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- W3199482021 abstract "This thesis focuses on the study of classes of operators. Two different families of classes of operators are mainly studied.- The first classes we study are classes of operators on Hilbert spaces that generalize the classes dollarC_{ho}dollar of Nagy and Foias. For dollar(ho_n)_ndollar a sequence of non-zero complex numbers, we define the class dollarC_{(ho_n)}(H)dollar as the set of operators dollarT in mathcal{L}(H)dollar that are said to possess a dollar(ho_n)dollar-dilation: there exists a Hilbert space K and a unitary operator dollarU in mathcal{L}(K)dollar with dollarH subset Kdollar and dollarT^n=ho_n P_H U^n|_Hdollar for every dollarn geq 1dollar (dollarP_H in mathcal{L}(K)dollar being the orthogonal projection from K onto its closed subspace H). These classes can be associated with an holomorphic map dollarf_{(ho_n)}dollar as well as a quasi-norm dollarw_{(ho_n)}dollar. These three objects are tied together and we use them to characterize, describe, and give several spectral properties of operators belonging to this class.We give multiple relationships between multiple classes of this form, generalize many results that were known for classes dollarC_{(ho)}dollar, and give several examples and cases that exhibit new behaviours. We also bring a new geometric meaning behind a relationship between quasi-norms dollarw_{ho}dollar and extend the computations of dollarw_{ho}(T)dollar for operators T that are zeroes of a degree two polynomial. The second main part of our study concerns classes of L^p-projections.An L^p-projection on a Banach space X, for dollar1leq p leq +inftydollar, is an idempotent operator P satisfying dollar |f|_X = |(|P(f)|_X, |(I-P)(f)|_X) |_{ell_{p}}dollar for all f in X. This is anL^p version of the equality dollar|f|^2=|Q(f)|^2 + |(I-Q)(f)|^2dollar, valid for orthogonal projections on Hilbert spaces. We are interested into relationships between L^p-projections on a Banach space X and L^p-projections on a subspace F, on a quotient X/F, or on a subspace of a quotient G/F. These questions are given an answer on Banach spaces with additional properties, depending on the value of p. We also introduce a notion of maximal L^p-projections for X, that is L^p-projections defined on a subspace G of X that cannot be extended to L^p-projections on larger subspaces, and study their properties, especially on finite dimensional Banach spaces. A characterization of L^{infty}-projections on every space L^{infty}(Omega) is obtained as well using new methods, generalizing previously known results." @default.
- W3199482021 created "2021-09-27" @default.
- W3199482021 creator A5008223923 @default.
- W3199482021 date "2021-03-08" @default.
- W3199482021 modified "2023-09-23" @default.
- W3199482021 title "Dilations of operators and L^p-projections" @default.
- W3199482021 hasPublicationYear "2021" @default.
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