The PI will explore and apply logic, set-theoretic and combinatorial methods to the geometry of Banach spaces. The isomorphic classification of the complemented subspaces of the spaces of p-power integrable functions, especially those contained in some nice subspaces, will be considered. Necessary and sufficient conditions for bounded linear operators from these spaces to factor through spaces with simpler and better structures will be studied. The investigator will also study operators from spaces with certain asymptotic structures. Characterizations of subspaces and quotients of separable Banach spaces with shrinking unconditional basis and subsequences of unconditional sequences will be investigated.
Banach space theory is about vector spaces in which there is a very natural notion of distance. These spaces are of fundamental importance in many areas, including mathematical models in quantum mechanics. The understanding of the geometry of Banach spaces has been and will continue to be useful in many areas of mathematics and engineering. In particular, the use of logic, set-theoretic and combinatorial methods will provide rich information about the geometric structure of Banach spaces.
In this project, the PI employed methods of logic and set theory to study the subspaces of spaces of p-integral functions, especially the complemented subspaces and the "small" subspaces of these spaces. Necessary and sufficient conditions for operators from spaces with certain asymptotic structures to factor through spaces with simpler and better strctures have been found. The PI also found intrinsic characterizations of subspaces and quotients of spaces with shrinking unconditional basis. A normalized weakly null sequence in a reflexive space of which block bases generate every Haar basis for p-integeral function space with p in the reflexive range. The PI also gave a complete classification of commutators on certain classical sequence spaces. Those results provided tools for studying operator theory, frame theory and theoretical computer science. The project has resulted the following ten journal publications. W. B. Johnson, Bentuo Zheng, "Subspaces and quotients of spaces with shrinking unconditional bases", Israel Journal of Mathematics, p. , vol. 185, (2011). Published, E. Odell, Bentuo Zheng, "On the unconditional subsequence property", Journal of Functional Analysis, p. 604, vol. 258, (2010). Published, E. Odell, Th. Schlumprecht, B. Sari, Bentuo Zheng, "Systems formed by translates of one element in Lp(R)", Transactions of the American Mathematical Society, p. 6, vol. 363, (2011). Published, Rui Liu, Bentuo Zheng, "A characterization of Schauder frames which are near-Schauder bases", Journal of Fourier Analysis and Applications, p. 791, vol. 16, (2010). Published, Dongyang Chen and Bentuo Zheng, "Remarks on Lipschitz $p$-summing operators", Proceedings of the American Mathematical Society, p. 2891, vol. 139, (2011). Published, Dongyang Chen and Bentuo Zheng, "Three space problems for the approximation property", Acta Math Sinica, p. , vol. , (2011). Submitted, D. Chen, W. B. Johnson and B. Zheng, "Commutators on $(sumell_q)_{ell_p}$", Studia Math., p. 175, vol. 206, (2011). Published, F. Botelho, J. Jamison and B. Zheng, "Strict isometries of arbitrary orders", Linear Algebra and its Applications, p. 3303, vol. 436, (2012). Published, F. Botelho, J. Jamison and B. Zheng, "Isometries on spaces of vector valued Lipschitz functions", Positivity, p. , vol. , (2011). Published, 10.1007/s11117-011-0148-2 J. Garcia-Pacheco, "Geometric properties on non-complete spaces", Quaestiones Mathematicae, p. 489, vol. 34, (2011). The PI gave several talks at conferences held by the American Mathematical Society and other univerities. He also served as SemCzar for the Workshop in Analysis and Probability at Texas A&M University. He organized the seminar session, encouraged and provided chances for graduate students and young researchers to give presentations. The PI collaborated with a wide range of mathematicians with ages from early thirties to over seventies.
