arXiv subjects

Raphaël Clouâtre

Publications and source records attributed to Raphaël Clouâtre.

41 records · Page 3Linked to original sources

arXiv2014

On the unilateral shift as a Hilbert module over the disc algebra

We study the unilateral shift (of arbitrary countable multiplicity) as a Hilbert module over the disc algebra and the associated extension groups. In relation with the problem of determining whether this module is projective, we consider a special class of extensions, which we call "polynomial". We show that the subgroup of polynomial extensions of a contractive module by the adjoint of the unilateral shift is trivial. The main tool is a function theoretic decomposition of the Sz.-Nagy--Foias model space for completely non-unitary contractions.

math.OA↗

arXiv2014

Unitary equivalence and similarity to Jordan models for weak contractions of class $C_0$

We obtain results on the unitary equivalence of weak contractions of class $C_0$ to their Jordan models under an assumption on their commutants. In particular, our work addresses the case of arbitrary finite multiplicity. The main tool is the theory of boundary representations due to Arveson. We also generalize and improve previously known results concerning unitary equivalence and similarity to Jordan models when the minimal function is a Blaschke product.

math.OA↗

arXiv2014

Spectral and homological properties of Hilbert modules over the disc algebra

We study general Hilbert modules over the disc algebra and exhibit necessary spectral conditions for the vanishing of certain associated extension groups. In particular, this sheds some light on the problem of identifying the projective Hilbert modules. Part of our work also addresses the classical derivation problem.

math.FA↗

arXiv2014

Completely bounded isomorphisms of operator algebras and similarity to complete isometries

A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $ϕ:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $ϕ$ is similar to a completely contractive map $ϕ'$. Motivated by classification problems for Hilbert space contractions, we are interested in making the inverse $ϕ'^{-1}$ completely contractive as well whenever the map $ϕ$ has a completely bounded inverse. We show that there exist invertible operators $X$ and $Y$ such that the map $$ XaX^{-1}\mapsto Yϕ(a)Y^{-1} $$ is completely contractive and is "almost" isometric on any given finite set of elements from $\mathcal{A}$ with non-zero spectrum. Although the map cannot be taken to be completely isometric in general, we show that this can be achieved if $\mathcal{A}$ is completely boundedly isomorphic to either a $C^*$-algebra or a uniform algebra. In the case of quotient algebras of $H^\infty$, we translate these conditions in function theoretic terms and relate them to the classical Carleson condition.

math.OA↗

arXiv2011

Similarity results for operators of class C_0

If T is a multiplicity-free contraction of class C_0 with minimal function m_T, then it is quasisimilar to the Jordan block S(m_T). In case m_T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m_T) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor u of m_T, the result also holds in the more general setting where the roots have bounded multiplicities.

math.FA↗