Banach spaces with few operators and multiplier results

• Main Author: Wark, H. M.
• Published: University of Oxford 1997
• Subjects: 510; Banach spaces
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362117

The construction of a non-separable reflexive Banach space on which every operator is the sum of a scalar multiple of the identity operator and an operator of separable range is presented. Using a result of Rao, a sufficient condition is given for Banach spaces with smooth norms to be decomposable. It is shown that operators on Banach spaces of co-dimension one in their biduals are the sum of a scalar multiple of the identity operator and a weakly compact operator. The Banach spaces of bounded operators L(1¹, 1^p) (1<p<ꝏ) and L(1^p, 1^r), 1 < p ≤ r ≤ p¹ < ꝏ, where 1/p + 1/p¹ = 1, are shown to be primary. The spaces of bounded diagonal operators and compact diagonal operators on a seminormalized Schauder basis β, the multiplier algebras L<sup>d</sub>(X, β) and K_d(X, β), are introduced and studied. New examples of these multiplier algebras are presented and a theorem of Sersouri is extended. A necessary and sufficient condition is given for c_o to embed in K_d(X, β). A sufficient condition is given on a semi-normalized Schauder basis β of a reflexive hereditarily indecomposable Banach space Y to ensure that K_d(Y, β) has the RNP. It is shown that the algebra L_d(X, β) is semisimple and that on the algebra K_d(X, β) derivations are automatically continuous. By representing diagonal operators as stochastic processes a general method of constructing multiplier algebras is given. A non trivial multiplier invariance for the normalized Haar basis of L¹[0,1] is proved.