/pdf/cyclic-vectors-and-invariant-subspaces-for-the-backward-1m4duugftq.pdf

Analyse harmonique des opérateurs de l'espace de Hilbert

Cyclic vectors and invariant subspaces for the backward shift operator

An analogue to the Sz.-Nagy-Foia5 dilation theory is presented for several commuting operators on a Hilbert space. The dilation theory of Hilbert space contractions, a reference of which is the excellent monograph [9], has not yet found a complete counterpart valid for several (commuting) Hilbert space operators. Apart from the results of this type already contained in [9] and other results of general character (i.e., Arveson's or Stinespring's extension theorem), there are not too many contributions specialized to the multiparameter dilation theory. Among these contributions, we quote the works [7, 3-5, 10], which are closer to our topics. The aim of this paper is to analyse some positivity conditions for commuting multioperators, which ensure the unitary equivalence of these objects to some standard models consisting of backwards multishifts. We emphasize, in particular, a situation that seems to satisfy the requirements of an appropriate extension for the dilation theory of Hilbert space contractions and that exploits the results already obtained in [4, 10]. Our finer methods also imply better statements of some assertions from [10] and provide an answer to Question 4.5 from [5]. Let n > 1 be an integer, and let Zn be the set of all n-tuples of nonnegative integers (i.e., the multi-indices of length n). If a = (a, ...I, an) E Zn+, we set, as usual, lal =a,+ + -+a, and a! =a1! .. !. For a,ficZ+ wewrite a + f = (a1 + f1j ... , an + fin), and a < fi whenever aci < /i (i = 1, ... , n). Let H be a Hilbert space, and let 2(H) be the algebra of all bounded linear operators on H. An element T = (T,, ... , T,) E 2(H)n such that T,, ... , T, mutually commute will be designated as a commuting multioperator (briefly, a c.m.). Let T E Y(H)n be a c.m. We define the operator MT: Y(H) -* J(H) by the formula

/pdf/standard-models-for-some-commuting-multioperators-53drchw0yp.pdf

Standard models for some commuting multioperators

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

/pdf/an-index-theory-for-quantum-dynamical-semigroups-1yoik236g7.pdf

An Index Theory For Quantum Dynamical Semigroups

In this paper we provide an analytical procedure which leads to a system of (n − 2)2 polynomial equations whose solutions give the parametrization of the complex n × n Hadamard matrices. It is shown that in general the Hadamard matrices depend on a number of arbitrary phases and a lower bound for this number is given. The moduli equations define interesting geometrical objects whose study will shed light on the parametrization of the Hadamard matrices, as well as on some interesting geometrical varieties defined by them.

http://iopscience.iop.org/article/10.1088/0305-4470/37/20/008/pdf

Some results on the parametrization of complex Hadamard matrices

In what follows the term operator will mean a bounded linear operator T which acts on some infinite dimensional complex Banach space B. The spectrum of T will be denoted by g(T). An invariant subspace for T will always mean here a closed subspace M c B such that T M c M. We say that M is hyperinvariant for T if it is invariant for every operator tha t commutes with T. A subspace M is called nontrivial if M ~ { 0 ) and M ~ B . J. Wermer [27] proved that if T is an invertible operator which satisfies

background

/pdf/operators-which-are-annihilated-by-analytic-functions-and-1qs4yw5897.pdf

Analyse harmonique des opérateurs de l'espace de Hilbert

Citations

Cyclic vectors and invariant subspaces for the backward shift operator

Standard models for some commuting multioperators

An Index Theory For Quantum Dynamical Semigroups

Some results on the parametrization of complex Hadamard matrices

Operators which are annihilated by analytic functions and invariant subspaces

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