A sphere theorem for odd-dimensional submanifolds of spheres

We establish a topological sphere theorem from the point of view of submanifold geometry for odd-dimensional submanifolds of a unit sphere. We give examples which show that our result is optimal. Moreover, we note the assumption that the dimension is odd is essential.

     

Triangulations and homology of Riemann surfaces

We derive an algorithmic way to pass from a triangulation to a homology basis of a (Riemann) surface. The procedure will work for any surfaces with finite triangulations. We will apply this construction to Riemann surfaces to show that every compact hyperbolic Riemann surface has a homology basis consisting of curves whose lengths are bounded linearly by the genus of and by the homological systole.

This work got started by comments presented by Y. Imayoshi in his lecture at the 37th Taniguchi Symposium which took place in Katinkulta near Kajaani, Finland, in 1995.

     

A short proof of an index theorem

We give a -theoretical proof of an index theorem for Dirac-Schrödinger operators on a noncompact manifold.

     

Algebraic structure in the loop space homology Bockstein spectral sequence

Let be a finite, -dimensional, -connected CW complex. We prove the following theorem:

If is an odd prime, then the loop space homology Bockstein spectral sequence modulo is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.

     

On Bredon homology of elementary amenable groups

We show that for elementary amenable groups the Hirsch length is equal to the Bredon homological dimension. This also implies that countable elementary amenable groups admit a finite-dimensional model for of dimension less than or equal to the Hirsch length plus one. Some remarks on groups of type are also made.

     

A criterion on weighted boundedness for rough multilinear oscillatory singular integrals

In this paper the authors give a criterion on the weighted boundedness of the multilinear oscillatory singular integral operators with rough kernels.

     

Compact sets of compact operators in absence of

We characterize the compactness of a subset of compact operators between Banach spaces when the domain space does not have a copy of

     

Cuntz-Krieger algebras and endomorphisms of finite direct sums of type I factors

A correspondence between algebra endomorphisms of a finite sum of copies of the algebra of all bounded operators on a Hilbert space and representations of certain norm closed -subalgebras of bounded operators generated by a finite collection of partial isometries is introduced. Basic properties of this correspondence are investigated after developing some operations on bipartite graphs that usefully describe aspects of this relationship.

     

Symplectic Dolbeault operators on Kähler manifolds

For a Kähler manifold $M$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar{\partial }$ and $\bar{\partial }^*$ , arise from Dirac operators on the canonical complex spinors on $M$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$ . We give a thorough analysis of these operators on $\mathbb{C } P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.     

Uniqueness of the Kontsevich-Vishik trace

Let be a closed manifold. We show that the Kontsevich-Vishik trace, which is defined on the set of all classical pseudodifferential operators on , whose (complex) order is not an integer greater than or equal to , is the unique functional which (i) is linear on its domain, (ii) has the trace property and (iii) coincides with the -operator trace on trace class operators.

Also the extension to even-even pseudodifferential operators of arbitrary integer order on odd-dimensional manifolds and to even-odd pseudodifferential operators of arbitrary integer order on even-dimensional manifolds is unique.

     

On homological properties of singular braids

Homology of objects which can be considered as singular braids, or braids with crossings, is studied. Such braids were introduced in connection with Vassiliev's theory of invariants of knots and links. The corresponding algebraic objects are the braid-permutation group of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid which embeds into the singular braid group . The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group

where is an infinite loop space and is a double loop space.

     

An example in the theory of -operators

-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact -operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper.

     

Powers and roots of Toeplitz operators

We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in which commute with Toeplitz operators whose symbols are of the form .

     

Multi-linear operators given by singular multipliers

We prove estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer (1991), as well as the bilinear Hilbert transform and other operators with large groups of modulation symmetries.

     

Harmonic calculus on fractals--A measure geometric approach II

Riesz potentials of fractal measures in metric spaces and their inverses are introduced. They define self-adjoint operators in the Hilbert space and the former are shown to be compact.

In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal.

     

Sharp Gaussian bounds and -growth of semigroups associated with elliptic and Schrödinger operators

We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrödinger operators, including Schrödinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrödinger operators , we have the -estimate (for large ):

where is the spectral bound of The same estimate holds for elliptic and Schrödinger operators on general domains.

     

On approximations of rank one -perturbations

Approximations of rank one -perturbations of self-adjoint operators by operators with regular rank one perturbations are discussed. It is proven that in the case of arbitrary not semibounded operators such approximations in the norm resolvent sense can be constructed without any renormalization of the coupling constant. Approximations of semibounded operators are constructed using rank one non-symmetric regular perturbations.

     

Hochschild homology criteria for trivial algebra structures

We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.

     

Products of roots of the identity

It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five -th roots of the identity for every 2$">. For invertible normal operators four factors suffice in general.

     

On the Helton class of -hyponormal operators

In this paper we show that the Helton class of -hyponormal operators has scalar extensions. As a corollary we get that each operator in the Helton class of -hyponormal operators has a nontrivial invariant subspace if its spectrum has its interior in the plane.