The Cayley transform of the generator of a polynomially stable $$C_0$$-semigroup

In this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable $$C_0$$ C 0 -semigroup. To estimate the decay rate of the Cayley transform, we develop an integral condition on resolvents for polynomial stability. Using this integral condition, we relate polynomial stability to Lyapunov equations. We also study robustness of polynomial stability for a certain class of structured perturbations.

Dimensional Reduction and Scattering Formulation for Even Topological Invariants

Strong invariants of even-dimensional topological insulators of independent Fermions are expressed in terms of an invertible operator on the Hilbert space over the boundary. It is given by the Cayley transform of the boundary restriction of the half-space resolvent. This dimensional reduction is routed in new representation for the K-theoretic exponential map. It allows to express the invariants via the reflection matrix at the Fermi energy, for the scattering set-up of a wire coupled to the half-space insulator.

The Cayley transform in complex, real and graded K-theory

We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele [Formula: see text]-theory to [Formula: see text]-theory for graded [Formula: see text]-algebras with a real structure. Isomorphisms between [Formula: see text]-theory and complex or real [Formula: see text]-theory for ungraded [Formula: see text]-algebras are a special case of this map. In all cases, our map is compatible with the computational techniques required in physical and geometrical applications, in particular, index pairings and Kasparov products. We provide applications to real [Formula: see text]-theory and topological phases of matter.