A complex scalar λ is said to be an extended eigenvalue of a bounded linear operator T on a complex Banach space if there is a nonzero operator X such that T X = λXT. Such an operator X is called an extended eigenoperator of T corresponding to the extended eigenvalue λ.The purpose of this paper is to give a description of the extended eigenvalues for the discrete Cesàro operator C 0 , the finite continuous Cesàro operator C 1 and the infinite continuous Cesàro operator C ∞

A characterization of uniform continuity for maps between unit balls of real Banach spaces is given in terms of universal properties. Also, it is shown that the classes of compact and weakly compact composition operators induced by such maps agree.

It is shown that if the Deddens algebra DT associated with a quasinilpotent operator T on a complex Banach space is closed and localizing then T has a nontrivial closed hyperinvariant subspace.Ministerio de Economía y CompetitividadJunta de Andalucí

Let(X,d)be a pointed compact metric space, let0<α<1, and letφ:X→Xbe a base point preserving Lipschitz map. We prove that the essential norm of the composition operatorCφinduced by the symbolφon the spaceslip0(X,dα)andLip0(X,dα)is given by the formula‖Cφ‖e=limt→0 sup⁡0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α)whenever the dual spacelip0(X,dα)∗has the approximation property. This happens in particular whenXis an infinite compact subset of a finite-dimensional normed linear space.

No abstract