A Pseudo-daugavet Property for Narrow Projections in Lorentz Spaces
نویسنده
چکیده
Let X be a rearrangement-invariant space. An operator T : X → X is called narrow if for each measurable setA and each ε > 0 there exists x ∈ X with x = χA, ∫ xdμ = 0 and ‖Tx‖ < ε. In particular all compact operators are narrow. We prove that if X is a Lorentz function space Lw,p on [0,1] with p > 2, then there exists a constant kX > 1 so that for every narrow projection P on Lw,p ‖Id− P‖ ≥ kX . This generalizes earlier results on Lp and partially answers a question of E. M. Semenov. Moreover we prove that every rearrangement-invariant function space X with an absolutely continuous norm contains a complemented subspace isomorphic to X which is the range of a narrow projection and a non-narrow projection, which gives a negative answer to a question of A.Plichko and M.Popov.
منابع مشابه
THE DAUGAVET PROPERTY OF C-ALGEBRAS AND NON-COMMUTATIVE Lp-SPACES
where IX is the identity map on X . The study of the Daugavet property was inaugurated by I. Daugavet [4] in 1961; he proved that every compact operator on C([0, 1]) satisfies (1.1). Consequently, C([0, 1]) has the DP. Investigation of the Daugavet property continued in the seventies and eighties. In particular, J. Holub proved in [7] and [8] (see also [11]) that for any Hausdorff compact topol...
متن کاملFrom the Lorentz Transformation Group in Pseudo-Euclidean Spaces to Bi-gyrogroups
The Lorentz transformation of order $(m=1,n)$, $ninNb$, is the well-known Lorentz transformation of special relativity theory. It is a transformation of time-space coordinates of the pseudo-Euclidean space $Rb^{m=1,n}$ of one time dimension and $n$ space dimensions ($n=3$ in physical applications). A Lorentz transformation without rotations is called a {it boost}. Commonly, the ...
متن کاملThe Daugavet Property of the Space of Lipschitz Functions
where Id is the identity operator on C[0, 1]. This equation is now called Daugavet equation. The Banach space X is said to have the Daugavet property when all compact operators on X satisfy the Daugavet equation. More information about the Daugavet spaces can be found in [Werner, 2001]. In the same paper was also posed the question, whether the Banach space of Lipschitz functions on unit square...
متن کاملBi-Gyrogroup: The Group-Like Structure Induced by Bi-Decomposition of Groups
The decomposition $Gamma=BH$ of a group $Gamma$ into a subset B and a subgroup $H$ of $Gamma$ induces, under general conditions, a group-like structure for B, known as a gyrogroup. The famous concrete realization of a gyrogroup, which motivated the emergence of gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary mbox{...
متن کاملThe Daugavet equation for operators on function spaces
We prove the norm identity ‖Id + T ‖ = 1 + ‖T ‖, which is known as the Daugavet equation, for weakly compact operators T on natural function spaces such as function algebras and L-predual spaces, provided a non-discreteness assumption is met. We also consider c0-factorable operators and operators on CΛ-spaces.
متن کامل