25 research outputs found

    Unitary equivalence to truncated Toeplitz operators

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    In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen

    Mixed commutators and little product BMO

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    We consider iterated commutators of multiplication by a symbol function and tensor products of Hilbert or Riesz transforms. We establish mixed BMO classes of symbols that characterize boundedness of these objects in LpL^p. Little BMO and product BMO, big Hankel operators and iterated commutators are the base cases of our results. We use operator theoretical methods and existing profound results on iterated commutators for the Hilbert transform case, while the general result in several variables is obtained through the construction of a Journ\'e operator that models the behavior of the multiple Hilbert transform. Upper estimates for commutators with paraproduct free Journ\'e operators as well as weak factorisation results are proven

    Quality of care for hypertension in the United States

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    BACKGROUND: Despite heavy recent emphasis on blood pressure (BP) control, many patients fail to meet widely accepted goals. While access and adherence to therapy certainly play a role, another potential explanation is poor quality of essential care processes (QC). Yet little is known about the relationship between QC and BP control. METHODS: We assessed QC in 12 U.S. communities by reviewing the medical records of a randomly selected group of patients for the two years preceding our study. We included patients with either a diagnosis of hypertension or two visits with BPs of ≥140/90 in their medical records. We used 28 process indicators based on explicit evidence to assess QC. The indicators covered a broad spectrum of care and were developed through a modified Delphi method. We considered patients who received all indicated care to have optimal QC. We defined control of hypertension as BP < 140/90 in the most recent reading. RESULTS: Of 1,953 hypertensive patients, only 57% received optimal care and 42% had controlled hypertension. Patients who had received optimal care were more likely to have their BP under control at the end of the study (45% vs. 35%, p = .0006). Patients were more likely to receive optimal care if they were over age 50 (76% vs. 63%, p < .0001), had diabetes (77% vs. 71%, p = .0038), coronary artery disease (87% vs. 69%, p < .0001), or hyperlipidemia (80% vs. 68%, p < .0001), and did not smoke (73% vs. 66%, p = .0005). CONCLUSIONS: Higher QC for hypertensive patients is associated with better BP control. Younger patients without cardiac risk factors are at greatest risk for poor care. Quality measurement systems like the one presented in this study can guide future quality improvement efforts

    Propriétés spectrales des opérateurs de Toeplitz

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    La première partie de la thèse réunit des résultats classiques sur l espace de Hardy, les espaces modèles et l espace de Bergman. Puis sur cette base, nous exposons des travaux relatifs aux opérateurs de Toeplitz, en particulier, nous présentons la description du spectre et du spectre essentiel de ces opérateurs sur l espace de Hardy et de Bergman. La première partie de notre recherche tire son inspiration de deux faits établis pour un opérateur de Toeplitz T. Premièrement, sur l espace de Hardy, la norme de T, la norme essentielle de T et la norme infinie du symbole de T sont égales. Nous étudions ce cas d égalité sur l espace de Bergman pour les opérateurs de Toeplitz à symbole quasihomogène et radial. Deuxièmement, sur l espace de hardy, le spectre et le spectre essentiel sont fortement liés à l image du symbole de T. Nous étudions le cas d égalité entre le spectre et l image essentielle du symbole pour les symboles quasihomogènes et radials. Pour répondre à ces deux questions, nous utilisons la transformée de Berezin, les coefficients de Mellin et la moyenne du symbole. La dernière partie de la thèse s interesse au théorème de Szegö qui donne un lien entre les valeurs propres d une suite de matrices de Toeplitz de taille n, et le symbole de cette suite de matrice. Nous donnons un résultat du même type sur l espace de Bergman pour les symboles harmoniques sur le disque et continus sur le cercle. Enfin, nous étudions une généralisation de ce théorème en compressant l opérateur de Toeplitz sur une suite d espaces modèles de dimension finie.This thesis deals with the spectral properties of the Toeplitz operators in relation to their associated symbol. In the first part, we give some classical results about Hardy space, model spaces and Bergman space. Afterwards, we expose some results about Toeplitz operator on the Hardy space. In particular, we discuss their spectrum and essential spectrum. Our work is inspired from two facts which have been proved on the Hardy space. First, considering a Toeplitz operator T, the norm, essential norm, spectral radius of T and the supremum of its symbol are equal. Secondly, on the Hardy space, spectrum, essential spectrum and essential range are strongly related. We answer the question of the equality between the norms, the spectral radius and the supremum of the symbol and between spectrum and essential range on the Bergman space. We look at these two properties on the Bergman space when the symbol is radial or quasihomogeneous. We answer these questions using the Berezin transform, the Mellin coefficients and the mean value of the symbol. The last part deals with the classical Szegö theorem which underline a link between the eigenvalues of a Toeplitz matrix sequence and its symbol. We give a result of the same type on Bergman space considering harmonic symbol wich have a continuous extension. We give a generalization, considering the sequence of the compressions of a Toeplitz operator on a sequence of model spaces.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF

    Products of Toeplitz operators on the Bergman space

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    When is the product of two Toeplitz operators again a Toeplitz operator in the context of the Bergman space in the unit disk? This problem was satisfactorily solved by A. Brown and P. R. Halmos [J. Reine Angew. Math. 213 (1963/1964), 89–102 in 1963 for the Hardy space H2. P. R. Ahern and Zˇ . Cˇ ucˇkovic´ obtained a corresponding result for the Bergman space with the additional assumption that the symbols of the operators are bounded and harmonic. In this paper the authors deal with some special types of operators with symbols eip'(r) where p is an integer, (r, ) are polar coordinates in the complex plane and ' belongs to L1(D), D being the unit disk. These symbols give rise to Toeplitz operators that are only densely defined, possibly unbounded. So any such function is called a T-function if the associated Toeplitz operator is bounded. Theorem 6.1 of the authors gives a necessary and sufficient condition for a product of two Toeplitz operators given by two T-functions eip'1(r) and e−is'2(r) to be again a Toeplitz operator assuming p s > 0 are integers. They also provide examples to show that even in the class of such special operators, general products need not be Toeplitz without additional hypotheses

    Products of Toeplitz operators on the Bergman space

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    When is the product of two Toeplitz operators again a Toeplitz operator in the context of the Bergman space in the unit disk? This problem was satisfactorily solved by A. Brown and P. R. Halmos [J. Reine Angew. Math. 213 (1963/1964), 89–102 in 1963 for the Hardy space H2. P. R. Ahern and Zˇ . Cˇ ucˇkovic´ obtained a corresponding result for the Bergman space with the additional assumption that the symbols of the operators are bounded and harmonic. In this paper the authors deal with some special types of operators with symbols eip'(r) where p is an integer, (r, ) are polar coordinates in the complex plane and ' belongs to L1(D), D being the unit disk. These symbols give rise to Toeplitz operators that are only densely defined, possibly unbounded. So any such function is called a T-function if the associated Toeplitz operator is bounded. Theorem 6.1 of the authors gives a necessary and sufficient condition for a product of two Toeplitz operators given by two T-functions eip'1(r) and e−is'2(r) to be again a Toeplitz operator assuming p s > 0 are integers. They also provide examples to show that even in the class of such special operators, general products need not be Toeplitz without additional hypotheses

    Higher order Journé commutators and characterizations of multi-parameter BMO

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    International audienceWe characterize L p boundedness of iterated commutators of multiplication by a symbol function and tensor products of Riesz and Hilbert transforms. We obtain a two-sided norm estimate that shows that such operators are bounded on L p if and only if the symbol belongs to the appropriate multi-parameter BMO class. We extend our results to a much more intricate situation; commutators of multiplication by a symbol function and paraproduct-free Journé operators. We show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi-parameter BMO class. In this sense the tensor products of Riesz transforms are a representative testing class for Journé operators. Previous results in this direction do not apply to tensor products and only to Journé operators which can be reduced to Calderón-Zygmund operators. Upper norm estimate of Journé commutators are new even in the case of no iterations. Lower norm estimates for iterated commutators only existed when no tensor products were present. In the case of one dimension, lower estimates were known for products of two Hilbert transforms, and without iterations. New methods using Journé operators are developed to obtain these lower norm estimates in the multi-parameter real variable setting

    A Szegö type theorem for truncated Toeplitz operators.

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    Truncated Toeplitz operators are compressions of multiplication operators on L 2 to model spaces (that is, subspaces of H 2 which are invariant with respect to the backward shift). For this class of operators we prove certain Szegö type theorems concerning the asymptotics of their compressions to an increasing chain of finite dimensional model spaces. The Toeplitz operators are compressions of multiplication operators on the space L 2 (T) to the Hardy space H 2 ; the multiplier is called the symbol of the operator. With respect to the standard exponential basis, their matrices are constant along diagonals; if we truncate such a matrix considering only its upper left finite corner, we obtain classical Toeplitz matrices. It does not come as a surprise that there are connections between the asymptotics of these Toeplitz matrices and the whole Toeplitz operator, or its symbol. A central result is Szegö's strong limit theorem and its variants (see, for instance, [4] and the references within), which deal with the asymptotics of the eigenvalues of the Toeplitz matrix. On the other hand, certain generalizations of Toeplitz matrices have attracted a great deal of attention in the last decade, namely compressions of multiplication operators to subspaces of the Hardy space which are invariant under the backward shift. These " model spaces " are of the form H 2 ⊖uH 2 with u an inner function, and the compressions are called truncated Toeplitz operators. They have been formally introduced in [11]; see [8] for a more recent survey. Although classical Toeplitz matrices have often been a starting point for investigating truncated Toeplitz operators , the latter may exhibit surprising properties. It thus seems natural to see whether an analogue of Szegö's strong limit theorem can be obtained in this more general context. Viewed as truncated Toeplitz operators, the Toeplitz matrices act on model spaces corresponding to the inner functions u(z) = z n , and Szegö's theorem is about the asymptotical situation when n → ∞. The natural generalization is then to consider a sequence of zeros (λ j) in D, and to let the truncations act on the model space corresponding to the finite Blaschke product associated to λ j , 1 ≤ j ≤ n
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