5월10일(금) 16:00~16:50 5E102
연사: 이한주(동국대)
제목:Bishop-Phelps theorem and recent development
초록: A bounded linear operator 𝑇 from a Banach space 𝑋 to a Banach space 𝑌 is said to
be norm-attaining if there is an element x in the closed unit ball 𝐵! of 𝑋 such that
|𝑇| = |𝑇𝑥| . If 𝑌 is a scalar (real or complex) field, then such a 𝑇 is called a
norm-attaining functional.
The celebrated Bishop-Phelps theorem says that norm-attaining functionals are
dense in the dual space 𝑋∗. This theorem has far-reaching applications. Bishop-
Phelps asked in the same paper if the set of norm-attaining operators is dense in
the space 𝐿(𝑋, 𝑌) of bounded linear operators from 𝑋 to 𝑌. Even though the
answer is negative in general, there has been several attempts to find a proper
solution to this question.
In this talk, I briefly review various approach to find the proper solution of the
Bishop-Phelps question. These include Lindenstrauss’ and Bourgain’s approach.
These are also related to the various geometric properties of norm such as the
Radon-Nikodym property, differentiability and uniform convexity of norm.
Finally, I will introduce new quantitative approach, so-called “Bishop-Phelphs-
Bollobas property”, which is a stronger property than the Bishop-Phelps property.
In particular, the Bishop-Phelps-Bollobas theorem holds for 𝐿(𝐿!, 𝐿!), where
1 ≤ 𝑝, 𝑞 < ∞ .