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Register a new userA Lebesgue-type decomposition on one side for sesquilinear forms
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Rosario Corso
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Sesquilinear forms which are not necessarily positive may have a different behavior, with respect to a positive form, on each side. For this reason a Lebesgue-type decomposition on one side is provided for generic forms satisfying a boundedness condition.
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The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation theorems of sesquilinear forms, such as Katos theorems. The associated operators correspond to classical frame operators or weakly-defined multipliers in the bounded context. In general some properties of them, such as the invertibility and the resolvent set, are related to properties of the sesquilinear forms. As an upshot of this approach new features of sequences (or pairs of sequences) which are semi-frames (or reproducing pairs) are obtained.
A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial $mathbb{C}^n$-bundle over a smooth $m$-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our goal is to classify such forms up to $GL(n,mathbb{C})$ gauge equivalence. We achieve this classification in the special case of $m=4$ and $n=2$ by means of geometric and topological invariants (e.g. Lorentzian metric, spin/spin$^c$ structure, electromagnetic covector potential) naturally contained within the sesquilinear form - a purely analytic object. Essential to our approach is the interplay of techniques from analysis, geometry, and topology.
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Let $G $ be a noncompact semisimple Lie group with finite centre. Let $X=G/K$ be the associated Riemannian symmetric space and assume that $X$ is of rank one. The spectral projections associated to the Laplace-Beltrami operator are given by $P_{lambda}f =fast Phi_{lambda}$, where $Phi_{lambda}$ are the elementary spherical functions on $X$. In this paper, we prove an Ingham type uncertainty principle for $P_{lambda}f$. Moreover, similar results are obtained in the case of spectral projections associated to Dunkl Laplacian.
Two sesquilinear forms $Phi:mathbb C^mtimesmathbb C^mto mathbb C$ and $Psi:mathbb C^ntimesmathbb C^nto mathbb C$ are called topologically equivalent if there exists a homeomorphism $varphi :mathbb C^mto mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $Phi(x,y)=Psi(varphi (x),varphi (y))$ for all $x,yin mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix $A$; that is, a direct sum $SAS^*=Roplus J_{n_1}oplusdotsoplus J_{n_p}$, in which $S$ and $R$ are nonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ singular Jordan block. In this paper, we prove that $Phi$ and $Psi$ are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands $J_{n_i}$ and replacement of $Rinmathbb C^{rtimes r}$ by a nonsingular matrix $Rinmathbb C^{rtimes r}$ such that $R$ and $R$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained.
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