2009/07/15 by Douglas, Ronald G., Foias, Ciprian, Sarkar, Jaydeb · 1 citation
#46C07 #46E22 #46M20 #47A13 #47A20 #47A45 #47B32 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.0907.2487
Let H2m be the Drury-Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function (z, w) ∈ Bm × Bm \raro (1 - )-1. We investigate for which multipliers θ: \mathbbBm \raro \cll(\cle, \cle_*) the quotient module \clhθ is similar to H2m ⊗ \clf for some Hilbert space \clf, where Mθ is the corresponding multiplication operator in \cll(H2m ⊗ \cle, H2m ⊗ \cle_*) for Hilbert spaces \cle and \cle_* and \clhθ is the quotient module (H2m ⊗ \cle_*)/ clos [Mθ(H2m ⊗ \cle)]. We show that a necessary condition is the existence of a multiplier ψ in \clm(\cle_*, \cle) such that θψθ= θ. Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of H2m ⊗ \cle for a Hilbert space \cle, which is valid for the case of m=1. The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a finite resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator tuple or algebra is not necessarily commuting. In this case the converse to the similarity result is always valid.