A Banach space with the Schur and the Daugavet property

We show that a minor refinement of the Bourgain-Rosenthal construction of a Banach space without the Radon-Nikodým property which contains no bounded -trees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such spaces; in particular, we show that the Daugavet property is not inherited by ultraproducts.

     

Banach spaces with the Daugavet property

A Banach space is said to have the Daugavet property if every operator of rank satisfies . We show that then every weakly compact operator satisfies this equation as well and that contains a copy of . However, need not contain a copy of . We also study pairs of spaces and operators satisfying , where is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with is as small as possible and give characterisations in terms of a smoothness condition.

     

Banach spaces with the Daugavet property, and the centralizer

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:

(1)

Every member of R has the Daugavet property.

(2)

It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product lie in R.

(3)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C(K,(Y,τ)) (of all Y-valued τ-continuous functions on K) is a member of R.

(4)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C(K,Y)-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R.

(5)

All dual Banach spaces without minimal M-summands are members of R.

     

The largest linear space of operators satisfying the Daugavet equation in

We find the largest linear space of bounded linear operators on that, being restricted to any , , satisfy the Daugavet equation.

     

L-orthogonality in Daugavet centers and narrow operators

We study the presence of L-orthogonal elements in connection with Daugavet centers and narrow operators. We prove that if $\mathrm{dens}(Y)?{\omega }_1$ and $G:X?Y$ is a Daugavet center with separable range then, for every non-empty $w^{?}$-open subset W of $B_{X^{??}}$, it follows that $G^{??}(W)$ contains some L-orthogonal to Y. In the context of narrow operators, we show that if X is separable and $T:X?Y$ is a narrow operator, then given $y\in B_X$ and any non-empty $w^{?}$-open subset W of $B_{X^{??}}$ then W contains some L-orthogonal u so that $T^{??}(u)=T(y)$. In the particular case that $T^{?}(Y^{?})$ is separable, we extend the previous result to $\mathrm{dens}(X)={\omega }_1$. Finally, we prove that none of the previous results holds in larger density characters (in particular, a counterexample is shown for ${\omega }_2$ under the assumption $2^c={\omega }_2$).     

On a new geometric property for Banach spaces

In this paper we study a geometric property for Banach spaces called condition (*), introduced by de Reynaet al in [3], A Banach space has this property if for any weakly null sequencex _n of unit vectors inX, ifx _* ⁿ is any sequence of unit vectors inX ^* that attain their norm at x_n’s, then . We show that a Banach space satisfies condition (*) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.     

The Riesz decomposition property for the space of regular operators

If and are Banach lattices such that is separable and has the countable interpolation property, then the space of all continuous regular operators has the Riesz decomposition property. This result is a positive answer to a conjecture posed by A. W. Wickstead.

     

Compact lines and the Sobczyk property

We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta.     

A new characterisation of the analytic Radon-Nikodym property

We show that a separable complex Banach space has the analytic Radon-Nikodym property if and only if there exists , such that the space consisting of all -bounded -valued analytic martingales is separable.

     

An example of an asymptotically Hilbertian space which fails the approximation property

Following Davie's example of a Banach space failing the approximation property (1973), we show how to construct a Banach space which is asymptotically Hilbertian and fails the approximation property. Moreover, the space is shown to be a subspace of a space with an unconditional basis which is ``almost' a weak Hilbert space and which can be written as the direct sum of two subspaces all of whose subspaces have the approximation property.

     

Invertibility and the fredholm property of difference operators

We obtain conditions for the invertibility and the Fredholm property of the difference operator (Dx)(n)=x(n) -U(n)x(n − 1),n ε ℤ, in the Banach space l _p (ℤ, X),p ε [1, ∞], of vector sequences, whereX is a Banach space andU is a bounded operator function. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 816–827, June, 2000.     

The strong Schur property in Banach lattices

We prove that in the class of discrete Banach lattices the strong Schur property is equivalent to the disjoint strong Schur property (Theorem 3.1). Roughly speaking the strong Schur property holds iff an appropriate condition concerning sequences with positive pairwise disjoint terms is satisfied.      

An image problem for compact operators

Let be a separable Banach space and a sequence of closed subspaces of satisfying for all . We first prove the existence of a dense-range and injective compact operator such that each is a dense subset of , solving a problem of Yahaghi (2004). Our second main result concerns isomorphic and dense-range injective compact mappings between dense sets of linearly independent vectors, extending a result of Grivaux (2003).

     

Hahn-Banach extension operators and spaces of operators

Let be Banach spaces and let be closed operator ideals. Let be a Banach space having the Radon-Nikodým property. The main results are as follows. If is a Hahn-Banach extension operator, then there exists a set of Hahn-Banach extension operators , , such that , where . If is an ideal in for all equivalently renormed versions of , then there exist Hahn-Banach extension operators and such that .

     

The decomposition theorem for functions satisfying the law of large numbers

LetB be a Banach space with the Radon-Nikodym property and (S, , ) a probability space. Then anf: SB satisfies the strong law of large numbers if and only if there exists a Bochner integrable functionf ₁ and a Pettis integrable functionf ₂,f ₂f ₂=0 in the Glivenko-Cantelli norm, such thatf=f ₁+f ₂. The composition is unique.     

Banach-Saks property in Marcinkiewicz spaces

We catalogue all Marcinkiewicz function and sequence spaces with the Banach-Saks property and present necessary and sufficient conditions for a wide subclass of spaces to possess the p-Banach-Saks property, 1<p<∞. We apply our results to several open problems.     

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coeffcients in Banach-valued Lp-spaces are studied.To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations,the suffcient conditions are founded.By using these facts,the unique continuation properties are established.In the application part,the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.     

Modulation space estimates for the fractional integral operators

We obtain the boundedness for the fractional integral operators from the modulation Hardy space μp,q to the modulation Hardy space μr,q for all 0 < p < ∞. The result is an extension of the known result for the case 1 < p < ∞ and it contains a larger range of r than those in the classical result of the Lp → Lr boundedness in the Lebesgue spaces. We also obtain some estimates on the modulation spaces for the bilinear fractional operators.     

Quasisimilarity of power bounded operators and Blum-Hanson property

We construct a power bounded operator on a Hilbert space which is not quasisimilar to a contraction. To this aim, we solve an open problem from operator ergodic theory showing that there are power bounded Hilbert space operators without the Blum-Hanson property. We also find an example of a power bounded operator quasisimilar to a unitary operator which is not similar to a contraction, thus answering negatively open questions raised by Kérchy and Cassier. On the positive side, we prove that contractions on ?p spaces (1?p<∞) possess the Blum-Hanson property.