-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.

     

An application of Ramsey's Theorem to the Banach Contraction Principle

One of the most fundamental fixed-point theorems is Banach's Contraction Principle, of which the following conjecture is a generalization.

Generalized Banach Contraction Conjecture (GBCC). Let be a self-map of a complete metric space , and let . Let be a positive integer. Assume that for each pair , . Then has a fixed point.

Unlike Banach's original theorem (the case ), the above hypothesis does not compel to be continuous. In this paper we use Ramsey's Theorem from combinatorics to establish the GBCC for arbitrary in the case when is assumed to be continuous, and also derive a result which enables us to prove the GBCC when without the assumption of continuity; it is known that the case includes instances where is not continuous.

     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

On the irreducibility of the iterates of

Let be a field and suppose that is irreducible in . We discuss the following question: under what conditions are all iterates of irreducible over ?

     

The strong radical and finite-dimensional ideals

Let be a semi-prime Banach algebra with strong radical (intersection of its two-sided modular maximal ideals). A minimal left or right ideal of is infinite-dimensional if and only if . Thus all minimal one-sided ideals in are finite-dimensional if is strongly semi-simple.

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

A question of B. Plotkin about the semigroup of endomorphisms of a free group

Let be a free group of finite rank , let be the semigroup of endomorphisms of , and let be the group of automorphisms of .

Theorem. If is an automorphism of , then there is an such that for all .

     

Index of B-Fredholm operators and generalization of a Weyl theorem

The aim of this paper is to show that if and are commuting B-Fredholm operators acting on a Banach space , then is a B-Fredholm operator and , where means the index. Moreover if is a B-Fredholm operator and is a finite rank operator, then is a B-Fredholm operator and We also show that if is isolated in the spectrum of , then is a B-Fredholm operator of index if and only if is Drazin invertible. In the case of a normal bounded linear operator acting on a Hilbert space , we obtain a generalization of a classical Weyl theorem.     

Translation invariant Julia sets

We show that if the Julia set of a rational function is invariant under translation by one and infinity is a periodic or preperiodic point for , then must either be a line or the Riemann sphere.

     

The f-depth of an ideal on a module

Let be an ideal of a Noetherian local ring and a finitely generated -module. The f-depth of on is the least integer such that the local cohomology module is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules.

     

Composition of inner mappings on the ball

Suppose that is an inner map and that . We show that the identity

holds with an abstract boundary value . If the natural compatibility condition is satisfied, then . Here, denotes the image of the surface measure on under . In particular, is inner if and are inner and . Furthermore, we characterize the boundedness of composition operators on Hardy spaces in terms of the absolute continuity of .

     

Set-theoretic complete intersections on binomials

Let be an affine toric variety of codimension over a field of any characteristic. We completely characterize the affine toric varieties that are set-theoretic complete intersections on binomials. In particular we prove that in the characteristic zero case, is a set-theoretic complete intersection on binomials if and only if is a complete intersection. Moreover, if are binomials such that , then . While in the positive characteristic case, is a set-theoretic complete intersection on binomials if and only if is completely -glued.

These results improve and complete all known results on these topics.

     

Invariant ideals of abelian group algebras under the multiplicative action of a field. I

Let be a division ring and let be a finite-dimensional -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . Our goal is to completely describe the semiprime -stable ideals of . As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.

     

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group with principal isotropy group and cohomogeneity such that . Moreover, we prove that the Euler characteristic of a compact Riemannian manifold or with positive sectional curvature is positive if admits an effective isometric action of a torus , i.e., if the symmetry rank of is .

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

Spectral synthesis for

Let be a locally compact group, the Fourier algebra of and the von Neumann algebra generated by the left regular representation of . We introduce the notion of -spectral set and -Ditkin set when is an -invariant linear subspace of , thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of -spectral sets and -Ditkin sets, and an injection theorem for -spectral sets.

     

On classes of maps which preserve finitisticness

We shall prove the following: Let be a refinable map between paracompact spaces. Then is finitistic if and only if is finitistic. Let be a hereditary shape equivalence between metric spaces. Then if is finitistic, is finitistic.

     

A matrix-valued Choquet-Deny theorem

Let be a positive matrix-valued measure on a locally compact abelian group such that is the identity matrix. We give a necessary and sufficient condition on for the absence of a bounded non-constant matrix-valued function on satisfying the convolution equation . This extends Choquet and Deny's theorem for real-valued functions on .

     

Schur orthogonality relations and invariant sesquilinear forms

Important connections between the representation theory of a compact group and are summarized by the Schur orthogonality relations. The first part of this work is to generalize these relations to all finite-dimensional representations of a connected semisimple Lie group The second part establishes a general framework in the case of unitary representations of a separable locally compact group. The key step is to identify the matrix coefficient space with a dense subset of the Hilbert-Schmidt endomorphisms on .

     

On a semilinear Schrödinger equation with critical Sobolev exponent

We consider the semilinear Schrödinger equation , , where , are periodic in for , 0$">, is of subcritical growth and 0 is in a gap of the spectrum of . We show that under suitable hypotheses this equation has a solution . In particular, such a solution exists if and .