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We consider the canonical Wiener-Hopf factorisation of $2 \times 2$ symmetric matrices $\mathcal M$ with respect to a contour $\Gamma$. For the case that the quotient $q$ of the two diagonal elements of $\mathcal M$ is a rational function, we show that due to the symmetric nature of the matrix $\mathcal M$, the second column in each of the two matrix factors that arise in the factorisation is determined in terms of the first column in each of these matrix factors, by multiplication by a rational matrix, and we give a method for determining the second columns of these factors. We illustrate our method with two examples in the context of a Riemann-Hilbert approach to obtaining solutions to the Einstein field equations., Comment: 20 pages

This paper considers paired operators in the context of the Lebesgue Hilbert space $L^2$ on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Inclusion relations between such kernels are considered in detail, and the results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators., Comment: 22 pages

We review the basic properties of paired operators and their adjoints, the transposed paired operators, with particular reference to commutation relations, and we study the properties of their kernels, bringing out their similarities and also, somewhat surprisingly, their stark differences. Various notions expressing different invariance properties are also reviewed and we extend to paired operators some known invariance results., Comment: 24 pages

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix $M$ depending on the Weyl coordinates $\rho$ and $v$, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the $(\rho,v)$ plane on which some elements of $M(\rho,v)$ tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces., Comment: 23 pages; v2: matches published version

This paper considers paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space $H^2$. The kernels of such operators, together with their analytic projections, which are generalizations of Toeplitz kernels, are studied. Results on near-invariance properties, representations, and inclusion relations for these kernels are obtained. The existence of a minimal Toeplitz kernel containing any projected paired kernel and, more generally, any nearly $S^*$-invariant subspace of $H^2$, is derived. The results are applied to describing the kernels of finite-rank asymmetric truncated Toeplitz operators., Comment: 26 pages; some minor corrections

This paper is concerned with paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space. By considering when such operators commute, generalizations of the Brown--Halmos results for Toeplitz operators are derived. Further, the kernels of such operators are described, giving results on invariant and nearly-invariant subspaces, together with a generalization of Coburn's theorem on Toeplitz kernels., Comment: 17 pages. Minor revisions

Explicit solutions to the non-linear field equations of some gravitational theories can be obtained, by means of a Riemann-Hilbert approach, from a canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices. In this paper we describe other types of factorisation from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann-Hilbert problem and allows to construct solutions that could not have been obtained by Wiener-Hopf factorisation of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener-Hopf factorisation, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein-Rosen wave and gravitational pulse wave solutions., Comment: 29 pages; v2:various improvements; v3: matches published version; a typo corrected

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators., Comment: 17 pages

The property of being shift invariant and being reflexive or transitive in the case of the space of (asymmetric) truncated Toeplitz operators, and the space of (asymmetric) dual truncated operators is investigated. Most of the results obtained are new even for the symmetric case. A characterization of asymmetric dual truncated Toeplitz operators is also given.

We relate dual-band general Toeplitz operators to block truncated Toeplitz operators and, via equivalence after extension, with Toeplitz operators with $4 \times 4$ matrix symbols. We discuss their norm, their kernel, Fredhomlness, invertibility and spectral properties in various situations, focusing on the spectral properties of the dual-band shift, which turns out to be considerably complex, leading to new and nontrivial connections with the boundary behaviour of the associated inner function., Comment: 30 pages. Title changed from the previous version. Simpler proofs found for some of the later results

Following Beurling's theorem the natural compressions of the multiplication operator in the classical $L^2$ space are compressions to model spaces and to their orthogonal complements. Two possibly different model spaces are considered hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.

We consider kernels of unbounded Toeplitz operators in $H^p(\mathbb C^+)$ in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in $H^p(\mathbb C^+)$, we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of $z$. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols., Comment: 31 pages, 2 figures

For an inner function u we discuss the dual operator for the well-known compressed shift. We establish conditions for two dual compressed shifts to be unitarily equivalent/similar and we describe the invariant subspace structure for the dual., Comment: 14 pages

Truncated Toeplitz operators in a model space are C--symmetric with respect to a natural conjugation in that space. We show that this and another conjugation associated to an orthogonal decomposition possess unique properties and we study their relations with asymmetric truncated Toeplitz operators in terms of C--symmetry. New connections with Hankel operators are established through this approach.

Conjugations in space $L^2$ of the unit circle commuting with multiplication by $z$ or intertwining multiplications by $z$ and $\bar z$ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces.

Conjugations commuting with $\mathbf{M}_z$ and intertwining $\mathbf{M}_z$ and $\mathbf{M}_{\bar z}$ in $L^2(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space $H^2(\mathcal{H})$ or a model space $K_{\Theta}=H^2(\mathcal{H})\ominus\Theta H^2(\mathcal{H})$, where $\Theta$ is a pure operator valued inner function.

Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on $L^2(\mathbb T) \oplus L^2(\mathbb T)$ and, if their symbols are invertible in $L^\infty(\mathbb T)$, to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson's corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows us to construct explicit solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results., Comment: 48 pages, 7 figures; v2: added section 10; matches published version

It is shown that the kernel of a Toeplitz operator with $2\times 2$ symbol $G$ can be described exactly in terms of any given function in a very wide class, its image under multiplication by $G$, and their left inverses, if the latter exist. As a consequence, under many circumstances the kernel of a block Toeplitz operator may be described as the product of a space of scalar complex-valued functions by a fixed column vector of functions. Such kernels are said to be of scalar type, and in this paper they are studied and described explicitly in many concrete situations. Applications are given to the determination of kernels of truncated Toeplitz operators for several new classes of symbols., Comment: 34 pages, 1 figure. Some typos corrected and clarifying comments added

We review some classical and more recent results concerning kernels of Toeplitz operators and their relations with model spaces, which are themselves Toeplitz kernels of a special kind. We highlight the fundamental role played by the existence of maximal vectors for every nontrivial Toeplitz kernel., Comment: 20 pages, based on talk given at IWOTA 2017. This version incorporates some minor corrections

Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in $L^p(\mathbb R)$., Comment: 26 pages

We construct rotating extremal black hole and attractor solutions in gravity theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. By employing a vectorial Riemann-Hilbert factorization method we explicitly factorize the corresponding monodromy matrices, which have second order poles in the spectral parameter. In the underrotating case we identify elements of the Geroch group which implement Harrison-type transformations which map the attractor geometries to interpolating rotating black hole solutions. The factorization method we use yields an explicit solution to the linear system, from which we do not only obtain the spacetime solution, but also an explicit expression for the master potential encoding the potentials of the infinitely many conserved currents which make this sector of gravity integrable., Comment: Minor changes, final version, accepted by JHEP. 75 pages, 2 figures

Multipliers between kernels of Toeplitz operators are characterised in terms of test functions (so-called maximal vectors for the kernels); these maximal vectors may easily be parametrised in terms of inner and outer factorizations. Immediate applications to model spaces are derived. The case of surjective multipliers is also analysed. These ideas are applied to describing equivalences between two Toeplitz kernels., Comment: 21 pages. Final revisions made

This paper considers model spaces in an $H_p$ setting. The existence of unbounded functions and the characterisation of maximal functions in a model space are studied, and decomposition results for Toeplitz kernels, in terms of model spaces, are established., Comment: 26 pages. Accepted by Operators and Matrices

Truncated Toeplitz operators and their asymmetric versions are studied in the context of the Hardy space $H^p$ of the half-plane for $1

The local spectrum of a vertex set in a graph has been proven to be very useful to study some of its metric properties. It also has applications in the area of pseudo-distance-regularity around a set and can be used to obtain quasi-spectral characterizations of completely (pseudo-)regular codes. In this paper we study the relation between the local spectrum of a vertex set and the local spectrum of each of its subconstituents. Moreover, we obtain a new characterization for completely pseudo-regular codes, and consequently for completely regular codes, in terms of the relation between the local spectrum of an extremal set of vertices and the local spectrum of its antipodal set. We also present a new proof of the version of the Spectral Excess Theorem for extremal sets of vertices.

A graph is edge-distance-regular when it is distance-regular around each of its edges and it has the same intersection numbers for any edge taken as a root. In this paper we give some (combinatorial and algebraic) proofs of the fact that every edge-distance-regular graph $\G$ is distance-regular and homogeneous. More precisely, $\G$ is edge-distance-regular if and only if it is bipartite distance-regular or a generalized odd graph. Also, we obtain the relationships between some of their corresponding parameters, mainly, the distance polynomials and the intersection numbers.

The solvability of the Riemann-Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener-Hopf type factorization with bounded outer factors but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2-by-2 matrices., Comment: 32 pages

The very high energy (VHE; E > 100 GeV) blazar Markarian 501 has a well-studied history of extreme spectral variability and is an excellent laboratory for studying the physical processes within the jets of active galactic nuclei. However, there are few detailed multiwavelength studies of Markarian 501 during its quiescent state, due to its low luminosity. A short-term multiwavelength study of Markarian 501 was coordinated in March 2009, focusing around a multi-day observation with the Suzaku X-ray satellite and including {\gamma}-ray data from VERITAS, MAGIC, and the Fermi Gamma-ray Space Telescope with the goal of providing a well-sampled multiwavelength baseline measurement of Markarian 501 in the quiescent state. The results of these quiescent-state observations are compared to the historically extreme outburst of April 16, 1997, with the goal of examining variability of the spectral energy distribution between the two states. The derived broadband spectral energy distribution shows the characteristic double-peaked profile. We find that the X-ray peak shifts by over two orders of magnitude in photon energy between the two flux states while the VHE peak varies little. The limited shift in the VHE peak can be explained by the transition to the Klein-Nishina regime. Synchrotron self-Compton models are matched to the data and the implied Klein-Nishina effects are explored., Comment: Accepted for publication in ApJ, 25 pages, 6 figures, corresponding authors: Daniel Gall daniel-d-gall@uiowa.edu and Wei Cui cui@purdue.edu

