25 research outputs found

### Unitary equivalence to truncated Toeplitz operators

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

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 $L^p$. 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

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 â‰¥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

### PropriÃ©tÃ©s spectrales des opÃ©rateurs de Toeplitz

La premiÃ¨re partie de la thÃ¨se rÃ©unit des rÃ©sultats classiques sur l espace de Hardy, les espaces modÃ¨les et l espace de Bergman. Puis sur cette base, nous exposons des travaux relatifs aux opÃ©rateurs de Toeplitz, en particulier, nous prÃ©sentons la description du spectre et du spectre essentiel de ces opÃ©rateurs sur l espace de Hardy et de Bergman. La premiÃ¨re partie de notre recherche tire son inspiration de deux faits Ã©tablis pour un opÃ©rateur de Toeplitz T. PremiÃ¨rement, sur l espace de Hardy, la norme de T, la norme essentielle de T et la norme infinie du symbole de T sont Ã©gales. Nous Ã©tudions ce cas d Ã©galitÃ© sur l espace de Bergman pour les opÃ©rateurs de Toeplitz Ã symbole quasihomogÃ¨ne et radial. DeuxiÃ¨mement, sur l espace de hardy, le spectre et le spectre essentiel sont fortement liÃ©s Ã l image du symbole de T. Nous Ã©tudions le cas d Ã©galitÃ© entre le spectre et l image essentielle du symbole pour les symboles quasihomogÃ¨nes et radials. Pour rÃ©pondre Ã ces deux questions, nous utilisons la transformÃ©e de Berezin, les coefficients de Mellin et la moyenne du symbole. La derniÃ¨re partie de la thÃ¨se s interesse au thÃ©orÃ¨me de SzegÃ¶ 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 rÃ©sultat du mÃªme type sur l espace de Bergman pour les symboles harmoniques sur le disque et continus sur le cercle. Enfin, nous Ã©tudions une gÃ©nÃ©ralisation de ce thÃ©orÃ¨me en compressant l opÃ©rateur de Toeplitz sur une suite d espaces modÃ¨les 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 SzegÃ¶ 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

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), 89â€“102 in 1963 for the Hardy space H2. P. R. Ahern and ZË‡ . CË‡ ucË‡kovicÂ´ 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 eâˆ’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

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), 89â€“102 in 1963 for the Hardy space H2. P. R. Ahern and ZË‡ . CË‡ ucË‡kovicÂ´ 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 eâˆ’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 JournÃ© commutators and characterizations of multi-parameter BMO

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 JournÃ© 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 JournÃ© operators. Previous results in this direction do not apply to tensor products and only to JournÃ© operators which can be reduced to CalderÃ³n-Zygmund operators. Upper norm estimate of JournÃ© 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 JournÃ© operators are developed to obtain these lower norm estimates in the multi-parameter real variable setting

### A SzegÃ¶ type theorem for truncated Toeplitz operators.

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 SzegÃ¶ 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 SzegÃ¶'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 âŠ–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 SzegÃ¶'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 SzegÃ¶'s theorem is about the asymptotical situation when n â†’ âˆž. The natural generalization is then to consider a sequence of zeros (Î» j) in D, and to let the truncations act on the model space corresponding to the finite Blaschke product associated to Î» j , 1 â‰¤ j â‰¤ n