Additive Jordan isomorphisms of nest algebras on normed spaces

Let X be a real or complex Banach space. Let and be two nest algebras on X. Suppose that φ is an additive bijective mapping from onto such that φ(A²)=φ(A)² for every . Then φ is either a ring isomorphism or a ring anti-isomorphism. Moreover, if X is a real space or an infinite dimensional complex space, then there exists a continuous (conjugate) linear bijective mapping T such that either φ(A)=TAT⁻¹ for every or φ(A)=TA∗T−1 for every .     

Hyponormal operators with rank-two self-commutators

In this paper it is shown that if T∈L(H) satisfies

(i)

T is a pure hyponormal operator;

(ii)

[T^∗,T] is of rank two; and

(iii)

ker[T^∗,T] is invariant for T,

then T is either a subnormal operator or the Putinar's matricial model of rank two. More precisely, if T_|ker[T^∗,T] has a rank-one self-commutator then T is subnormal and if instead T_|ker[T^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.     

Toeplitz operators with unbounded symbols of several complex variables

In this note we construct a function φ in L²(Bn,dA) which is unbounded on any neighborhood of each boundary point of Bn such that Tφ is a trace class operator on Bergman space for several complex variables. In addition, we also discuss the compactness of Toeplitz operators with L¹ symbols.     

Mass transport generated by a flow of Gauss maps

Let A⊂Rd, d?2, be a compact convex set and let be a probability measure on A equivalent to the restriction of Lebesgue measure. Let be a probability measure on equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν=μ○T⁻¹ and T=φ⋅n, where is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth φ the level sets of φ are governed by the Gauss curvature flow , where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface.     

Commuting dual Toeplitz operators with pluriharmonic symbols

In this paper we completely characterize commuting dual Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball. We show that for φ and ψ pluriharmonic, SφSψ=Sψφ on only in the trivial case. Here the trivial case is φ or holomorphic.     

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.     

Remarks on common hypercyclic vectors

We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ⊆R₊×C which is not of zero three-dimensional Lebesgue measure, the family has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if and φ∈H^∞(D) is non-constant, then the family has a common hypercyclic vector, where Mφ:H²(D)→H²(D), Mφf=φf, and , providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family has a common hypercyclic vector, where Tbf(z)=f(z−b) acts on the Fréchet space H(C) of entire functions on one complex variable.     

Trace-preserving homomorphisms of semigroups

We consider those homomorphisms φ of semigroups of trace-class operators on a Hilbert space that preserve trace. If φ is a spatially induced isomorphism on a semigroup , that is φ(S)T=TS for an invertible operator T and for all S in , then φ clearly has this property. More generally, if T in the relation above is a densely defined, closed, injective operator with dense image, φ still preserves trace. We prove the converse of this statement under certain conditions. Using these results we prove simultaneous similarity theorems for semigroups of operators (on finite or infinite-dimensional spaces) whose members are individually similar to unitary or J-unitary operators.     

Rigidity in one-dimensional tiling spaces

Suppose Tφ and Tθ are tiling spaces arising from primitive nonperiodic substitutions φ and θ. Suppose Fφ and Fθ denote the corresponding inflation and substitution maps on the respective tiling spaces. We prove that Tφ and Tθ are homeomorphic if and only if there exist positive integers m and n such that and are topologically conjugate.     

Compact operators and Toeplitz algebras on multiply-connected domains

If Ω is a smoothly bounded multiply-connected domain in the complex plane and S belongs to the Toeplitz algebra τ of the Bergman space of Ω, we show that S is compact if and only if its Berezin transform vanishes at the boundary of Ω. We also show that every element S in T, the C^?-subalgebra of τ generated by Toeplitz operators with symbols in H^∞(Ω), has a canonical decomposition for some R in the commutator ideal CT; and S is in CT iff the Berezin transform vanishes identically on the set M₁ of trivial Gleason parts.     

Sub-Bergman Hilbert spaces in the unit disk, II

For a finite Blaschke product B let T_B denote the analytic multiplication operator (also called a Toeplitz operator) on the Bergman space of the unit disk. We show that the defect operators and both map the Bergman space to the Hardy space and the Hardy space to the Dirichlet space.     

Conjugate operators that preserve the nonconvergence of bounded martingales

We show that the conjugate T^∗ of an operator , with X and Y Banach spaces, satisfies the following dichotomy: either T^∗ preserves the nonconvergence of bounded martingales in Y^∗, or there exists a compact operator such that the kernel N(T^∗+K^∗) fails the Radon-Nikodým property.     

Fredholmness of Toeplitz operators and corona problems

A meromorphic analogue to the corona problem is formulated and studied and its solutions are characterized as being left-invertible in a space of meromorphic functions. The Fredholmness of Toeplitz operators with symbol G∈(L_∞(R))^2×2 is shown to be equivalent to that of a Toeplitz operator with scalar symbol , provided that the Riemann-Hilbert problem admits a solution such that the meromorphic corona problems with data are solvable. The Fredholm properties are characterized in terms of and the corresponding meromorphic left-inverses. Partial index estimates for the symbols and Fredholmness criteria are established for several classes of Toeplitz operators.     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Composition operators from the Hardy space to the nth weighted-type space on the unit disk and the half-plane

The boundedness of the composition operator C_φf(z)=f(φ(z)) from the Hardy space , where X is the upper half-plane or the unit disk D={z∈C:|z|<1} in the complex plane C, to the nth weighted-type space, where φ is an analytic self-map of X, is characterized.     

Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras

A nonlinear map φ between operator algebras is said to be a numerical radius isometry if w(φ(T−S))=w(T−S) for all T, S in its domain algebra, where w(T) stands for the numerical radius of T. Let and be two atomic nests on complex Hilbert spaces H and K, respectively. Denote the nest algebra associated with and the diagonal algebra. We give a thorough classification of weakly continuous numerical radius isometries from onto and a thorough classification of numerical radius isometries from onto .