E. M. Semenov

Symmetric Functionals and Singular Traces

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.

DISJOINTLY STRICTLY-SINGULAR INCLUSIONS BETWEEN REARRANGEMENT INVARIANT SPACES

A linear operator between two Banach spaces X and Y is strictly-singular (or Kato) if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion for Banach lattices introduced in [8] is the following one: an operator T from a Banach lattice X to a Banach space Y is said to be disjointly strictly-singular if there is no disjoint sequence of non-null vectors (xn)n∈N in X such that the restriction of T to the subspace [(xn)∞n=1] spanned by the vectors (xn)n∈N is an isomorphism. Clearly every strictly-singular operator is disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion Lq[0, 1]↪Lp[0, 1] for 1≤p<q<∞). In the special case of considering Banach lattices X with a Schauder basis of disjoint vectors both concepts coincide. The notion of disjointly strictly-singular has turned out to be a useful tool in the study of lattice structure of function spaces (cf. [7–9]). In general the class of all disjointly strictly-singular operators is not an operator ideal since it fails to be stable with respect to the composition on the right. The aim of this paper is to study when the inclusion operators between arbitrary rearrangement invariant function spaces E[0, 1] ≡ E on the probability space [0, 1] are disjointly strictly-singular operators.

Embeddings of Rearrangement Invariant Spaces that are not Strictly Singular

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L1([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space LΦ with Φ(x) = exp(x2)-1.

Multiplicator space and complemented subspaces of rearrangement invariant space

Abstract We show that the multiplicator space M (X) of an rearrangement invariant (r.i.) space X on [0,1] and the nice part N 0 ( X ) of X , that is, the set of all a ∈ X for which the subspaces generated by sequences of dilations and translations of a are uniformly complemented, coincide when the space X is separable. In the general case, the nice part is larger than the multiplicator space. Several examples of descriptions of M (X) and N 0 ( X ) for concrete X are presented.

Bernstein Widths and Super Strictly Singular Inclusions

The super strict singularity of inclusions between rearrangement invariant function spaces on [0, 1] is studied. Estimates of the Bernstein widths 𝛾n of the inclusions L∞ are given. It is showed that if the inclusion is strong and the order continuous part of exp E2 is not included in ⊂ then the inclusion ?? F is super strictly singular. Applications to the classes of Lorentz and Orlicz spaces are given.

Strictly singular operators in pairs of L p space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q ⊂ E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from Lp to Lq is found. There exists a strictly singular but not superstrictly singular operator on Lp, provided that p ≠ 2.

Banach limits: Invariance and functional characteristics

Banach limits invariant with respect to the Cesàro transform are studied. New functional characteristics of Banach limits are introduced and studied.

Estimation of a quadratic function and the -Banach–Saks property

Let E be a rearrangement-invariant Banach function space on [0, 1], and let Γ(E) denote the set of all p ≥ 1 such that any sequence {xn} in E converging weakly to 0 has a subsequence {yn} with supm m−1/p‖ ∑ 1≤k≤m yn‖ < ∞. The set Γi(E) is defined similarly, but only sequences {xn} of independent random variables are taken into account. It is proved (under the assumption Γ(E) = {1}) that if Γi(E) \ Γ(E) = ∅, then Γi(E) \ Γ(E) = {2}. §1 A classical Banach–Saks theorem (see [B, Chapter 12, Theorem 2]) states that if a sequence xn ∈ Lp[0, 1], 1 < p < ∞, converges weakly to zero, then there exists a sequence nk ∈ N and a number C > 0 such that ∥∥∥ m ∑ k=1 xnk ∥∥∥ Lp ≤ Cm for all m ∈ N (N denotes the set of positive integers). For p ∈ (2,∞) this estimate follows also from [KP]. The exponent max(1/2, 1/p) is sharp. It suffices to consider the Rademacher system xn(t) = sign sin 2πt, n ∈ N, for p ≥ 2 and any sequence of normalized Lp-functions with disjoint supports for p ≤ 2. This theorem leads to the following definitions (see [J, Be]). Let E be a Banach space, and let p ≥ 1. A bounded sequence {xn} in E is called a p-BS-sequence if there exists a subsequence {yn} ⊂ {xn}, such that sup m∈N m− 1 p ∥∥∥ m ∑ k=1 yk ∥∥∥ E < ∞. We shall say that E possesses the p-BS-property and write E ∈ BS(p) if any sequence converging weakly to zero contains a p-BS-subsequence. Obviously, every Banach space possesses the 1-BS-property. The set Γ(E) = {p : p ≥ 1, E ∈ BS(p)} is either [1, α] or [1, α), for some α ∈ [1,∞]. This set will be called the index set of the space E, and α is the Banach–Saks index of E; we write γ(E) = α if Γ(E) = [1, α] and γ(E) = α − 0 if Γ(E) = [1, α). A related notion was introduced in [R], where the coordinate Orlicz spaces with the p-BS-property were described. The Banach–Saks theorem says that γ(Lp) = min(p, 2) for 1 < p < ∞. It is known also that γ(L1) = γ(L∞) = 1. In lp, the sequences that converge weakly to zero have a 2000 Mathematics Subject Classification. Primary 46E30.

Some Geometrical Properties of Rearrangement Invariant Spaces

We study the set of rearrangement invariant spaces with the property that the unit sphere generated by a pair of elements from such a space is a parallelogram, as well as characteristic properties ofL 1andL ∞, in the class of Lorentz spaces. We also investigate a monotonicity property of rearrangement invariant spaces.

On the equivalence of the Lorentz and Marcinkiewicz norm on subsets of measurable functions

Abstract Sets of measurable functions whose truncations have equivalent Lorentz and Marcinkiewicz norm are characterized. Several examples are given.