121 research outputs found
Pinchings and Positive linear maps
We employ the pinching theorem, ensuring that some operators A admit any
sequence of contractions as an operator diagonal of A, to deduce/improve two
recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional
expectations onto a masa in the algebra of operators on a Hilbert space. We
also get a few results for sums in a unitary orbit
Wavelet representations and Fock space on positive matrices
We show that every biorthogonal wavelet determines a representation by
operators on Hilbert space satisfying simple identities, which captures the
established relationship between orthogonal wavelets and Cuntz-algebra
representations in that special case. Each of these representations is shown to
have tractable finite-dimensional co-invariant doubly-cyclic subspaces.
Further, motivated by these representations, we introduce a general Fock-space
Hilbert space construction which yields creation operators containing the
Cuntz--Toeplitz isometries as a special case.Comment: 32 pages, LaTeX ("amsart" document class), one EPS graphic file used
for shading, accepted March 2002 for J. Funct. Ana
Higher rank numerical ranges and low rank perturbations of quantum channels
For a positive integer , the rank- numerical range of an
operator acting on a Hilbert space \cH of dimension at least is the
set of scalars such that for some rank
orthogonal projection . In this paper, a close connection between low rank
perturbation of an operator and is established. In
particular, for it is shown that for any operator with \rank (F) \le r. In quantum
computing, this result implies that a quantum channel with a -dimensional
error correcting code under a perturbation of rank will still have a
-dimensional error correcting code. Moreover, it is shown that if is
normal or if the dimension of is finite, then can be
obtained as the intersection of for a collection of rank
operators . Examples are given to show that the result fails if is a
general operator. The closure and the interior of the convex set
are completely determined. Analogous results are obtained for
defined as the set of scalars such that for an infinite rank orthogonal projection . It is shown that
is the intersection of all for . If is not compact for any \mu \in \IC, then the closure
and the interior of coincide with those of the essential
numerical range of . The situation for the special case when is
compact for some \mu \in \IC is also studied.Comment: 21 page
Poncelet's Theorem, Paraorthogonal Polynomials and the Numerical Range of Compressed Multiplication Operators
There has been considerable recent literature connecting Poncelet's theorem
to ellipses, Blaschke products and numerical ranges, summarized, for example,
in the recent book [11]. We show how those results can be understood using
ideas from the theory of orthogonal polynomials on the unit circle (OPUC) and,
in turn, can provide new insights to the theory of OPUC.Comment: 46 pages, 4 figures; minor revisions from v1; accepted for
publication in Adv. Mat
Interactions in noncommutative dynamics
A mathematical notion of interaction is introduced for noncommutative
dynamical systems, i.e., for one parameter groups of *-automorphisms of \Cal
B(H) endowed with a certain causal structure. With any interaction there is a
well-defined "state of the past" and a well-defined "state of the future". We
describe the construction of many interactions involving cocycle perturbations
of the CAR/CCR flows and show that they are nontrivial. The proof of
nontriviality is based on a new inequality, relating the eigenvalue lists of
the "past" and "future" states to the norm of a linear functional on a certain
C^*-algebra.Comment: 22 pages. Replacement corrects misnumbering of formulas in section 4.
No change in mathematical conten
Classifying Linear Matrix Inequalities via Abstract Operator Systems
We systematically study how properties of abstract operator systems help
classifying linear matrix inequality definitions of sets. Our main focus is on
polyhedral cones, the 3-dimensional Lorentz cone, where we can completely
describe all defining linear matrix inequalities, and on the cone of positive
semidefinite matrices. Here we use results on isometries between matrix
algebras to describe linear matrix inequality definitions of relatively small
size. We conversely use the theory of operator systems to characterize special
such isometries
- …