12,342 research outputs found
A completely entangled subspace of maximal dimension
A completely entangled subspace of a tensor product of Hilbert spaces is a
subspace with no non-trivial product vector. K. R. Parthasarathy determined the
maximum dimension possible for such a subspace. Here we present a simple
explicit example of one such space. We determine the set of product vectors in
its orthogonal complement and see that it spans whole of the orthogonal
complement. This way we are able to determine the minimum dimension possible
for an unextendible product basis (UPB) consisting of product vectors which are
linearly independent but not necessarily mutually orthogonal.Comment: 8 page
Nilpotent Completely Positive Maps
We study the structure of nilpotent completely positive maps in terms of
Choi-Kraus coefficients. We prove several inequalities, including certain
majorization type inequalities for dimensions of kernels of powers of nilpotent
completely positive maps.Comment: 10 page
Pure Semigroups of Isometries on Hilbert C*-Modules
We show that pure strongly continuous semigroups of adjointable isometries on
a Hilbert C*-module are standard right shifts. By counter examples, we
illustrate that the analogy of this result with the classical result on Hilbert
spaces by Sz.-Nagy, cannot be improved further to understand arbitrary isometry
semigroups of isometries in the classical way.Comment: 18 pages; correction of an awful lot of typos; avoiding in some
places a conflict with the known terminology 'reducing subspace
Standard noncommuting and commuting dilations of commuting tuples
We introduce a notion called `maximal commuting piece' for tuples of Hilbert
space operators. Given a commuting tuple of operators forming a row contraction
there are two commonly used dilations in multivariable operator theory. Firstly
there is the minimal isometric dilation consisting of isometries with
orthogonal ranges and hence it is a noncommuting tuple. There is also a
commuting dilation related with a standard commuting tuple on Boson Fock space.
We show that this commuting dilation is the maximal commuting piece of the
minimal isometric dilation. We use this result to classify all representations
of Cuntz algebra O_n coming from dilations of commuting tuples.Comment: 18 pages, Latex, 1 commuting diagra
The Spatial Product of Arveson Systems is Intrinsic
We prove that the spatial product of two spatial Arveson systems is
independent of the choice of the reference units. This also answers the same
question for the minimal dilation the Powers sum of two spatial CP-semigroups:
It is independent up to cocycle conjugacy
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