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We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the principal pivot transform is matrix monotone by establishing Hermitian square representations for the imaginary part and the derivative.
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In this paper, some more properties of the generalized principal pivot transform are derived. Necessary and sufficient conditions for the equality between Moore-Penrose inverse of a generalized principal pivot transform and its complementary generalized principal pivot transform are presented. It has been shown that the generalized principal pivot transform preserves the rank of symmetric part of a given square matrix. These results appear to be more generalized than the existing ones. Inheritance property of $P_{dagger}$-matrix are also characterized for generalized principal pivot transform.
Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.
We study the problem of phase retrieval in which one aims to recover a function $f$ from the magnitude of its wavelet transform $|mathcal{W}_psi f|$. We consider bandlimited functions and derive new uniqueness results for phase retrieval, where the wavelet itself can be complex-valued. In particular, we prove the first uniqueness result for the case that the wavelet $psi$ has a finite number of vanishing moments. In addition, we establish the first result on unique reconstruction from samples of the wavelet transform magnitude when the wavelet coefficients are complex-valued
An analysis of the stability of the spindle transform, introduced in (Three dimensional Compton scattering tomography arXiv:1704.03378 [math.FA]), is presented. We do this via a microlocal approach and show that the normal operator for the spindle transform is a type of paired Lagrangian operator with blowdown--blowdown singularities analogous to that of a limited data synthetic aperture radar (SAR) problem studied by Felea et. al. (Microlocal analysis of SAR imaging of a dynamic reflectivity function SIAM 2013). We find that the normal operator for the spindle transform belongs to a class of distibutions $I^{p,l}(Deltacupwidetilde{Delta},Lambda)$ studied by Felea and Marhuenda (Microlocal analysis of SAR imaging of a dynamic reflectivity function SIAM 2013 and Microlocal analysis of some isospectral deformations Trans. Amer. Math.), where $widetilde{Delta}$ is reflection through the origin, and $Lambda$ is associated to a rotation artefact. Later, we derive a filter to reduce the strength of the image artefact and show that it is of convolution type. We also provide simulated reconstructions to show the artefacts produced by $Lambda$ and show how the filter we derived can be applied to reduce the strength of the artefact.
We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of $mathbb{Q}$. Our results contrast the classical case, where the analogous groups may not be abelian, and the free case, where free universal monodromy implies such notions would be trivial.
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