A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced and studied, and its relations with holomorphic $\Sigma$-factorization are discussed. It is applied to study and solve some scalar Riemann-Hilbert problems in $\Sigma$ and vectorial Riemann-Hilbert problems in $\mathbb{C}$, including Wiener-Hopf matrix factorization, as well as to study some properties of a class of Toeplitz operators with $2 \times 2$ matrix symbols., Comment: accepted for publication in Journal of Mathematical Analysis and Applications

We report new observations of the intermediate-frequency peaked BL Lacertae object 3C 66A with the MAGIC telescopes. The data sample we use were taken in 2009 December and 2010 January, and comprises 2.3 hr of good quality data in stereoscopic mode. In this period, we find a significant signal from the direction of the blazar 3C 66A. The new MAGIC stereoscopic system is shown to play an essential role for the separation between 3C 66A and the nearby radio galaxy 3C 66B, which is at a distance of only $6^\prime$. The derived integral flux above $100\eh{GeV}$ is 8.3% of Crab Nebula flux and the energy spectrum is reproduced by a power law of photon index $3.64 \pm 0.39_{\rm stat} \pm 0.25_{\rm sys}$. Within errors, this is compatible with the one derived by VERITAS in 2009. From the spectra corrected for absorption by the extragalactic background light, we only find small differences between the four models that we applied, and constrain the redshift of the blazar to $z < 0.68$., Comment: Accepted for publication in ApJ

We report on the detection with the MAGIC telescopes of very high energy gamma-rays from IC 310, a head-tail radio galaxy in the Perseus galaxy cluster, observed during the interval November 2008 to February 2010. The Fermi satellite has also detected this galaxy. The source is detected by MAGIC at a high statistical significance of 7.6sigma in 20.6 hr of stereo data. The observed spectral energy distribution is flat with a differential spectral index of -2.00 \pm 0.14. The mean flux above 300 GeV, between October 2009 and February 2010, (3.1 \pm 0.5)x10^{-12} cm^{-2} s^{-1}, corresponds to (2.5 \pm 0.4)% of Crab Nebula units. Only an upper limit, of 1.9% of Crab Nebula units above 300 GeV, was obtained with the 2008 data. This, together with strong hints (>3sigma) of flares in the middle of October and November 2009, implies that the emission is variable. The MAGIC results favour a scenario with the very high energy emission originating from the inner jet close to the central engine. More complicated models than a simple one-zone SSC scenario, e.g. multi-zone SSC, external Compton or hadronic, may be required to explain the very flat spectrum and its extension over more than three orders of magnitude in energy., Comment: 6 pages, 5 figures, submitted to ApJL

We report on the observation of the region around supernova remnant G65.1+0.6 with the stand-alone MAGIC-I telescope. This region hosts the two bright GeV gamma-ray sources 1FGL J1954.3+2836 and 1FGL J1958.6+2845. They are identified as GeV pulsars and both have a possible counterpart detected at about 35 TeV by the Milagro observatory. MAGIC collected 25.5 hours of good quality data, and found no significant emission in the range around 1 TeV. We therefore report differential flux upper limits, assuming the emission to be point-like (<0.1 deg) or within a radius of 0.3 deg. In the point-like scenario, the flux limits around 1 TeV are at the level of 3 % and 2 % of the Crab Nebula flux, for the two sources respectively. This implies that the Milagro emission is either extended over a much larger area than our point spread function, or it must be peaked at energies beyond 1 TeV, resulting in a photon index harder than 2.2 in the TeV band., Comment: 8 pages, 3 figures, 1 table

Cygnus X-3 is a microquasar consisting of an accreting compact object orbiting around a Wolf-Rayet star. It has been detected at radio frequencies and up to high-energy gamma rays (above 100 MeV). However, many models also predict a very high energy (VHE) emission (above hundreds of GeV) when the source displays relativistic persistent jets or transient ejections. Therefore, detecting such emission would improve the understanding of the jet physics. The imaging atmospheric Cherenkov telescope MAGIC observed Cygnus X-3 for about 70 hours between 2006 March and 2009 August in different X-ray/radio spectral states and also during a period of enhanced gamma-ray emission. MAGIC found no evidence for a VHE signal from the direction of the microquasar. An upper limit to the integral flux for energies higher than 250 GeV has been set to 2.2 x 10-12 photons cm-2 s-1 (95% confidence level). This is the best limit so far to the VHE emission from this source. The non-detection of a VHE signal during the period of activity in the high-energy band sheds light on the location of the possible VHE radiation favoring the emission from the innermost region of the jets, where absorption is significant. The current and future generations of Cherenkov telescopes may detect a signal under precise spectral conditions., Comment: 14 pages, 11 figures, This manuscript has been accepted by ApJ (2010)

