Vector valued logarithmic residues and the extraction of elementary factors

An analysis is presented of the circumstances under which, by the extraction of elementary factors, an analytic Banach algebra valued function can be transformed into one taking invertible values only. Elementary factors are generalizations of the simple scalar expressions λ – α, the building blocks of scalar polynomials. In the Banach algebra situation they have the form e – p + (λ – α)p with p an idempotent. The analysis elucidates old results (such as on Fredholm operator valued functions) and yields new insights which are brought to bear on the study of vector-valued logarithmic residues. These are contour integrals of logarithmic derivatives of analytic Banach algebra valued functions. Examples illustrate the subject matter and show that new ground is covered. Also a long standing open problem is discussed from a fresh angle.analytic vector-valued function;annihilating family of idempotents;elementary factor;generalizations of analytic functions;idempotent;integer combination of idempotents;logarithmic residue;plain function;resolving family of traces;topological algebras

Exact semi-relativistic model for ionization of atomic hydrogen by electron impact

We present a semi-relativistic model for the description of the ionization process of atomic hydrogen by electron impact in the first Born approximation by using the Darwin wave function to describe the bound state of atomic hydrogen and the Sommerfeld-Maue wave function to describe the ejected electron. This model, accurate to first order in $Z/c$ in the relativistic correction, shows that, even at low kinetic energies of the incident electron, spin effects are small but not negligible. These effects become noticeable with increasing incident electron energies. All analytical calculations are exact and our semi-relativistic results are compared with the results obtained in the non relativistic Coulomb Born Approximation both for the coplanar asymmetric and the binary coplanar geometries.Comment: 8 pages, 6 figures, Revte

Absolute differential cross sections for electron-impact excitation of CO near threshold: II. The Rydberg states of CO

Absolute differential cross sections for electron-impact excitation of Rydberg states of CO have been measured from threshold to 3.7 eV above threshold and for scattering angles between 20° and 140°. Measured excitation functions for the b 3Σ+, B 1Σ+ and E 1π states are compared with cross sections calculated by the Schwinger multichannel method. The behaviour of the excitation functions for these states and for the j 3Σ+ and C 1Σ+ states is analysed in terms of negative-ion states. One of these resonances has not been previously reported

Asymptotics of block Toeplitz determinants and the classical dimer model

We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice ($t=0$) and the triangular lattice ($t=1$), and we obtain the asymptotics for $0<t\le 1$. For $0<t<1$ we apply the Szeg\"o Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form

Logarithmic residues, Rouché’s theorem, and spectral regularity: The C∗-algebra case

AbstractUsing families of irreducible Hilbert space representations as a tool, the theory of analytic Fredholm operator valued function is extended to a C∗-algebra setting. This includes a C∗-algebra version of Rouché’s Theorem known from complex function theory. Also, criteria for spectral regularity of C∗-algebras are developed. One of those, involving the (generalized) Calkin algebra, is applied to C∗-algebras generated by a non-unitary isometry

Logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encoutered in previous investigations. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. The set of sums of idempotens and the set of logarithmic residues have an intriguing topological structure.Banach algebra;Logarithmic residues;sums of idempotents

Logarithmic residues of analytic Banach algebra valued functions possessing a simply meromorphic inverse

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter.Logarithmic residues;Cauchy domains;analytic Banach algebra valued function;meromorphic inverse

Logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues and sums of idempotents in the Banach algebra generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encoutered in previous investigations. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. The set of sums of idempotens and the set of logarithmic residues have an intriguing topological structure

Logarithmic residues of analytic Banach algebra valued functions possessing a simply meromorphic inverse

A logarithmic residue is a contour integral of a logarithmic derivative (left or right) of an analytic Banach algebra valued function. For functions possessing a meromorphic inverse with simple poles only, the logarithmic residues are identified as the sums of idempotents. With the help of this observation, the issue of left versus right logarithmic residues is investigated, both for connected and nonconnected underlying Cauchy domains. Examples are given to elucidate the subject matter

Logarithmic residues in Banach algebras

Let f be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivative f′f-1 around a Cauchy domain D vanishes. Does it follow that f takes invertible values on all of D? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues