Related papers: Constrained dilation and $\Gamma$-contractions

Suppose that $f\colon X\to\mathrm{Spec}\, R$ is a minimal model of a complete local Gorenstein 3-fold, where the fibres of $f$ are at most one dimensional, so by [VdB1d] there is a noncommutative ring $\Lambda$ derived equivalent to $X$.…

Let $M$ be a $3$-manifold, compact with boundary and $\Gamma$ its fundamental group. Consider a complex reductive algebraic group G. The character variety $X(\Gamma,G)$ is the GIT quotient $\mathrm{Hom}(\Gamma,G)//G$ of the space of…

We study two subspace systems in a separable infinite-dimensional Hilbert space up to (bounded) isomorphism. One of the main result of this paper is the following: Isomorphism classes of two subspace systems given by graphs of bounded…

Let the complex reflection group $G(m,p,n)$ act on the unit polydisc $\mathbb D^n$ in $\mathbb C^n.$ A $\boldsymbol\Theta_n$-contraction is a commuting tuple of operators on a Hilbert space having…

Given a contraction A on a Hilbert space H, an operator T on H is said to be A-invariant if <Tx,x>=<TAx,Ax> for every x in H such that ||Ax||=||x||. In the special case in which both defect indices of A are equal to 1, we show that every…

We prove the three embeddedness results as follows. $({\rm i})$ Let $\Gamma_{2m+1}$ be a piecewise geodesic Jordan curve with $2m+1$ vertices in $\mathbb{R}^n$, where $m$ is an integer $\geq2$. Then the total curvature of…

A $(\gamma,n)$-gonal pair is a pair $(S,f)$, where $S$ is a closed Riemann surface and $f:S \to R$ is a degree $n$ holomorphic map onto a closed Riemann surface $R$ of genus $\gamma$. If the signature of $(S,f)$ is of hyperbolic type, then…

Let $T$ and $V$ be two Hilbert space contractions and let $X$ be a linear bounded operator. It was proved by C. Foias and J.P. Williams that in certain cases the operator block matrix $R(X;T,V)$ (defined in the text) is similar to a…

Let $G$ be a simply connected, solvable Lie group and $\Gamma$ a lattice in $G$. The deformation space $\mathcal{D}(\Gamma,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(\Gamma,G)$ of all lattice…

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension of $\Gamma$ is the smallest size of…

Let $X$ be a product of locally compact rank one Hadamard spaces and $\Gamma$ a discrete group of isometries which contains two elements projecting to a pair of independent rank one isometries in each factor. In [arXiv:1308.5584] we gave a…

Valiant's Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets $\mathcal{F}$ and $\mathcal{G}$ of tensors (a.k.a. constraint functions or signatures) are related by a…

The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality…

Let S be a flat surface of genus g with cone type singularities. Given a bipartite graph G isoradially embedded in S, we define discrete analogs of the 2^{2g} Dirac operators on S. These discrete objects are then shown to converge to the…

I investigate contractions via Kac-Moody formalism. In particular, I show how the symmetry algebra of the standard 2-D Kepler system, which was identified by Daboul and Slodowy as an infinite-dimensional Kac-Moody loop algebra, and was…

We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a…

Let \Sigma = \Sigma _{g,1} be a compact surface of genus g at least 3 with one boundary component, \Gamma its mapping class group and M = H_1(\Sigma , Z) the first integral homology of \Sigma . Using that \Gamma is generated by the Dehn…

Using works of T.~Ando and L.~Gurvits, the well-known theorem of P.R.~Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces recast as a result for $d$-tuples of contractive Hilbert…

Let $\Gamma$ be a bounded Jordan curve and $\Omega_i,\Omega_e$ its two complementary components. For $p\in (1, \infty),\,s\in(0,1)$ we define the two spaces $\mathcal{B}_{p,p}^s(\Omega_{i,e})$ as the set of harmonic functions $u$…

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping $\phi:\Gamma\to\Gamma'$ between graphs from this class is shown to be an isomorphism provided that the following…