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Documents  Albiac, Fernando | enregistrements trouvés : 5

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Nigel Kalton played a prominent role in the development of a holomorphic functional calculus for unbounded sectorial operators. He showed, in particular, that such a calculus is highly unstable under perturbation: given an operator $D$ with a bounded functional calculus, fairly stringent conditions have to be imposed on a perturbation $B$ for $DB$ to also have a bounded functional calculus. Nigel, however, often mentioned that, while these results give a fairly complete picture of what is true at a pure operator theoretic level, more should be true for special classes of differential operators. In this talk, I will briefly review Nigel's general results before focusing on differential operators with perturbed coefficients acting on $L_p(\mathbb{R}^{n})$. I will present, in particular, recent joint work with $D$. Frey and A. McIntosh that demonstrates how stable the functional calculus is in this case. The emphasis will be on trying, as suggested by Nigel, to understand what makes differential operators so special from an operator theoretic point of view. Nigel Kalton played a prominent role in the development of a holomorphic functional calculus for unbounded sectorial operators. He showed, in particular, that such a calculus is highly unstable under perturbation: given an operator $D$ with a bounded functional calculus, fairly stringent conditions have to be imposed on a perturbation $B$ for $DB$ to also have a bounded functional calculus. Nigel, however, often mentioned that, while these ...

47F05 ; 47A60 ; 42B30 ; 42B37

Multi angle  Multi-norms and Banach lattices
Dales, H. Garth (Auteur de la Conférence) | CIRM (Editeur )

I shall discuss the theory of multi-norms. This has connections with norms on tensor products and with absolutely summing operators. There are many examples, some of which will be mentioned. In particular we shall describe multi-norms based on Banach lattices, define multi-bounded operators, and explain their connections with regular operators on lattices. We have new results on the equivalences of multi-norms. The theory of decompositions of Banach lattices with respect to the canonical 'Banach-lattice multi-norm' has a pleasing form because of a substantial theorem of Nigel Kalton that I shall state and discuss. I shall also discuss brie y a generalization that gives 'pmulti-norms' (for $1\leq p\leq1$) and an extension of a representation theorem of Pisier that shows that many pmulti-norms are 'sous-espaces de treillis'. The theory is based on joint work with Maxim Polyakov (deceasead), Hung Le Pham (Wellington), Matt Daws (Leeds), Paul Ramsden (Leeds), Oscar Blasco (Valencia), Niels Laustsen (Lancaster), Timur Oikhberg (Illinois), and Vladimir Troitsky (Edmonton).

multi-norms - equivalences - absolutely summing operators - tensor products
I shall discuss the theory of multi-norms. This has connections with norms on tensor products and with absolutely summing operators. There are many examples, some of which will be mentioned. In particular we shall describe multi-norms based on Banach lattices, define multi-bounded operators, and explain their connections with regular operators on lattices. We have new results on the equivalences of multi-norms. The theory of decompositions of ...

46B15 ; 46B28 ; 46B42 ; 47L10

We study the Daugavet equation
$\parallel Id+T\parallel$ $=1$ $+$ $\parallel T\parallel$
for Lipschitz operators on a Banach space. For this we introduce a substitute for the concept of slice for the case of non-linear Lipschitz functionals and transfer some results about the Daugavet and the alternative Daugavet equations previously known only for linear operators to the non-linear case.

numerical radius - numerical index - Daugavet equation - Daugavet property - SCD space - Lipschitz operator
We study the Daugavet equation
$\parallel Id+T\parallel$ $=1$ $+$ $\parallel T\parallel$
for Lipschitz operators on a Banach space. For this we introduce a substitute for the concept of slice for the case of non-linear Lipschitz functionals and transfer some results about the Daugavet and the alternative Daugavet equations previously known only for linear operators to the non-linear case.

numerical radius - numerical index - Daugavet equation - ...

46B04 ; 46B80 ; 46B22 ; 47A12

I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not.
In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then $\mathbb{Z}^{(2)}$ becomes the well-known Kalton-Peck $Z_2$ space, and one gets from here that there are quite natural nontrivial twisted sums $0\to Z_2\to\mathbb{Z}^{(4)}\to Z_2 \to0$ of $Z_2$ with itself. The twisted sum space $\mathbb{Z}^{(4)}$ does not embeds in, or is a quotient of, a twisted Hilbert space and does not contain $\ell_2$ complemented. We will also construct another nontrivial twisted sum of $Z_2$ with itself that contains $\ell_2$ complemented. These results have some connection with the nowadays called Kalton calculus [3, 4], and thus several recent advances [2] in this theory that combines twisted sums and interpolation theory will be shown.

Banach space - twisted sum - complex interpolation - Hilbert space
I'd like to share with the audience the Kaltonian story behind [1], started in 2004, including the problems we wanted to solve, and could not.
In that paper we show that Rochberg's generalized interpolation spaces $\mathbb{Z}^{(n)}$ [5] can be arranged to form exact sequences $0\to\mathbb{Z}^{(n)}\to\mathbb{Z}^{(n+k)}\to\mathbb{Z}^{(k)} \to 0$. In the particular case of Hilbert spaces obtained from the interpolation scale of $\ell_p$ spaces then ...

46M18 ; 46B70 ; 46B20

One of my recent main interests has been the characterization of boundedness of (integral) operators between two $L^p$ spaces equipped with two different measures. Some recent developments have indicated a need of "Banach spaces and their applications" also in this area of Classical Analysis. For instance, while the theory of two-weight $L^2$ inequalities is already rich enough to deal with a number of singular operators (like the Hilbert transform), the $L^p$ theory has been essentially restricted to positive operators so far. In fact, a counterexample of $F$. Nazarov shows that the common "Sawyer testing" or "David-Journé $T(1)$" type characterization will fail, in general, in the two-weight $L^p$ world. What comes to rescue is what we so often need to save the $L^2$ results in an Lp setting: $R$-boundedness in place of boundedness! Even in the case of positive operators, it turns out that a version of "sequential boundedness" is useful to describe the boundedness of operators from $L^p$ to $L^q$ when $q < p$. - This is about my recent joint work with T. Hänninen and K. Li, as well as the work of my student E. Vuorinen.

two-weight inequalities - boundedness - singular operators
One of my recent main interests has been the characterization of boundedness of (integral) operators between two $L^p$ spaces equipped with two different measures. Some recent developments have indicated a need of "Banach spaces and their applications" also in this area of Classical Analysis. For instance, while the theory of two-weight $L^2$ inequalities is already rich enough to deal with a number of singular operators (like the Hilbert ...

42B25 ; 47G40

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