Basic ideas about noncommuting coordinates are summarized, and then coordinate noncommutativity, as it arises in the Landau problem, is investigated. I review a quantum solution to the Landau problem, and evaluate the coordinate commutator in a truncated state space of Landau levels. Restriction to the lowest Landau level reproduces the well known commutator of planar coordinates. Inclusion of ...

In this paper we use the notion of slice monogenic functions [2] to define a new functional calculus for an n-tuple T of not necessarily commuting operators. This calculus is different from the one discussed in [5] and it allows the explicit construction of the eigenvalue equation for the n-tuple T based on a new notion of spectrum for T . Our functional calculus is consistent with the Riesz-Du...

We show that, given a general mixed state for a quantum system, there are no physical means for broadcasting that state onto two separate quantum systems, even when the state need only be reproduced marginally on the separate systems. This result generalizes and extends the standard no-cloning theorem for pure states. 1995 PACS numbers: 03.65.Bz, 89.70.+c, 02.50.-r Typeset using REVTEX 1 The fl...

The common fixed point results for Banach operator pair with generalized nonexpansive mappings in q-normed space have been obtained in the present work. As application, some more general best approximation results have also been determined without the assumption of linearity or affinity of mappings. These results unify and generalize various existing known results with the aid of more general c...

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satissed in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.

We define a free holomorphic function to be a function that is locally a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an OkaWeil theorem for free analytic functions.

Abstract. We consider functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B ∞,1 (R), then we have the following Lipschitz type estimate in the trace norm: ‖f(A1, B1)− f(A2, B2)‖S1 ≤ const(‖A1 −A2‖S1 + ‖B1 −B2‖S1). However, the condition f ∈ B ∞,1 (R) does not imply the Lipschitz ...

In this paper we study constrained eigenvalue optimization of noncommutative (nc) polynomials, focusing on the polydisc and the ball. Our three main results are as follows: (1) an nc polynomial is nonnegative if and only if it admits a weighted sum of hermitian squares decomposition; (2) (eigenvalue) optima for nc polynomials can be computed using a single semidefinite program (SDP) – this shar...

Lochlainn O’Raifeartaigh and his work are recalled; the connection between fluid mechanics – his last research topic – and noncommuting gauge fields is explained. O’Raifeartaigh Memorial Meeting, Dublin, Ireland, September 2002 Lochlainn O’Raifeartaigh, Fluids, and Noncommuting Fields 3 Lochlainn O’Raifeartaigh was an eminent theoretical/mathematical physicist, the most recent member of a cohor...