Professor Bunyamin Sari invites you to attend the

Master's Defense of Philip Azad

WHEN: Wednesday, June 21, 2023

2:00pm in GAB 461

"The Haar basis of Lp[0, 1] for 1 ≤ p < ∞ is two-player Reproducible"

The reproducibility of a Schauder basis was a notion developed by Lindenstrauss
and Pelczynski. When a Banach space with a basis is embedded via
an isometric isomorphism into another space with a basis then the embedded
basis can be approximated by a block basis of the image space, in other words
the basis of the original space is "reproduced" in the image space. They proved
that the Haar bases of Lp[0, 1] is precisely reproducible or 1-reproducible, that
is, the constant of equivalence can be arbitrarily close to 1.
A two-player game version of reproducibility was developed by Alspach and
Sari, which they call two-player subsequential reproducibility. This notion relaxes
the condition of isometry from the embedding, thus the embedding is only
required to be isomorphic but the Player II is limited to choose the image of
subsequences of the basis. This notion played a key role in their solution of the
Elastic Banach space problem. In this thesis, first we prove that the Haar basis
do not possess the stronger property of two-player subsequential reproducibility
by showing that if Player I chooses a fast enough sequence that forces the Player
II to choose a lacunary sequence then the infinite subsequence of Haar functions
so chosen is equivalent to the canonical bases of ℓp.
Secondly, we prove that the Haar bases of Lp[0, 1] is two-player reproducible.
In this notion, the subsequence condition for Player II is removed, which is more
natural and it is a direct strategic analog of the original notion of reproducibility.
This notion is concurrently and independently developed by Lechner, Motakis,
Muller and Schlumprecht and already found applications in factorization of
operators.
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