On the isolated points of the surjective spectrum of a bounded operator

For a bounded operator acting on a complex Banach space, we show that if is not surjective, then is an isolated point of the surjective spectrum of if and only if , where is the quasinilpotent part of and is the analytic core for . Moreover, we study the operators for which . We show that for each of these operators , there exists a finite set consisting of Riesz points for such that and is connected, and derive some consequences.

     

On Drazin invertibility

The left Drazin spectrum and the Drazin spectrum coincide with the upper semi--Browder spectrum and the -Browder spectrum, respectively. We also prove that some spectra coincide whenever or satisfies the single-valued extension property.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

On exceptional eigenvalues of the Laplacian for

An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

On the plane section of an integral curve in positive characteristic

If is an integral curve and an algebraically closed field of characteristic 0, it is known that the points of the general plane section of are in uniform position. From this it follows easily that the general minimal curve containing is irreducible. If char, the points of may not be in uniform position. However, we prove that the general minimal curve containing is still irreducible.

     

On endomorphism rings of local cohomology modules

Let be a local complete ring. For an -module the canonical ring map is in general neither injective nor surjective; we show that it is bijective for every local cohomology module if for every ( an ideal of ); furthermore the same holds for the Matlis dual of such a module. As an application we prove new criteria for an ideal to be a set-theoretic complete intersection.

     

Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let and be complete local noetherian rings with maximal ideals and such that is isomorphic to for every . Does it follow that and are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

     

On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups and , and a -adapted ring (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings and is monomial, i.e., maps class sums in to class sums in up to multiplication with roots of unity. As a consequence, and have identical character tables if and only if the centers of their integral group rings and are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

     

Almost positive curvature on the Gromoll-Meyer sphere

Gromoll and Meyer have represented a certain exotic 7-sphere as a biquotient of the Lie group . We show for a 2-parameter family of left invariant metrics on that the induced metric on has strictly positive sectional curvature at all points outside four subvarieties of codimension which we describe explicitly.

     

Global co-stationarity of the ground model from a new countable length sequence

Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.

     

Weyl's theorem for perturbations of paranormal operators

A bounded linear operator on a Banach space is said to satisfy ``Weyl's theorem' if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if is a paranormal operator on a Hilbert space, then satisfies Weyl's theorem for every algebraic operator which commutes with .

     

A counterexample for boundedness of pseudo-differential operators on modulation spaces

We prove that pseudo-differential operators with symbols in the class ( ) are not always bounded on the modulation space ().

     

Infinite index subalgebras of depth two

An algebra extension is right depth two in this paper if its tensor-square is --isomorphic to a direct summand of any (not necessarily finite) direct sum of with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite-dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory (Kadison and Szlachányi, 2003) but extends the main theorem of depth two theory, as for example in (Kadison and Nikshych, 2001). That is, a right depth two extension has right bialgebroid over its centralizer . The main theorem: An extension is right depth two and right balanced if and only if is -Galois with respect to left projective, right -bialgebroid .

     

Tauberian type theorem for operators with interpolation spectrum for Hölder classes

We consider an invertible operator on a Banach space whose spectrum is an interpolating set for Hölder classes. We show that if , , with and , then for all , assuming that satisfies suitable regularity conditions. When is a Hilbert space and (i.e. is a contraction), we show that under the same assumptions, is unitary and this is sharp.

     

On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .

     

A Banach-Stone theorem for Riesz isomorphisms of Banach lattices

Let and be compact Hausdorff spaces, and , be Banach lattices. Let denote the Banach lattice of all continuous -valued functions on equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism such that is non-vanishing on if and only if is non-vanishing on , then is homeomorphic to , and is Riesz isomorphic to . In this case, can be written as a weighted composition operator: , where is a homeomorphism from onto , and is a Riesz isomorphism from onto for every in . This generalizes some known results obtained recently.

     

-identities for the Grassmann algebra: The conjecture of Henke and Regev

Let be an algebraically closed field of characteristic 0, and let be the infinite dimensional Grassmann (or exterior) algebra over . Denote by the vector space of the multilinear polynomials of degree in , ..., in the free associative algebra . The symmetric group acts on the left-hand side on , thus turning it into an -module. This fact, although simple, plays an important role in the theory of PI algebras since one may study the identities satisfied by a given algebra by applying methods from the representation theory of the symmetric group. The -modules and are canonically isomorphic. Letting be the alternating group in , one may study and its isomorphic copy in with the corresponding action of . Henke and Regev described the -codimensions of the Grassmann algebra , and conjectured a finite generating set of the -identities for . Here we answer their conjecture in the affirmative.

     

Groups with a character of large degree

Let be a finite group of order and a simple -module of dimension . For some nonnegative number , we have . If is small, then the character of has unusually large degree. We fix and attempt to classify such groups. For we give a complete classification. For any other fixed we show that there are only finitely many examples.

     

A short proof of Hara and Nakai's theorem

We give a short proof of the following theorem of Hara and Nakai: for a finitely bordered Riemann surface , one can find an upper bound of the corona constant of that depends only on the genus and the number of boundary components of .