Prequojections and their duals

Convergence of arithmetic means of operators in Fréchet spaces

“…So, every complemented subspace of a prequojection is also a prequojection. The problem of the existence of prequojections which are not quojections arose in a natural way in [6] and was solved in [7], [15], [32], [34]; see the survey paper [31] for further information. An example of a space of continuous functions on a suitable topological space which is a nonquojection prequojection is given in [2].…”

Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces

“…By the surjectivity of there exists ∈ such that =̂. By (25) it follows that ( )̂/ = ( / ), for ∈ N. Moreover, / → 0 as → ∞ by assumption. So, the continuity of yields that lim → ∞ (( )̂/ ) = 0 in the Banach space .…”

“…All of these papers employ the same method, which consists in the construction of the dual of a prequojection, rather than the prequojection itself, which is often difficult to describe (see the survey paper [24] for further information). However, in [25] an alternative method for constructing prequojections is presented which has the advantage of being direct. For an example of a concrete space (i.e., a space of continuous functions on a suitable topological space), which is a nontrivial prequojection, see [26].…”

Weak$^*$ closures and derived sets in dual Banach spaces

“…The study of weak * derived sets was initiated by Banach and continued by many authors, see [17,2,18,34,9,21,23], and references therein. Weak * derived sets and their relations with weak * closures found applications in many areas: the structure theory of Fréchet spaces (see [1,3,5,19,20,22,24]), Borel and Baire classification of linear operators, including the theory of ill-posed problems ( [28,31,32,33]), Harmonic Analysis ( [12,16,18,29]), theory of biorthogonal systems ( [10,30]; I have to mention that the historical information on weak * sequential closures in [10] is inaccurate). The survey [25] contains a historical account and an up-to-date-in-2000 information on weak * sequential closures.…”