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Register a new userOn c.n.c. commuting contractive tuples
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The characteristic function has been an important tool for studying completely non unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space $clh$. We show that the characteristic function, which is now an operator valued analytic function on the open Euclidean unit ball in $mathbb{C}^n$, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.
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An operator $T$ is called a 3-isometry if there exists operators $B_1(T^*,T)$ and $B_2(T^*,T)$ such that [Q(n)=T^{*n}T^n=1+nB_1(T^*,T)+n^2 B_2(T^*,T)] for all natural numbers $n$. An operator $J$ is a Jordan operator of order $2$ if $J=U+N$ where $U$ is unitary, $N$ is nilpotent order $2$, and $U$ and $N$ commute. An easy computation shows that $J$ is a $3$-isometry and that the restriction of $J$ to an invariant subspace is also a $3$-isometry. Those $3$-isometries which are the restriction of a Jordan operator to an invariant subspace can be identified, using the theory of completely positive maps, in terms of a positivity condition on the operator pencil $Q(s).$ In this article, we establish the analogous result in the multi-variable setting and show, by modifying an example of Choi, that an additional hypothesis is necessary. Lastly we discuss the joint spectrum of sub-Jordan tuples and derive results for 3-symmetric operators as a corollary.
We determine when contractive idempotents in the measure algebra of a locally compact group commute. We consider a dynamical version of the same result. We also look at some properties of groups of measures whose identity is a contactive idempotent.
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple $A$ is fully compressed if and only if it is multiplicity-free and the Shilov ideal is trivial, which occurs if and only if $A$ is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.
Let $q = e^{i theta} in mathbb{T}$ (where $theta in mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, i.e., $u$ and $v$ are unitaries such that $vu = quv$. In this paper we find the optimal constant $c = c_theta$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that [ c_theta = frac{4}{|u_theta+u_theta^*+v_theta+v_theta^*|}, ] where $u_theta, v_theta$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q$-commuting unitaries. The techniques that we develop allow us to give new and simple dilation theoretic proofs of well known results regarding the continuity of the field of rotations algebras. In particular, for the so-called Almost Mathieu Operator $h_theta = u_theta+u_theta^*+v_theta+v_theta^*$, we recover the fact that the norm $|h_theta|$ is a Lipshitz continuous function of $theta$, as well as the result that the spectrum $sigma(h_theta)$ is a $frac{1}{2}$-Holder continuous function in $theta$ with respect to the Hausdorff metric. In fact, we obtain this Holder continuity of the spectrum for every selfadjoint $*$-polynomial $p(u_theta,v_theta)$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.
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