Rational matrix solutions to the Leech equation: The Ball-Trent approach revisited

Using spectral factorization techniques, a method is given by which rational matrix solutions to the Leech equation with rational matrix data can be computed explicitly. This method is based on an approach by J.A. Ball and T.T. Trent, and generalizes techniques from recent work of T.T. Trent for the case of polynomial matrix data.Comment: 15 page

Recent developments on equivalence after extension and Schur coupling

Two Banach space operators $U:\mathcal{X}_1\to{\mathcal{X}}_2$ and $V:\mathcal{Y}_1\to{\mathcal{Y}}_2$ are said to be (a) {\em equivalent after extension} if there exist Banach spaces $\mathcal{X}_0$ and $\mathcal{Y}_0$ such that $U \dotplus I_{\mathcal{X}_0}$ and $V\dotplus \mathcal{Y}_0$ are equivalent and (b) {\em Schur coupled} in case there exists an operator matrix $\left[\begin{smallmatrix} A&B\\C&D\end{smallmatrix}\right] :\left[\begin{smallmatrix} \mathcal{X}_1\\ \mathcal{Y}_1\end{smallmatrix}\right]\to \left[\begin{smallmatrix} \mathcal{X}_2\\ \mathcal{Y}_2\end{smallmatrix}\right]$ with $A$ and $D$ invertible and $$U=A-B D^{-1}C,\quad V=D-CA^{-1}B.$$ In the 1990s Bart and Tsekanovskii \cite{BT92,BT94} studied the relation between these two notions, and the notion of {\em matricial coupling} which coincides with equivalence after extension, and proved that Schur coupling implies equivalence after extension. The converse question, whether equivalence after extension implies Schur coupling, was answered affirmatively only for rather special classes of operators, until recently \cite{tHR13,T14,tHMR15}. In this talk we discuss some of these recent developments. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{9} \bibitem{BT92} H. Bart and V.E. Tsekanovskii, Matricial coupling and equivalence after extension, in: {\em Operator Theory and Complex Analysis}, OT {\bf 59}, 1992, pp.\ 143–-160. \bibitem{BT94} H. Bart and V.E. Tsekanovskii, Complementary Schur complements, {\em Linear Algebra Appl.} {\bf 197} (1994) 651-–658. \bibitem{tHMR15} S. Ter Horst, M. Messerschmidt, and A.C.M. Ran, Equivalence after extension for compact operators on banach spaces, {\em J. Math.\ Anal.\ Appl.} {\bf 431} (2015), 136–-149. \bibitem{tHR13} S. ter Horst and A.C.M. Ran, Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces, \emph{Linear Algebra Appl.} {\bf 439} (2013), 793--805. \bibitem{T14} D. Timotin, Schur coupling and related equivalence relations for operators on a Hilbert space, {\em Linear Algebra Appl.} {\bf 452} (2014), 106--119. \end{thebibliography

Equivalence after extension for compact operators on Banach spaces

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions on the singular values for compact Hilbert space operators that are equivalent after extension, we prove the necessity of similar relationships between the $s$-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary $s$-functions. We investigate equivalence after extension for operators on $\ell^{p}$-spaces. We show that two operators that act on different $\ell^{p}$-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension, however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces $X$ and $Y$, we investigate consequences of an operator on $X$ being equivalent after extension to a compact operator on $Y$. We show that, in this case, a closed finite codimensional subspace of $Y$ must embed into $X$, and that certain general Banach space properties must transfer from $X$ to $Y$. We also show that no operator on $X$ can be equivalent after extension to an operator on $Y$, if $X$ and $Y$ are essentially incomparable Banach spaces

Equivalence after extension and Schur coupling for relatively regular operators

It was recently shown in Ter Horst et al. (Bull Lond Math Soc 51:1005–1014, 2019) that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regular operators, the latter being operators with complementable kernels and ranges. In this paper we analyse the relations EAE and SC for the class of relatively regular operators, leading to an equivalent Banach space operator problem from which we derive new cases where EAE and SC coincide and provide a new example for which EAE and SC do not coincide and where the Banach spaces are not essentially incomparable.The National Research Foundation of South Africa (NRF) and the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS).http://link.springer.com/journal/202021-08-26hj2020Mathematics and Applied Mathematic

Cohort profile of Acutelines:a large data/biobank of acute and emergency medicine

Purpose Research in acute care faces many challenges, including enrolment challenges, legal limitations in data sharing, limited funding and lack of singular ownership of the domain of acute care. To overcome these challenges, the Center of Acute Care of the University Medical Center Groningen in the Netherlands, has established a de novo data, image and biobank named ‘Acutelines’.Participants Clinical data, imaging data and biomaterials (ie, blood, urine, faeces, hair) are collected from patients presenting to the emergency department (ED) with a broad range of acute disease presentations. A deferred consent procedure (by proxy) is in place to allow collecting data and biomaterials prior to obtaining written consent. The digital infrastructure used ensures automated capturing of all bed-side monitoring data (ie, vital parameters, electrophysiological waveforms) and securely importing data from other sources, such as the electronic health records of the hospital, ambulance and general practitioner, municipal registration and pharmacy. Data are collected from all included participants during the first 72 hours of their hospitalisation, while follow-up data are collected at 3 months, 1 year, 2 years and 5 years after their ED visit.Findings to date Enrolment of the first participant occurred on 1 September 2020. During the first month, 653 participants were screened for eligibility, of which 180 were approached as potential participants. In total, 151 (84%) provided consent for participation of which 89 participants fulfilled criteria for collection of biomaterials.Future plans The main aim of Acutelines is to facilitate research in acute medicine by providing the framework for novel studies and issuing data, images and biomaterials for future research. The protocol will be extended by connecting with central registries to obtain long-term follow-up data, for which we already request permission from the participant.Trial registration number NCT04615065