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Register a new userContinuous and holomorphic semicocycles in Banach spaces
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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 which have generator, we establish a sufficient condition for differentiablity with respect to the time variable, and hence for the semicocycle to satisfy a linear evolution problem, giving rise to the notion of `generator of a semicocycle. Bounds on the growth of a semicocycle with respect to the time variable are given in terms of this generator. Special consideration is given to the case of holomorphic semicocycles, for which we prove an exact correspondence between certain uniform continuity properties of a semicocyle and boundedness properties of its generator.
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We consider holomorphic semicocycles on the open unit ball in a Banach space taking values in a Banach algebra. We establish criteria for a semicocycle to be linearizable, that is, cohomologically equivalent to one independent of the spatial variable.
We develop a novel and unifying setting for phase retrieval problems that works in Banach spaces and for continuous frames and consider the questions of uniqueness and stability of the reconstruction from phaseless measurements. Our main result states that also in this framework, the problem of phase retrieval is never uniformly stable in infinite dimensions. On the other hand, we show weak stability of the problem. This complements recent work [9], where it has been shown that phase retrieval is always unstable for the setting of discrete frames in Hilbert spaces. In particular, our result implies that the stability properties cannot be improved by oversampling the underlying discrete frame. We generalize the notion of complement property (CP) to the setting of continuous frames for Banach spaces (over $mathbb{K}=mathbb{R}$ or $mathbb{K}=mathbb{C}$) and verify that it is a necessary condition for uniqueness of the phase retrieval problem; when $mathbb{K}=mathbb{R}$ the CP is also sufficient for uniqueness. In our general setting, we also prove a conjecture posed by Bandeira et al. [5], which was originally formulated for finite-dimensional spaces: for the case $mathbb{K}=mathbb{C}$ the strong complement property (SCP) is a necessary condition for stability. To prove our main result, we show that the SCP can never hold for frames of infinite-dimensional Banach spaces.
This paper studies holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is a solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the non-commutative case non-linearizable semicocycles are shown to exist. Simple conditions for linearizability are derived and are shown to be sharp.
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 functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
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 $A(mathcal{I}):=left{t in {0,1}^{mathbb{N}} colon sum_n t(n)x_n textrm{ is } mathcal{I}textrm{-convergent}right}$. In the category case, we assume that $mathcal{I}$ has the Baire property and $sum_n x_n$ is not unconditionally convergent, and we deduce that $A(mathcal{I})$ is meager. We also study the smallness of $A(mathcal{I})$ in the measure case when the Haar probability measure $lambda$ on ${0,1}^{mathbb{N}}$ is considered. If $mathcal{I}$ is analytic or coanalytic, and $sum_n x_n$ is $mathcal{I}$-divergent, then $lambda(A(mathcal{I}))=0$ which extends the theorem of Dindov{s}, v{S}alat and Toma. Generalizing one of their examples, we show that, for every ideal $mathcal{I}$ on $mathbb{N}$, with the property of long intervals, there is a divergent series of reals such that $lambda(A(Fin))=0$ and $lambda(A(mathcal{I}))=1$.
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