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We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to a projector Q such that Range(Q) has co-dimension m in Range(P). We prove that if A is an asymptotically hyperbolic and essentially splitting path the differential operator F(u) = du/dt - Au is Fredholm. Moreover if A is also essentially hyperbolic the Fredholm index coincides with minus the spectral flow of A.
We study some fundamental properties of semicocycles over semigroups of self-mappings of a domain in a Banach space. We prove that any semicocycle over a jointly continuous semigroup is itself jointly continuous. For semicocycles over semigroups whic
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for
Assume that $mathcal{I}$ is an ideal on $mathbb{N}$, and $sum_n x_n$ is a divergent series in a Banach space $X$. We study the Baire category, and the measure of the set $A(mathcal{I}):=left{t in {0,1}^{mathbb{N}} colon sum_n t(n)x_n textrm{ is } mat
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box di
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $ell_1$. We present many examples and several properties of this class. We give some applicatio