Area of research

My research interests circles around functional Analysis, analysis on Banach spaces and various geometric aspects of Banach spaces. Currently I am working on Higher order smoothness in Banach spaces, the geometric properties determined by the intersection of balls and the higher duals of the spaces. My recent interest includes the geometric properties of the Operator ideals in spaces of bounded linear Operators between Banach spaces and Hahn-Banach type extension of linear maps between Banach spaces (see [2,6]). These Operator ideals appears from various notions of compactness in Banach spaces.

The theory of operators on Banach spaces is quite different from operators on Hilbert spaces. The non existence of orthonornal basis makes it difficult to identify a typical operator (bounded linear) inside the space of bounded linear operators. I work on operators between L1-preduals and their extensions when the said operator is considered compact or weakly compact. In [5], Lindenstrauss characterized L1-preduals spaces in terms of extension of compact and weakly compact operators. In this connection he studied Banach spaces which are Pλ, introduced by M.M. Day. Various types of operators and their extensions are studied in [5]. We aim to continue a similar study for the operators (between X and Y) which are from a special ideal of B(X,Y) and a subclass of K(X,Y). Here B(X,Y) and K(X,Y) represent bounded and compact operators between X and Y. In [6] the author presented a comprehensive study on various aspects of Hahn-Banach theorem and its analogues. Hahn-Banach smoothness is one of these analogues. At present we are engaged in searching solutions for the following problems.

• What are the examples of Banach spaces which are weakly Hahn-Banach smooth but not Hahn-Banach smooth?

• Let Y be a subspace of X. Suppose that Y has property-(U) in X and that for all  f∈ Y*, there exists F ∈ N A(X) such that ∥F ∥ = ∥f ∥ and F |Y = f .  Is Y weakly Hahn-Banach smooth?

By N A(X) we mean the set {f ∈ X* : ∃ x ∈ SX s.t. f (x) = ∥f ∥}.

My last few students have completed their doctoral dissertations on simultaneous approximation and its various geometric implications in Banach spaces. A few well-known researchers in this direction are A.L. Garkavi, Dan Amir, I.G. Tsarkov, Libor Vesely, Jaroslav Mach. Enormous study have been carried out in this field in last few decades. Two objects play crucial role in this study: the generalized radius and generalized centre  of a closed bounded set. In literature they are also known as Chebyshev radius and Chebyshev centre respectively.

My alumni have made contributions towards characterizations of L1-preduals using the existence of Chebyshev centre and some identity involving the distance of the set of restricted Chebyshev centre to the underlying subset. The aforesaid identity was first observed by P.W. Smith and J.D. Ward in 1975 for the spaces of type C(K). It is characterized when the state space of a closed subspace A of C(K) is a Choquet simplex in terms of generalized centre of all possible subsets of A consisting of 4 points.  It is found that similar characterizations to finite co-dimensional subspaces which are strongly proximinal, due to Godefroy and Indumathi remains true for finite co-dimensional subspaces of A(K) which have property-(P1). Property-(P1) is a set-valued analogue of strong proximinality. Here K is a Choquet simplex and A(K) represents the real valued affine continuous functions on K.

In 1997, Libor Vesely introduced the notion of Generalized centre in Banach spaces (see [4]). This concept represents a significant adjustment to the Principle of local reflexivity. Subsequently, this idea was refined in a variety of ways after becoming an area of investigation. It is evedent that a dual space and an L1-predual come under this category. We extended Vesely's concept to arbitrary closed bounded convex subsets.

We obtain a number of problem while investigating on simultaneous approximation in Banach spaces. Presumably the community of Banach space theorists is more or less aware of the obstacles while concluding anything on the following problems. 

• Let X be a Banach space where any closed and bounded subset of X has same Chebyshev radius when considering subset of X and of X**. Is X necessarily 1-complemented in X**?

• Let X be an L1-predual and A be a finite subset of X**. Does X admit restricted Chebyshev centre for A?

• Let X, Y be two Banach spaces which are isomorphic. Suppose that d(X,Y)=1. Consider X as a space that falls under the category of (GC). Is Y a member of the same family as well?

We call a subspace is 1-complemented if it is a range of norm-1 projection. Here d(X,Y) represents the Banach-Mazur distance between X and Y.

Including the above mentioned problems my current research in the field of simultaneous approximation is to explore the existence of best n-nets (see [1]) for different closed bounded sets in Banach spaces along with the existence of f-centre of closed bounded sets and their consequences. The notion f-centre was introduced by Vesely in his paper in 1997.

References

1. D. Amir, J. Mach, K. Saatkamp, Existence of Chebyshev centers, best n-nets and best compact approximants, Trans. Amer. Math. Soc. 271(2) (1982) 513–524.

2. K. Donor, Extension of Positive Operators and Korovkin Theorems, LNM 904, Springer-Verlag, Berlin-New York, 1982. xii+182 pp.

3. J. Lindenstrauss, Some Aspects of the theory of Banach spaces, Advances in Math.,  5, (1970) 159-180.

4. J. Lindenstruass, Extension of compact operators, Mem. Amer. Math. Soc. 48(1964), 112.

5. P. W. Smith and J. D. Ward, Restricted centers in C(Ω), Proc. Amer. Math. Soc. 48 (1975), 165–172,

6. M. A. Sofi, Some problems in functional analysis inspired by Hahn-Banach type theorems, Ann. Funct. Anal.,  5, (2014), 1-29

7. L. Veselý, Generalized centers of finite sets in Banach spaces. Acta Math Univ Comenian(N.S.). 66(1) (1997) 83–115.