6,994 research outputs found
Convex domains and K-spectral sets
Let be an open convex domain of the complex plane. We study
constants K such that is K-spectral or complete K-spectral for each
continuous linear Hilbert space operator with numerical range included in
. Several approaches are discussed.Comment: the introduction was changed and some remarks have been added. 26
pages ; to appear in Math.
On Matsaev's conjecture for contractions on noncommutative -spaces
We exhibit large classes of contractions on noncommutative -spaces which
satisfy the noncommutative analogue of Matsaev's conjecture, introduced by
Peller, in 1985. In particular, we prove that every Schur multiplier on a
Schatten space induced by a contractive Schur multiplier on
associated with a real matrix satisfy this conjecture. Moreover, we deal with
analogue questions for -semigroups. Finally, we disprove a conjecture of
Peller concerning norms on the space of complex polynomials arising from
Matsaev's conjecture and Peller's problem. Indeed, if denotes the shift on
and the shift on the Schatten space , the norms
\bnorm{P(S)}_{\ell^p \xra{}\ell^p} and \bnorm{P(\sigma)\ot
\Id_{S^p}}_{S^p(S^p) \xra{}S^p(S^p)} can be different for a complex polynomial
.Comment: 31 pages; minor corrections; to appear in Journal of Operator Theor
Lectures on Nehari's Theorem on the Polydisk
We give a leisurely proof of a result of Ferguson--Lacey (math.CA/0104144)
and Lacey--Terwelleger (math.CA/0601192) on a Nehari theorem for "little"
Hankel operators on a polydisk. If H_b is a little Hankel operator with symbol
b on product Hardy space we have || H_b || \simeq || b ||_{BMO} where BMO is
the product BMO space identified by Chang and Fefferman. This article begins
with the classical Nehari theorem, and presents the necessary background for
the proof of the extension above. The proof of the extension is an induction on
parameters, with a bootstrapping argument. Some of the more technical details
of the earlier proofs are now seen as consequences of a paraproduct theory.Comment: 35 pages. 65 Reference
User-friendly tail bounds for sums of random matrices
This paper presents new probability inequalities for sums of independent,
random, self-adjoint matrices. These results place simple and easily verifiable
hypotheses on the summands, and they deliver strong conclusions about the
large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for
the norm of a sum of random rectangular matrices follow as an immediate
corollary. The proof techniques also yield some information about matrix-valued
martingales.
In other words, this paper provides noncommutative generalizations of the
classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff,
Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of
application, ease of use, and strength of conclusion that have made the scalar
inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's
inequality has been moved to a separate note; other martingale bounds are
described in Caltech ACM Report 2011-0
Noncommutative Choquet theory
We introduce a new and extensive theory of noncommutative convexity along
with a corresponding theory of noncommutative functions. We establish
noncommutative analogues of the fundamental results from classical convexity
theory, and apply these ideas to develop a noncommutative Choquet theory that
generalizes much of classical Choquet theory.
The central objects of interest in noncommutative convexity are
noncommutative convex sets. The category of compact noncommutative sets is dual
to the category of operator systems, and there is a robust notion of extreme
point for a noncommutative convex set that is dual to Arveson's notion of
boundary representation for an operator system.
We identify the C*-algebra of continuous noncommutative functions on a
compact noncommutative convex set as the maximal C*-algebra of the operator
system of continuous noncommutative affine functions on the set. In the
noncommutative setting, unital completely positive maps on this C*-algebra play
the role of representing measures in the classical setting.
The continuous convex noncommutative functions determine an order on the set
of unital completely positive maps that is analogous to the classical Choquet
order on probability measures. We characterize this order in terms of the
extensions and dilations of the maps, providing a powerful new perspective on
the structure of completely positive maps on operator systems.
Finally, we establish a noncommutative generalization of the
Choquet-Bishop-de Leeuw theorem asserting that every point in a compact
noncommutative convex set has a representing map that is supported on the
extreme boundary. In the separable case, we obtain a corresponding integral
representation theorem.Comment: 81 pages; minor change
- …