In the recent years the research activity of members of the Department has covered a wide range of topics concerning functional analysis and its relationships with other fields. Below we present the most important results obtained in recent years.
Applications of interpolation theory to functional analysis. A modern approach by interpolation of Banach spaces is presented in [DMM1] to prove the abstract type Littlewood inequalities for inclusion maps between Banach symmetric sequence spaces. This extends the famous analogues in L_{p}-spaces due to Littlewood, Orlicz, Bennett and Carl. These results have many different applications, e.g. to eigenvalue distributions of compact operators [DMM2], local theory of Banach spaces and theory of interpolation functors [M], s-numbers in finite-dimensional Schatten classes [DMM3].
Approximation theory. The study of generalized Franklin systems was undertaken and results were presented in the series of papers (see e.g.[GK1]). Non-linear m-term greedy approximation with respect to the Haar system and other wavelet systems was studied. In particular stability of greedy approximation in the space BV was obtained in [BDKPW]. Important results about quasi-greedy bases we obtained in [GK2],[W].
Local theory of Banach spaces. In [MT1] the geometry of random sections and projections of symmetric convex bodies was investigated. Relation between optimal radii of Euclidean balls inscribed in sections and superscribed on projections of symmetric convex bodies was given in [MT2]. A lower bound for Banach-Mazur distances between symmetric polytopes generated by subgaussian vectors was given in [LMOT].
Methods of the theory of locally convex spaces and their applications to classical analysis. Derived functors on locally convex spaces are applied to the problem of parameter dependence of solutions of linear partial differential operators [BD] and to the problem which composition operators on the space of real analytic functions have closed range [DL]. The structure of the corresponding spaces of functions or distributions is analyzed [DV].
Sobolew spaces. In [PW1] the unconditional structure of Sobolev spaces and spaces of functions of bounded variation are studied. Faliure of local unconditional structure of Soblev spaces in L^{1}-norms and spaces of functions with bounded variation are dscussed [PW2]. The bounded approximation property of the space of functons with bounded variation is established in [ACDP]. The relation between singularities of vector measure and constrains on directions of its Fourier transform is investigated in [RW].
Topological algebras. Several papers deal with ideals in F-algebras. In [Z1] it is shown that a unital F-algebra has all left (right) maximal ideals closed if and only if it is a Q-algebra, i.e. the group of its invertible elements is open. In [Z2] it is shown that a unital F-algebra has all one-sided ideals closed if and only if it is both left and right Noetherian. In [Z3] it is constructed an m-convex B_{0}-algebra in which all left but not all right ideals are closed. Other results concern topologically invertible elements and operators on locally convex spaces and their hyperinvariant subspaces.
Handbook of the Geometry of Banach Spaces Members of the Department contributed four survey articles to the Handbook of the Geometry of Banach Spaces describing "state of the art" in presented areas, [HB1], [HB2], [HB3], [HB4].
[ACDP] | G. Alberti, M. Csörneyi, A.Pełczyński, D. Preiss, BV has the Bounded Approximation Property, Journal of Geometric Analysis 15 (2005),1-7. |
[BDKPW] | P. Bechler, R. Devore, A. Kamont, G. Petrova, P. Wojtaszczyk, Greedy wavelet projections are bounded on BV. Trans. Amer. Math. Soc., 359 (2007), 619-635. |
[BD] | J. Bonet, P. Domański, The splitting of exact sequences of PLS-spaces and smooth depepndence of solutions of linear partial differential equations, Adv. Math., 217 (2008), 561-585. |
[DMM1] | A. Defant, M. Mastyło, C. Michels, Summing inclusion maps between symmetric sequence spaces, Trans. Amer. Math. Soc., 354 (2002), 4473-4492. |
[DMM2] | A. Defant, M. Mastyło, C. Michels, Eigenvalues estimates for operators on symmetric Banach sequence spaces, Proc. Amer. Math. Soc. 132 (2003), 513-521. |
[DMM3] | A. Defant, M. Mastyło, C. Michels, Summing norms of identities between unitary ideals, Math. Z. 252 (2006), 863-882. |
[DL] | P. Domański, M. Langenbruch, Coherent analytic sets and composition of real analytic functions, J. reine angew. Math., 582 (2005), 41-59. |
[DV] | P. Domański, D. Vogt, The space of real analytic functions has no basis, Studia Math., 142 (2000), 187-200. |
[GK1] | G. G. Gevorkyan, A. Kamont, General Franklin systems as bases in H^{1}[0,1]. Studia Math., 167 (2005), 259-292. |
[GK2] | G. G. Gevorkyan, A. Kamont, Two remarks on quasi-greedy bases in the space L_{1}. (Russian) Izv. Nats. Akad. Nauk Armenii Mat., 40 (2005), no. 1, 5-17. |
[HB1] | T. Figiel, P. Wojtaszczyk, Special bases in function spaces, Handbook of the geometry of Banach spaces, vol I, North Holland, W.B.Johnson and J. Lindenstrauss editors, Amsterdam 2003, 561-590. |
[LMOT] | R. Latała, P. Mankiewicz, K. Oleszkiewicz, N. Tomczak-Jaegermann, Banach-Mazur distances and projections on random subgaussian polytopes, Discrete Comput. Geom., 38 (2007), 29-50. |
[HB2] | P. Mankiewicz, N. Tomczak-Jaegermann, Quotients of finite-dimensional Banach spaces; random phenomena, Handbook of the geometry of Banach spaces, vol II, North Holland, W.B.Johnson and J. Lindenstrauss editors, Amsterdam 2003, 1201-1246. |
[MT1] | P. Mankiewicz and N. Tomczak-Jaegermann, Geometry of Families of Random Projections of symmetric convex bodies, Geom. Funct. Anal., 11 (2001), 1282-1326. |
[MT2] | P. Mankiewicz, N. Tomczak-Jaegermann, Low Dimensional sections versus projections of convex bodies, Israel J. of Math., 153 (2006), 45-60. |
[M] | M. Mastyło, Interpolation methods of means and orbits, Studia Math. 17 (2005), 153-175. |
[HB3] | A. Pełczyński, M. Wojciechowski, Sobolev Spaces, ibidem, 1361-1425. |
[HB4] | P. Wojtaszczyk, Spaces of analytic functions with integral norm, ibidem, 1671-1702. |
[PW1] | A. Pełczyński, M. Wojciechowski, Spaces in several variables in L^{1} norm are non isomorphic to Banach lattices, Ark. Mat., 40 (2002) 363-382. |
[PW2] | A. Pełczyński, M. Wojciechowski, Spaces of functions with bounded variation and sobolev spaces without local unconditional structure, J. reine angew. Math., 558 (2003), 109-157. |
[RW] | M. Roginskaya, M. Wojciechowski, Singularity of vector valued measures in terms of Fourier transform, J. Fourier Analysis and Applications, 12, (2006), 213 - 223. |
[W] | P. Wojtaszczyk, Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107 (2000), 293-314. |
[Z1] | W. Żelazko, When a unital F-algebra has all left (right) ideals closed, Studia Math., 175 (2006), 279-284. |
[Z2] | W. Żelazko, A characterization of F-algebras with all one-sided ideals closed, Studia Math., 168 (2005), 135-145. |
[Z3] | W. Żelazko, An m-convex B_{0}-algebra with all left but not all right ideals closed, Coll. Math., 194 (2006), 317-324. |