Diagonal operators between vector-valued sequence spaces and measure of compactness

Lipi Rani Acharya Indian Institute of Technology, Kanpur, India e-mail: lipi@iitk.ac.in Diagonal operators between vector-valued sequence spaces and measure of compactness Abstract: In the literature, vector valued sequence spaces λ(X), defined with the help of a Banach space X and a scalar valued sequence space λ, have been studied quite extensively. This paper deals with the study of the diagonal operator D between these spaces, defined with the help of a sequence {Ti} of bounded linear operators acting between the underlying Banach spaces. The relationships of the measures of compactness of D with those of Ti’s, in terms of entropy numbers, approximation numbers and Kolmogorov numbers have been obtained. Using these results the compactness of D in terms of compactness of Ti’s has been characterized. Joint work with Manjul Gupta. Saturday, 2:00-2:20, Rm 215 PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP In the literature, vector valued sequence spaces λ(X), defined with the help of a Banach space X and a scalar valued sequence space λ, have been studied quite extensively. This paper deals with the study of the diagonal operator D between these spaces, defined with the help of a sequence {Ti} of bounded linear operators acting between the underlying Banach spaces. The relationships of the measures of compactness of D with those of Ti’s, in terms of entropy numbers, approximation numbers and Kolmogorov numbers have been obtained. Using these results the compactness of D in terms of compactness of Ti’s has been characterized. Joint work with Manjul Gupta. Saturday, 2:00-2:20, Rm 215 PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP Maŕıa D. Acosta Universidad de Granada, Granada, Spain e-mail: dacosta@ugr.es, macosta1@memphis.edu An “isomorphic” version of James’s theorem Abstract: James’s theorem states that every Banach space is reflexive if every functional on it attains its norm. There are (even classical) non-reflexive Banach spaces for which the set of norm attaining functionals has non-empty interior. Namioka posed the question if reflexivity holds anytime that for every equivalent norm, the above condition is satisfied. We present a quite general partial answer to the above question obtained in a joint work with V. Montesinos. Monday, 2:40-3:00, Rm 10 PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP James’s theorem states that every Banach space is reflexive if every functional on it attains its norm. There are (even classical) non-reflexive Banach spaces for which the set of norm attaining functionals has non-empty interior. Namioka posed the question if reflexivity holds anytime that for every equivalent norm, the above condition is satisfied. We present a quite general partial answer to the above question obtained in a joint work with V. Montesinos. Monday, 2:40-3:00, Rm 10 PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP