A characterization of k-hyponormality via weak subnormality

An operator T acting on a Hilbert space is said to be weakly subnormal if there exists an extension acting on such that for all . When such partially normal extensions exist, we denote by m.p.n.e.(T) the minimal one. On the other hand, for k?1, T is said to be k-hyponormal if the operator matrix is positive. We prove that a 2-hyponormal operator T always satisfies the inequality T∗[T∗,T]T?‖T‖2[T∗,T], and as a result T is automatically weakly subnormal. Thus, a hyponormal operator T is 2-hyponormal if and only if there exists B such that BA∗=A∗T and is hyponormal, where A:=[T∗,T]1/2. More generally, we prove that T is (k+1)-hyponormal if and and only if T is weakly subnormal and m.p.n.e.(T) is k-hyponormal. As an application, we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix.     

Extending compact topologies to compact Hausdorff topologies in ZF

Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:

(1)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compactT₂space.

(2)

For every family(Ai)_i∈Iof sets there exists a family(Ti)_i∈Isuch that for everyi∈I(Ai,Ti)is a compact, scattered, T₂space.

(3)

For every set X, every compactR₁topology (itsT₀-reflection isT₂) on X can be enlarged to a compactT₂topology.

We show:

(a)

(1) implies every infinite set can be split into two infinite sets.

(b)

(2) iff AC.

(c)

(3) and “there exists a free ultrafilter” iff AC.

We also show that if the topology of certain compact T₁ spaces can be enlarged to a compact T₂ topology then (1) holds true. But in general, compact T₁ topologies do not extend to compact T₂ ones.     

Linear maps preserving the minimum and surjectivity moduli of Hilbert space operators

Let B(H) be the algebra of all bounded linear operators on a complex infinite-dimensional Hilbert space H. For every T∈B(H), let m(T) and q(T) denote the minimum modulus and surjectivity modulus of T respectively. Let ?:B(H)→B(H) be a surjective linear map. In this paper, we prove that the following assertions are equivalent:

(i)

m(T)=m(?(T)) for all T∈B(H),

(ii)

q(T)=q(?(T)) for all T∈B(H),

(iii)

there exist two unitary operators U,V∈B(H) such that ?(T)=UTV for all T∈B(H).

This generalizes the result of Mbekhta [7, Theorem 3.1] to the non-unital case.     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

•

T contains all weakly Lindelöf Banach spaces;

•

l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Singular continuous Floquet operator for periodic quantum systems

Consider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: e−iH₀Te−iκT|?〉〈?| where T is the period, κ the coupling constant, and H₀ is a pure point self-adjoint operator, bounded from below. Under some hypotheses on the vector ?, cyclic w.r.t. H₀ we prove the following:

•

If the gaps between the eigenvalues (λn) are such that λn+1−λn?Cn−γ for some γ∈]0,1[ and C>0, then the Floquet operator of the perturbed system is purely singular continuous T-a.e.

•

If H₀ is the Hamiltonian of the one-dimensional rotator on L²(R/T₀Z) and the ratio is irrational, then the Floquet operator is purely singular continuous as soon as κT≠0(2π).

We also establish an integral formula for the family .     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

A note on unbounded strongly measure zero subgroups of the Baer-Specker group

We show that it is consistent with ZFC that there exist:

(1)

An unbounded (with respect to _?∗) and strongly measure zero subgroup of ZN, but without the Menger property.

(2)

An unbounded (with respect to _?∗) and strongly measure zero subgroup of ZN with the Menger property which does not have the Rothberger property.

This answers the last two problems which remained from a classification project of Machura and Tsaban.     

Sequential properties of function spaces with the compact-open topology

The main results of the paper are:

(1)

If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S₂, and hence does not have the property AP;

(2)

For any zero-dimensional Polish X, the space Ck(X,2) is sequential if and only if X is either locally compact or the derived set X^′ is compact; and

(3)

All spaces of the form Ck(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.

     

Properties of the Holmes space

The following properties of the Holmes space H are established:

(i)

H has the Metric Approximation Property (MAP).

(ii)

The w^∗-closure of the set of extreme points of the unit ball BH^∗ of the dual space H^∗ is the whole ball BH^∗.

A family of compact subsets X⊂U of the Urysohn space is described such that the Lipschitz-free space F(X) has a finite-dimensional decomposition and is not complemented in H.     

On the minimal members of convex expectations

In this paper, we show that for a convex expectation E[⋅] defined on L¹(Ω,F,P), the following statements are equivalent:

(i)

E is a minimal member of the set of all convex expectations defined on L¹(Ω,F,P);

(ii)

E is linear;

(iii)

two-dimensional Jensen inequality for E holds.

In addition, we prove a sandwich theorem for convex expectation and concave expectation.     

Inversion arrangements and Bruhat intervals

Let W be a finite Coxeter group. For a given w∈W, the following assertion may or may not be satisfied:

(?)

The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w.

We present a type independent combinatorial criterion which characterises the elements w∈W that satisfy (?). A couple of immediate consequences are derived:

(1)

The criterion only involves the order ideal of w as an abstract poset. In this sense, (?) is a poset-theoretic property.

(2)

For W of type A, another characterisation of (?), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjöstrand. We obtain a short and simple proof of that result.

(3)

If W is a Weyl group and the Schubert variety indexed by w∈W is rationally smooth, then w satisfies (?).

     

Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

Colouring graphs with bounded generalized colouring number

Given a graph G and a positive integer p, χ_p(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χ_p(G) in terms of the k-colouring number of G for k=2^p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χ_k+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:

(a)

For every positive integer p, χ_p(G) is bounded by a constant for all G∈K.

(b)

For every positive integer k, is bounded by a constant for all G∈K.

It was proved by Nešet?il and Ossona de Mendez that (a) is equivalent to the following:

(c)

For every positive integer q, ∇_q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K.

Therefore (b) and (c) are also equivalent. We shall give a direct proof of this equivalence, by introducing ∇_q−(1/2)(G) and by showing that there is a function F_k such that . This gives an alternate proof of the equivalence of (a) and (c).     

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u_t=Δu+u^p+u^q(x^*,t) in B(R) where p,q>0,B(R)={x∈R^N:|x|<R} and x^*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.

(i)

If or p=q>2, then single point blow-up occurs whenever solutions blow up.

(ii)

If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.

(iii)

If p?1<q, total blow-up occurs whenever solutions blow up.

(iv)

If max(p,q)?1, every solution exists globally in time.

     

Subnormal Toeplitz operators and the kernels of their self-commutators

In this note we give a connection between subnormal Toeplitz operators and the kernels of their self-commutators. This is closely related to P.R. Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? Our main theorem is as follows: If φ∈L^∞ is such that φ and are of bounded type (that is, they are quotients of two analytic functions on the open unit disk) and if the kernel of the self-commutator of Tφ is invariant for Tφ then Tφ is either normal or analytic.     

Locally compact abelian groups admitting non-trivial quasi-convex null sequences

In this paper, we show that, for every locally compact abelian group G, the following statements are equivalent:

(i)

G contains no sequence such that {0}∪{±x_n∣n∈N} is infinite and quasi-convex in G, and x_n?0;

(ii)

one of the subgroups {g∈G∣2g=0} or {g∈G∣3g=0} is open in G;

(iii)

G contains an open compact subgroup of the form or for some cardinal κ.

     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.