A trace formula for Hankel operators

We show that if is an operator valued analytic function in the open right half plane such that the Hankel operator with symbol is of trace-class, then has continuous extension to the imaginary axis,

exists in the trace-class norm, and .

     

A semi-classical trace formula at a non-degenerate critical level

We study the semi-classical trace formula at a critical energy level for a h-pseudodifferential operator whose principal symbol has a unique non-degenerate critical point for that energy. This leads to the study of Hamiltonian systems near equilibrium and near the non-zero periods of the linearized flow. The contributions of these periods to the trace formula are expressed in terms of degenerate oscillatory integrals. The new results obtained are formulated in terms of the geometry of the energy surface and the classical dynamics on this surface.     

Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalize the Carey–Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.     

Regular colligations for several commuting operators in Banach space

Several important results of the theory of operator colligations, stated initially for strict commutative colligations in Hilbert space [6] [7], are extended to the more general setting of regular commutative colligations in Banach space, which are defined in the text.     

A trace formula for integro-differential operators on the finite interval

In this paper, we consider the eigenvalue problem for integro-differential operators with separated boundary conditions on the finite interval and find a trace formula for the integro-differential operator.     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.     

The inverse problem for the characteristic functions of several commuting operators

In this paper we derive conditions for an operator valued function to be the characteristic function of several commuting operators in a Hilbert space. We use the connection of this problem to some problems in partial differential equation to get a solution for a class of operator valued functions.     

A trace formula for multiplication operators on invariant subspaces of the Bergman space

Supposef is a bounded analytic function in the unit disk andM _f is the multiplication operator on an invariant subspaceI of the Bergman space. We show that wheren=dim(IzI).     

A local trace formula for resonances of perturbed periodic Schrödinger operators

We give a local trace formula for the pair (P₁(h)=P₀+W(hy),P₀), where P₀ is a periodic Schrödinger operator, W is a decreasing perturbation and h is a small positive parameter. We apply this result to establish the existence of ∼h⁻ⁿ resonances near some energy λ of σ(P₀).     

A trace formula

To my Father, William F. Eberlein.     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given.     

A trace formula for isometric pairs

It is well known that for every isometry , This fact for the shift operator is a basis for many important developments in operator theory and topology. In this paper we prove an analogous formula for a pair of isometries , namely

where is the complete anti-symmetric sum and is the Fredholm index of the pair . The major tool is what we call the fringe operator. Two examples are considered.

     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given. This research has been partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0008562.     

A trace estimate for p-hyponormal operators

We prove a generalization to p-hyponormal operators of Berger's and Shaw's estimate for the trace of the self-commutator of a hyponormal operator.