Infinite-dimensional difference operators are studied. Under the assumption that the coefficients of the operator have limits at infinity, limiting operators and associated polynomials are introduced. Under some specific conditions on the polynomials, the operator is Fredholm and has the zero index. Solvability conditions are obtained and the exponential behavior of solutions of the homogeneous...

PURPOSE We aimed to compare the intra- and interoperator variability of lobar volumetry and emphysema scores obtained by semi-automated and manual segmentation techniques in lung emphysema patients. MATERIALS AND METHODS In two sessions held three months apart, two operators performed lobar volumetry of unenhanced chest computed tomography examinations of 47 consecutive patients with chronic ...

A classical problem in operator theory has been to determine the spectrum of Toeplitz-like operators on Hilbert spaces of vector-valued holomorphic functions on the open unit ball in Cm. In this note we obtain necessary conditions for k-tuples of such operators to be Fredholm in the sense of Taylor and show they are sufficient in the case of the unit disk.

Correlation functions of the XXZ spin chain in the critical regime is studied at zerotemperature. They are exactly represented in the Fredholm determinant form and are related with an operator-valued Riemann–Hilbert problem. Analyzing this problem we prove that a two-point correlation function consisting of sufficiently separated spin operators is expressed by power-functions of the distance be...

Parasupersymmetric quantum mechanics is exploited to introduce a topological invariant associated with a pair of parameter dependent Fredholm (respectively elliptic diierential) operators satisfying two compatibility conditions. An explicit algebraic expression for this topological invariant is provided. The latter identiies the parasupersymmetric topological invariant with the sum of the analy...

Parasupersymmetric quantum mechanics is exploited to introduce a topological invariant associated with a pair of parameter dependent Fredholm (respectively elliptic differential) operators satisfying two compatibility conditions. An explicit algebraic expression for this topological invariant is provided. The latter identifies the parasupersymmetric topological invariant with the sum of the ana...

We explore the relationship between the Evans function, transmission coefficient and Fredholm determinant for systems of first-order linear differential operators on the real line. The applications we have in mind include linear stability problems associated with travelling wave solutions to nonlinear partial differential equations, for example reaction-diffusion or solitary wave equations. The...

For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

1. Compact operators on Banach spaces 2. Appendix: total boundedness and Arzela-Ascoli [Fredholm 1900/1903] treated compact operators as limiting cases of finite-rank operators. [1] [Riesz 1917] defined and made direct use of the compactness condition, more apt for Banach spaces. See [Riesz-Nagy 1952] for extensive discussion in the Hilbert-space situation, and many references to original paper...

In [EE1] and [EE2] we presented the solution to the index problem for a natural class of hypoelliptic differential operators on compact contact manifolds. The methods developed to deal with that problem have wider applicability to the index theory of hypoelliptic Fredholm operators. As an example of the power of the proof techniques we present here a new proof of a little known index theorem of...