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Abstract We derive estimates for approximation numbers of bounded linear operators between normed linear spaces. As special cases of our general results, approximation numbers of some weighted shift operators on p and those of isometries and projections of norm 1 are found. In the case of finite-rank operators, we obtain estimates for the smallest nonzero approximation number in terms of their generalized inverses. Also, we prove some results regarding the relation between approximation numbers and the closedness of the range of an operator. We recall that the closedness of the range is a necessary condition for the boundedness of a generalized inverse. We give examples illustrating the results and also show that certain inequalities need not hold. Keywords Generalized inverse · Generalized inverse of operator · Normed linear space · Approximation numbers Mathematics Subject Classification (2010) 15A09 · 47B06

1 Introduction Let X and Y be normed linear spaces, and BL(X, Y ) be the class of all bounded linear operators from X to Y . We use the notations BL(X) for BL(X, X) and X  for BL(X, C). We shall denote the set of all finite-rank operators F ∈ BL(X, Y ) with rank(F ) < k by Fk (X, Y ) and use the notation Fk (X) for Fk (X, X). Also,

K.P. Deepesh · S.H. Kulkarni (B) · M.T. Nair Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India e-mail: [email protected] K.P. Deepesh e-mail: [email protected] M.T. Nair e-mail: [email protected] R.B. Bapat et al. (eds.), Combinatorial Matrix Theory and Generalized Inverses of Matrices, DOI 10.1007/978-81-322-1053-5_12, © Springer India 2013

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we denote by p (n) the space Cn with norm  · p , 1 ≤ p ≤ ∞. We also use the notation  1, i = j, for i, j ∈ N, δij = 0, i = j, and for T ∈ BL(X, Y ), we denote by R(T ) the range of T . The concept of approximation numbers of operators from BL(X, Y ) is a generalization of the concept of singular values of compact operators between Hilbert spaces. For T ∈ BL(X, Y ) and k ∈ N, the kth approximation number sk (T ) of T is defined as   sk (T ) := inf T − F  : F ∈ Fk (X, Y ) . It is clear that T  = s1 (T ) ≥ s2 (T ) ≥ · · · ≥ 0 and if T is of finite rank, then sk (T ) = 0 for all k > rank(T ). Some studies on approximation numbers and their properties can be found in Pietsch [13–15]. Approximation numbers play an important role in the geometry of Banach spaces as they are used in defining certain subclasses (ideals) of operator spaces (see Pietsch [15]). The convergence properties of approximation numbers are found useful in estimating the error while solving operator equations (see Schock [17]). Computation of approximation numbers is a very difficult task, even in the case of operators between finite-dimensional spaces. There have been very few attempts in literature to estimate the approximation numbers of bounded linear operators between normed linear spaces. For example, Hutton, Morrell and Retherford [7] and Pietsch [14] contain methods of computing approxim