Small subspaces of $L_p$

SMALL SUBSPACES OF Lp

We prove that if X is a subspace of Lp (2 < p < ∞), then either X embeds isomorphically into `p ⊕ `2 or X contains a subspace Y, which is isomorphic to `p(`2). We also give an intrinsic characterization of when X embeds into `p⊕`2 in terms of weakly null trees in X or, equivalently, in terms of the “infinite asymptotic game” played in X. This solves problems concerning small subspaces of Lp ori...

Large and Small Subspaces of Hilbert Space

For example, Theorem 3 says that if V is a closed subspace of f2 and if V CQp for some p < 2, then V is finite-dimensional . On the other hand, the corollary to Theorem 4 states that there exist infinite-dimensional subspaces V of f 2 none of whose nonzero elements belongs to any f p -space (p < 2) . [For L2(0, 1) the results are somewhat different: (1) if V is a closed subspace of L 2(0, 1) an...

Common Invariant Subspaces from Small Commutators

We study the following question: suppose that A and B are two algebras of complex n×n matrices such that the ring commutator [A, B] = AB−BA is “small” for each A ∈ A and B ∈ B. Does this imply that A and B have a common non-trivial invariant subspace? This question is motivated by a series of papers studying the structure of “almost-commutative” algebras and, more generally, semigroups. A simpl...

Subspaces of Small Codimension of Finite-dimensional Banach Spaces

Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking values in a Hubert space, we deduce an inequality for entropy numbers of the operator and its dual. In this note we study the following problem: given an n-dimensional Banach space E and a Euclidean no...

Comment on frame's of subspaces