Maps preserving Fredholm or semi-Fredholm elements relative to some ideal

Research Article Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

Some Remarks on Perturbation Classes of Semi-Fredholm and Fredholm Operators

We show the existence of Banach spaces X, Y such that the set of strictly singular operators ᏿(X,Y) (resp., the set of strictly cosingular operators Ꮿ᏿(X,Y)) would be strictly included in F + (X,Y) (resp., F − (X,Y)) for the nonempty class of closed densely defined upper semi-Fredholm operators Φ + (X,Y) (resp., for the nonempty class of closed densely defined lower semi-Fredholm operators Φ − ...

On Orientability and Degree of Fredholm Maps

The degree of a map between two manifolds has played important roles in various mathematical areas. Certain orientability is always required in order to make sense of the concept of degree. In the case of finite-dimensional nonorientable manifolds, this goes back to Hopf, Olum, and Steenrod, after Brouwer’s pioneering work on orientable manifolds (cf. [9] and references therein). Elworthy and T...

Fredholm Determinants, Anosov Maps and Ruelle Resonances

I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding RuellePerron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for example, that the zeroes of the dynamical determinant describe the eigenvalues of the transfer operator and the Ruelle resonances and that, for C∞ Anosov diff...

A Note on Semi-fredholm Hilbert Modules

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.