A matrix-valued measure Θ reduces to measures of smaller size if there exists a constant invertible matrix M such that MΘM∗ is block diagonal. Equivalently, the real vector space A of all matrices T such that TΘ(X) = Θ(X)T ∗ for any Borel set X is nontrivial. If the subspace Ah of self-adjoints elements in the commutant algebra A of Θ is nontrivial, then Θ is reducible via a unitary matrix. In ...

Let Jα k be a real power of the integration operator Jk defined on Sobolev space W k p [0, 1]. We investigate the spectral properties of the operator Ak = ⊕n j=1 λjJ α k defined on ⊕n j=1W k p [0, 1]. Namely, we describe the commutant {Ak} ′, the double commutant {Ak} ′′ and the algebra AlgAk. Moreover, we describe the lattices LatAk and HypLatAk of invariant and hyperinvariant subspaces of Ak,...

In a previous paper, the authors obtained a model for a bi-isometry, that is, a pair of commuting isometries on complex Hilbert space. This representation is based on the canonical model of Sz. Nagy and the third author. One approach to describing the invariant subspaces for such a bi-isometry using this model is to consider isometric intertwining maps from another such model to the given one. ...

Let Γ y X be a measure preserving (m.p.) action of a discrete group Γ on a probability measure space (X,μ) and H ⊂ Γ a non-amenable subgroup with commutant H = {g ∈ Γ | gh = hg, ∀h ∈ H} infinite. We prove that if the action satisfies a malleability condition on HH, is weak mixing on H and has stable spectral gap on H (e.g. if the action is Bernoulli on HH), then any cocycle with values in a Pol...

In this paper, using a power series methodology a design procedure applicable to analytic nonlinear plants is described . The technique used is a generalization of the linear HOC> theory. In contrast. to previolls work on this topic (lIndiana J. Math., 36 (1987), pp. 693-7091. rOper. Theory Adv. Appl., 41 (1989), pp . 255-277]' [SIAM J. Control Optim., 27 (1989), pp. 842­ 860]), the authors are...

John A. Holbrook, David W. Kribs and Raymond Laflamme Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1. Institute for Quantum Computing, University of Waterloo, Waterloo, ON, CANADA N2L 3G1. Perimeter Institute for Theoretical Physics, 35 King St. North, Waterloo, ON, CANADA N2J 2W9.∗ Abstract The effect of noise on a quantum system can be describe...

We give a new proof of a result of Ozawa showing that if a von Neumann subalgebra Q of a free group factor LFn, 2 ≤ n ≤ ∞ has relative commutant diffuse (i.e. without atoms), then Q is amenable.