A note on biorthogonal systems
- PDF / 308,681 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 100 Downloads / 260 Views
A note on biorthogonal systems Petr Hájek1 · Michal Johanis2 Received: 9 July 2019 / Accepted: 1 December 2019 © Universidad Complutense de Madrid 2020
Abstract We consider the following problem (which is a generalisation of a folklore result Proposition 1 below): given a continuous linear operator T : X → Y , where Y is a Banach space with a (long) sub-symmetric basis, under which conditions can we find a continuous linear operator S : X → Y such that S(B X ) contains the basis of Y . As a tool we also consider a non-separable version of Theorem 2 below: Given an infinite subset A ⊂ X ∗ , under which conditions can we find a biorthogonal system in X × A of cardinality card A? Keywords Biorthogonal systems · Linear operators Mathematics Subject Classification 46B25 · 46B26 We are interested in a generalisation of the following two results into a non-separable and more general setting. The first one is a folklore result, see e.g. [3, Proposition 3.33]: Proposition 1 Let X be a Banach space, Y = p , 1 ≤ p < ∞, or Y = c0 , and suppose there is a non-compact operator T ∈ L(X ; Y ). Then there are S ∈ L(X ; Y ) and a normalised basic sequence {xn } ⊂ X such that S(xn ) = en , n ∈ N, where {en } is the canonical basis of Y . If X does not contain 1 , then {xn } may be chosen to be weakly null.
Supported by OP VVV CAAS CZ.02.1.01/0.0/0.0/16_019/000077.
B
Michal Johanis [email protected] Petr Hájek [email protected]
1
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Prague 6, Czech Republic
2
Faculty of Mathematics and Physics, Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
123
P. Hájek, M. Johanis
It turns out that the following theorem by the authors which deals with finding biorthogonal systems in preduals is a good tool for this problem. Theorem 2 ([3, Theorem 3.56]). Let X be a Banach space and let { f n } ⊂ X ∗ be a bounded sequence. The following statements are equivalent: (i) { f n } is not a relatively compact set. (ii) There are a subsequence {gn } of { f n } and an (infinite-dimensional) subspace Y ⊂ X such that {gn Y } ⊂ Y ∗ is a semi-normalised w ∗ -null sequence. (iii) There is a semi-normalised basic sequence {xn } ⊂ X which is biorthogonal to a subsequence of { f n }. Moreover, we may assume in addition that {xn } is either weakly null or equivalent to the canonical basis of 1 . This theorem has useful applications, see e.g. [3] or [6]. To generalise these results into a non-separable setting we first need to define certain properties. Let X be a normed linear space. For A ⊂ X ∗ and x ∈ X we denote supp A x = { f ∈ A; f (x) = 0}. Definition 3 Let X be a normed linear space and A ⊂ X ∗ . We say that A has • property C if supp A x is countable for each x ∈ X ; w∗
• property Z if f n → 0 for every sequence { f n } of distinct elements of A; • property B if for every ε > 0 there is k(ε) ≥ 0 such that card{ f ∈ A; | f (x)| > ε} ≤ k(ε) for any x ∈ B X . Let μ b
Data Loading...
