121 research outputs found

    Pinchings and Positive linear maps

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    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

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    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

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    For a positive integer kk, the rank-kk numerical range Λk(A)\Lambda_k(A) of an operator AA acting on a Hilbert space \cH of dimension at least kk is the set of scalars λ\lambda such that PAP=λPPAP = \lambda P for some rank kk orthogonal projection PP. In this paper, a close connection between low rank perturbation of an operator AA and Λk(A)\Lambda_k(A) is established. In particular, for 1≤r<k1 \le r < k it is shown that Λk(A)⊆Λk−r(A+F)\Lambda_k(A) \subseteq \Lambda_{k-r}(A+F) for any operator FF with \rank (F) \le r. In quantum computing, this result implies that a quantum channel with a kk-dimensional error correcting code under a perturbation of rank ≤r\le r will still have a (k−r)(k-r)-dimensional error correcting code. Moreover, it is shown that if AA is normal or if the dimension of AA is finite, then Λk(A)\Lambda_k(A) can be obtained as the intersection of Λk−r(A+F)\Lambda_{k-r}(A+F) for a collection of rank rr operators FF. Examples are given to show that the result fails if AA is a general operator. The closure and the interior of the convex set Λk(A)\Lambda_k(A) are completely determined. Analogous results are obtained for Λ∞(A)\Lambda_\infty(A) defined as the set of scalars λ\lambda such that PAP=λPPAP = \lambda P for an infinite rank orthogonal projection PP. It is shown that Λ∞(A)\Lambda_\infty(A) is the intersection of all Λk(A)\Lambda_k(A) for k=1,2,>...k = 1, 2, >.... If A−μIA - \mu I is not compact for any \mu \in \IC, then the closure and the interior of Λ∞(A)\Lambda_\infty(A) coincide with those of the essential numerical range of AA. The situation for the special case when A−μIA-\mu I 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

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    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

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    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

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    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
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