Context: Gamma-ray bursts are cosmological sources emitting radiation from the gamma-rays to the radio band. Substantial observational efforts have been devoted to the study of gamma-ray bursts during the prompt phase, i.e. the initial burst of high-energy radiation, and during the long-lasting afterglows. In spite of many successes in interpreting these phenomena, there are still several open key questions about the fundamental emission processes, their energetics and the environment. Aim: Independently of specific gamma-ray burst theoretical recipes, spectra in the GeV/TeV range are predicted to be remarkably simple, being satisfactorily modeled with power-laws, and therefore offer a very valuable tool to probe the extragalactic background light distribution. Furthermore, the simple detection of a component at very-high energies, i.e. at $\sim 100$\,GeV, would solve the ambiguity about the importance of various possible emission processes, which provide barely distinguishable scenarios at lower energies. Methods: We used the results of the MAGIC telescope observation of the moderate resdhift ($z\sim0.76$) \object{GRB\,080430} at energies above about 80\,GeV, to evaluate the perspective for late-afterglow observations with ground based GeV/TeV telescopes. Results: We obtained an upper limit of $F_{\rm 95%\,CL} = 5.5 \times 10^{-11}$\,erg\,cm$^{-2}$\,s$^{-1}$ for the very-high energy emission of \object{GRB\,080430}, which cannot set further constraints on the theoretical scenarios proposed for this object also due to the difficulties in modeling the low-energy afterglow. Nonetheless, our observations show that Cherenkov telescopes have already reached the required sensitivity to detect the GeV/TeV emission of GRBs at moderate redshift ($z \lesssim 0.8$), provided the observations are carried out at early times, close to the onset of their afterglow phase., Comment: 6 pages, 1 figure. Astronomy & Astrophysics, in press. Revision affects only the author list.

Context: Part of the very high energy $\gamma$-ray radiation coming from extragalactic sources is absorbed through the pair production process on the extragalactic background light photons. Extragalactic magnetic fields alter the trajectories of these cascade pairs and, in turn, convert cosmic background photons to gamma-ray energies by inverse Compton scattering. These secondary photons can form an extended halo around bright VHE sources. Aims: We searched for an extended emission around the bright blazars Mrk 421 and Mrk 501 using the MAGIC telescope data. Methods: If extended emission is present, the angular distribution of reconstructed gamma-ray arrival directions around the source is broader than for a point-like source. In the analysis of a few tens of hours of observational data taken from Mrk 421 and Mrk 501 we used a newly developed method that provides better angular resolution. This method is based on the usage of multidimensional decision trees. Comparing the measured shapes of angular distributions with those expected from a point-like source one can detect or constrain possible extended emission around the source. We also studied the influence of different types of systematic errors on the shape of the distribution of reconstructed gamma-ray arrival directions for a point source. Results: We present upper limits for an extended emission calculated for both sources for various source extensions and emission profiles. We discuss possible constraints on the extragalactic magnetic fields strength. We obtain upper limits on the extended emission around the Mrk~421 (Mrk~501) on the level of < 5% (< 4%) of the Crab Nebula flux above the energy threshold of 300 GeV. Using these results we discuss possible constraints on the extragalactic magnetic fields strength around a few times $10^{-15}$~G., Comment: 8 pages, 9 figures, accepted for publication in A&A

Between 2004 and 2009 a sample of 28 X-ray selected high- and intermediate-frequency peaked blazars with a X-ray flux larger than 2 uJy at 1 keV in the redshift range from 0.018 to 0.361 was observed with the MAGIC telescope at energies above 100 GeV. Seven among them were detected and the results of these observations are discussed elsewhere. Here we concentrate on the remaining 21 blazars which were not detected during this observation campaign and present the 3 sigma (99.7 %) confidence upper limits on their flux. The individual flux upper limits lie between 1.6 % and 13.6 % of the integral flux from the Crab Nebula. Applying a stacking method to the sample of non-detections with a total of 394.1 hours exposure time, we find evidence for an excess with a cumulative significance of 4.9 standard deviations. It is not dominated by individual objects or flares, but increases linearly with the observation time as for a constant source with an integral flux level of ~1.5 % of that observed from the Crab Nebula above 150 GeV., Comment: 14 pages, 6 figures, accepted by ApJ