Bounded operators with no non-trivial closed invariant subspace have been constructed by P. Enflo [6]. In fact, there exist bounded operators on the space 1 with no non-trivial closed invariant subset [12]. It is still unknown, however, if such operators exist on reflexive Banach spaces, or on the separable Hilbert space. The main result of this note (Theorem 1) asserts that the existence of an...

We prove that for a generic measure preserving transformation T , the closed group generated by T is a continuous homomorphic image of a closed linear subspace of L0(λ,R), where λ is Lebesgue measure, and that the closed group generated by T contains an increasing sequence of finite dimensional toruses whose union is dense.

This paper generalizes polyhedra to infinite dimensional separable Hilbert spaces as countable intersections of closed semispaces. We show that a polyhedron is the sum of convex proper subset, which is compact in the product topology, plus a closed pointed cone plus a closed subspace. In the final part the dual range space technique is extended to solve infinite dimensional LP problems.

The complemented subspace problem asks, in general, which closed subspaces M of a Banach space X are complemented; i.e. there exists a closed subspace N of X such that X = M ⊕N? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some new results on complemented subspaces, to present a history ...

Johnson and Zippin recently showed that if X is a weak∗-closed subspace of 1 and T : X → C(K ) is any bounded operator then T can be extended to a bounded operator T̃ : 1 → C(K ). We give a converse result: if X is a subspace of 1 such that 1/X has an unconditional finitedimensional decomposition (UFDD) and every operator T : X → C(K ) can be extended to 1 then there is an automorphism τ of 1 su...

The famous Lomonosov’s invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E “commutes” with a compact operator K 6= 0, i.e., TK = KT, then T has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide some applications to singlevalued linear operators.