The Overdeterminedness of a Class of Functional Equations

We prove a uniqueness theorem for a large class of functional equations in the plane, which resembles in form a classical result of Aczel. It is also shown that functional equations in this class are overdetermined in the sense of Paneah. This means that the solutions, if they exist, are determined by the corresponding relation being fulfilled not in the original domain of validity, but only at the points of a subset of the boundary of the domain of validity.Comment: 6 page

What type of dynamics arise in E_0-dilations of commuting quantum Markov process?

Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $\phi$ and $\theta$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $\alpha$ and $\beta$ such that $\phi_s \circ \theta_t (P_H A P_H) = P_H \alpha_s \circ \beta_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $\phi$ is not an automorphism semigroup then $\alpha$ is cocycle conjugate to the minimal *-endomorphic dilation of $\phi$, and that if $\phi$ is an automorphism semigroup then $\alpha$ is also an automorphism semigroup. In particular, we conclude that if $\phi$ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional) then $\alpha$ is a type I E_0-semigroup.Comment: 9 pages, minor corrections mad

Dilation theorems for contractive semigroups

This note records some dilation theorems about contraction semigroups on a Hilbert space - all of which fall into the categories "known" or "probably known" - that I proved while working on my PhD in mathematics (under the supervision of Baruch Solel). It is convenient to have them recorded for reference.Comment: 11 pages

A note on equicontinuity of families of operators and automorphisms

This note concerns uniform equicontinuity of families of operators on a separable Hilbert space H, and of families of maps on B(H). It is shown that a one parameter group of automorphisms is uniformly equicontinuous if and only if the group of unitaries which implements it is so. A "geometrical" necessary and sufficient condition is given for a family of operators to be uniformly equicontinuous.Comment: 8 pages, typos correcte

E_0-dilation of strongly commuting CP_0-semigroups

We prove that every strongly commuting pair of CP_0-semigroups has a minimal E_0-dilation. This is achieved in two major steps, interesting in themselves: 1: we show that a strongly commuting pair of CP_0-semigroups can be represented via a two parameter product system representation; 2: we prove that every fully coisometric product system representation has a fully coisometric, isometric dilation. In particular, we obtain that every commuting pair of CP_0-semigroups on B(H), H finite dimensional, has an E_0-dilation.Comment: 45 pages; a gap in section 6 has been filled; minimality prove

E-dilation of strongly commuting CP-semigroups (the nonunital case)

In a previous paper, we showed that every strongly commuting pair of CP_0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E_0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to a two-parameter contraction semigroup.Comment: 23 pages. Final version. Changes from v3: some corrections and added references. To appear in Houston J. Mat

A sneaky proof of the maximum modulus principle

A proof for the maximum modulus principle (in the unit disc) is presented. This proof is unusual in that it is based on linear algebra.Comment: 4 page

Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

Let $\mathcal{H}_d^{(t)}$ ($t \geq -d$, $t>-3$) be the reproducing kernel Hilbert space on the unit ball $\mathbb{B}_d$ with kernel $$k(z,w) = \frac{1}{(1-\langle z, w \rangle)^{d+t+1}} .$$ We prove that if an ideal $I \triangleleft \mathbb{C}[z_1, \ldots, z_d]$ (not necessarily homogeneous) has what we call the "approximate stable division property", then the closure of $I$ in $\mathcal{H}_d^{(t)}$ is $p$-essentially normal for all $p>d$. We then show that all quasi homogeneous ideals in two variables have the stable division property, and combine these two results to obtain a new proof of the fact that the closure of any quasi homogeneous ideal in $\mathbb{C}[x,y]$ is $p$-essentially normal for $p>2$.Comment: Some mistakes fixed and details added to the proof of the main result. 17 page

Efficient coordinate-descent for orthogonal matrices through Givens rotations

Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of computation. Here we propose a framework for optimizing orthogonal matrices, that is the parallel of coordinate-descent in Euclidean spaces. It is based on {\em Givens-rotations}, a fast-to-compute operation that affects a small number of entries in the learned matrix, and preserves orthogonality. We show two applications of this approach: an algorithm for tensor decomposition that is used in learning mixture models, and an algorithm for sparse-PCA. We study the parameter regime where a Givens rotation approach converges faster and achieves a superior model on a genome-wide brain-wide mRNA expression dataset.Comment: A shorter version of this paper will appear in the proceedings of the 31st International Conference for Machine Learning (ICML 2014

Unitary N-dilations for tuples of commuting matrices

We show that whenever a contractive $k$-tuple $T$ on a finite dimensional space $H$ has a unitary dilation, then for any fixed degree $N$ there is a unitary $k$-tuple $U$ on a finite dimensional space so that $q(T) = P_H q(U) |_H$ for all polynomials $q$ of degree at most $N$.Comment: 11 page