that the orbit algebra of a self-injective algebra (considered as an --bimodule) is also invariant under stable equivalences of Morita type, where the orbit algebra is the algebra of all stable --bimodule morphisms from the non-negative Auslander-Reiten translations of to .

     

Une propriété de continuité du temps local

Let denote the local time (at 0) associated with a martingale . The aim of this note is to prove that the mapping is continuous from into weak-.

     

On stable equivalences of Morita type for finite dimensional algebras

In this paper, we assume that algebras are finite dimensional algebras with 1 over a fixed field and modules over an algebra are finitely generated left unitary modules. Let and be two algebras (where is a splitting field for and ) with no semisimple summands. If two bimodules and induce a stable equivalence of Morita type between and , and if maps any simple -module to a simple -module, then is a Morita equivalence. This conclusion generalizes Linckelmann's result for selfinjective algebras. Our proof here is based on the construction of almost split sequences.

     

The alternative Dunford-Pettis property in -algebras and von Neumann preduals

A Banach space is said to have the alternative Dunford-Pettis property if, whenever a sequence weakly in with , we have for each weakly null sequence in X. We show that a -algebra has the alternative Dunford-Pettis property if and only if every one of its irreducible representations is finite dimensional so that, for -algebras, the alternative and the usual Dunford-Pettis properties coincide as was conjectured by Freedman. We further show that the predual of a von Neumann algebra has the alternative Dunford-Pettis property if and only if the von Neumann algebra is of type I.

     

On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring in characteristic 0$">, a finitely generated module has finite projective dimension if for some 0$"> and for some 0$">, being the frobenius map repeated times. They used the notion of ``complexity' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity' at all.

     

Catégorie de Lusternik-Schnirelmann et genre des -espaces

Soit un espace ayant le type d'homotopie rationnelle d'un produit de sphères impaires. Si, pour tout nombre premier , la LS-catégorie de tous les -localisés de est majorée par , nous montrons que la LS-catégorie de est majorée par . Si est un élément dans le genre de Mislin de , nous en déduisons: . Dans le cas d'un -espace de rang 2, nous avons exactement , pour tout espace dans le genre de .

     

Deformations of minimal Lagrangian submanifolds with boundary

Let be a special Lagrangian submanifold of a compact Calabi-Yau manifold with boundary lying on the symplectic, codimension 2 submanifold . It is shown how deformations of which keep the boundary of confined to can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near with boundary on is found to be finite dimensional and is parametrized over the space of harmonic 1-forms of satisfying Neumann boundary conditions. The second is that if is a symplectic, codimension 2 submanifold sufficiently near , then, under suitable conditions, there exists a minimal Lagrangian submanifold near with boundary on .

     

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group on a -dimensional handlebody . Whenever acts freely preserving orientation on a connected -manifold , there is a -equivariant imbedding of into . There are choices of closed and Seifert-fibered for which the image of is a handlebody of a Heegaard splitting of . Provided that the genus of is at least , there are similar choices with closed and hyperbolic.

     

Lomonosov's invariant subspace theorem for multivalued linear operators

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator on an infinite-dimensional normed space ``commutes' with a compact operator i.e., then has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide an application to single-valued linear operators.

     

Characteristic classes and transfer relations in cobordism

Decompositions of products of the Ray elements by free generators of small dimensions in the symplectic cobordism ring are obtained. In particular it is stated that most of the -dimensional generators, for small, after multiplication by the Ray elements , , land in the ideal generated by Ray elements of low dimension.

     

The critical point equation on a three-dimensional compact manifold

On a compact -dimensional manifold , a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation (CPE), given by . It has been conjectured that a solution of the CPE is Einstein. Restricting our considerations to and assuming that there exist at least two distinct solutions of the CPE throughout the paper, we first prove that, if the second homology of vanishes, then is diffeomorphic to (Theorem 2). Secondly, we prove that the same conclusion holds if we have a lower Ricci curvature bound or the connectedness of a certain surface of (Theorem 3). Finally, we also prove that, if two connected surfaces of are disjoint, is isometric to a standard -sphere (Theorem 4).

     

Characterizations of paracompactness and Lindelöfness by the separation property

The separation property in our title is that, for two spaces and , any two disjoint closed copies of in are separated by open sets in . It is proved that a Tychonoff space is paracompact if and only if this separation property holds for the space and every Tychonoff space which is a perfect image of (where denotes the Stone-Cech compactification of ). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.

     

The class equation and counting in factorizable monoids

For orders and conjugacy in finite group theory, Lagrange's Theorem and the class equation have universal application. Here, the class equation (extended to monoids via standard group action by conjugation) is applied to factorizable submonoids of the symmetric inverse monoid. In particular, if is a monoid induced by a subgroup of the symmetric group , then the center (all elements of that commute with every element of ) is if and only if is transitive. In the case where is both transitive and of order either or (for prime), formulas are provided for the order of as well as the number and sizes of its conjugacy classes.

     

A characterization of compactly generated metric groups

Recall that a topological group is: (a) -compact if where each is compact, and (b) compactly generated if is algebraically generated by some compact subset of . Compactly generated groups are -compact, but the converse is not true: every countable non-finitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. We prove that a metric group is compactly generated if and only if is -compact and for every open subgroup of there exists a finite set such that algebraically generates . As a corollary, we obtain that a -compact metric group is compactly generated provided that one of the following conditions holds: (i) has no proper open subgroups, (ii) is dense in some connected group (in particular, if is connected itself), (iii) is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element (eventually constant sequences are not excluded here). Therefore, a countable metric group can be generated by a (possibly eventually constant) sequence converging to its identity element in each of the cases (i), (ii) and (iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.

     

The Nevanlinna counting functions for Rudin's orthogonal functions

and denote the Hardy spaces on the open unit disc . Let be a function in and . If is an inner function and , then is orthogonal in . W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function such that is not an inner function and is orthogonal in . In this paper, the following is shown: is orthogonal in if and only if there exists a unique probability measure on [0,1] with supp such that for nearly all in where is the Nevanlinna counting function of . If is an inner function, then is a Dirac measure at .

     

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if , are finite modules over a Gorenstein local ring of codimension at most , then the vanishing of for is equivalent to the vanishing of for . Furthermore, if has no embedded deformation, then such vanishing occurs if and only if or has finite projective dimension.

     

On commuting operator exponentials

Let , be bounded operators on a Banach space with -congruence-free spectra such that . E. M. E. Wermuth has shown that . Ch. Schmoeger later established this result, using inner derivations and, in a second paper, has shown that: for in a complex unital Banach algebra, if the spectrum of is -congruence-free and , then (and thus, answering an open problem raised by E. M. E. Wermuth). In this paper we use the holomorphic functional calculus to give alternative simple proofs of both of these results. Moreover, we use the Borel functional calculus to give new proofs of recent results of Ch. Schmoeger concerning normal operator exponentials on a complex Hilbert space, under a weaker hypothesis on the spectra.