On a generalized punctured neighborhood theorem in $\mathcal{L}(X)$

On the Punctured Neighbourhood Theorem

Let X, Y, Z be Banach spaces and X S(z) −→Y T (z) −→Z an analytically dependant sequence of operators satisfying T (z)S(z) = 0. We study properties of the function z 7→ dim Ker T (z)/ Im S(z). Let X, Y be complex Banach spaces. Denote by L(X,Y ) the set of all bounded linear operators from X to Y . If Y = X then we write for short L(X) = L(X, X). Recall the well-known punctured neighbourhood th...

A Fixed Point Theorem in Generalized D-metric Spaces

On a generalized Sturm theorem

Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose here is a Sturm theorem for indefinite systems with Dirichlet boundary conditions of the form p2m du dx + p2m−2(x) du dx + · · ·+ p1(x) du dx + p0(x)u = 0, where pi is a smooth path of matrices on the comple...

The generalized Lefschetz number of homeomorphisms on punctured disks

We compute the generalized Lefschetz number of orientation-preserving selfhomeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canoni...

On Generalized Stone’s Theorem

It is known that the generator of a strictly continuous one parameter unitary group in the multiplier algebra of a C∗ -algebra is affiliated to that C∗ -algebra . We show that under natural non degeneracy conditions, this self adjoint unbounded operator lies indeed in the (unbounded) multiplier algebra of the Pedersen’s ideal of the C∗ -algebra .