Evgeny M. Semenov

Strictly singular operators on L_p spaces and interpolation

We study the class Vp of strictly singular non-compact operators on Lp spaces. This allows us to obtain interpolation results for strictly singular operators on Lp spaces. Given 1 ≤ p < q ≤ ∞, it is shown that if an operator T bounded on Lp and Lq is strictly singular on Lr for some p ≤ r ≤ q, then it is compact on Ls for every p < s < q.

Strictly Singular Embeddings

An operator A mapping a Banach space E into a Banach space F is called strictly singular (or Kato) if any restriction of A to an infinite-dimensional subspace of E is not an isomorphism. The paper deals with the problem of describing all couples of rearrangement-invariant spaces E↪F for which the embedding operator is strictly singular.

Lebesgue constants of the Walsh system and Banach limits

We study the properties of the Lebesgue constants of the Walsh system Ln(W), n ∈ N, and apply the results to the theory of Banach limits. We show that the sequence

Characteristic functions of Banach limits

\left\{ {\frac{{{L_n}\left( W \right)}}{{{{\log }_2}n}},n \geqslant 2} \right\}

Strictly singular inclusions into \(L^1 + L^\infty\)

does not belong to the space of almost convergent sequences ac, which reveals their extremely irregular behavior. Several results of the opposite nature are obtained for some special means of these constants.

Interpolation and extrapolation of strictly singular operators between L-p spaces

We calculate the norms of multipliers for the Haar system in some rearrangement invariant spaces for which the Haar system is not an absolute basis.