Universal elements for non-linear operators and their applications

We prove that under certain topological conditions on the set of universal elements of a continuous map T acting on a topological space X, that the direct sum T and M_g is universal, where M_g is multiplication by a generating element of a compact topological group. We use this result to characterize R_+-supercyclic operators and to show that whenever T is a supercyclic operator and z_1,...,z_n are pairwise different non-zero complex numbers, then the operator z_1T\oplus ... \oplus z_n T is cyclic. The latter answers affirmatively a question of Bayart and Matheron.

Structure of spectra of linear operators in Banach spaces

Descriptive characterizations of the point, the continuous, and the residual spectra of operators in Banach spaces are put forward. In particular, necessary and sufficient conditions for three disjoint subsets of the complex plane to be the point spectrum, the continuous spectrum, and the residual spectrum of a linear continuous operator in a separable Banach space are obtained.

Time operators for one parameter semigroups

We develop two simple approaches to the construction of time operators for semigroups of continuous linear operators in Hilbert spaces provided that the generators of these semigroups are normal operators. The first approach enables us to give explicit formulas (in the spectral representations) both for the time operators and for their eigenfunctions. The other approach provides no explicit formula. However, it enables us to find necessary and sufficient conditions for the existence of time operators for semigroups of continuous linear operators in separable Hilbert spaces with normal generators. Time superoperators corresponding to unitary groups are also discussed.

Hypercyclic tuples of operators on $C^n$ and $R^n$

A tuple $(T_1,\dots,T_n)$ of continuous linear operators on a topological vector space $X$ is called hypercyclic if there is $x\in X$ such that the the orbit of $x$ under the action of the semigroup generated by $T_1,\dots,T_n$ is dense in $X$. This concept was introduced by N.~Feldman, who have raised 7 questions on hypercyclic tuples. We answer those 4 of them, which can be dealt with on the level of operators on finite dimensional spaces. In
particular, we prove that the minimal cardinality of a hypercyclic tuple of operators on $\C^n$ (respectively, on $\R^n$) is $n+1$ (respectively, $\frac n2+\frac{5+(-1)^n}{4}$), that there are non-diagonalizable tuples of operators on $\R^2$ which possess an orbit being neither dense nor nowhere dense and construct a hypercyclic 6-tuple of operators on $\C^3$ such that every operator commuting with each member of the tuple is non-cyclic.

Hypercyclic operators on countably dimensional spaces

According to Grivaux, the group GL(X) of invertible linear operators on a separable infinite dimensional Banach space X acts transitively on the set s (X) of countable dense linearly independent subsets of X. As a consequence, each A? s (X) is an orbit of a hypercyclic operator on X. Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm. We show that for a separable infinite dimensional Fréchet space X, GL(X) acts transitively on s (X) if and only if X possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator.

Completely bounded disjointness preserving operators between Fourier algebras

In this paper, we characterize surjective completely bounded disjointness preserving linear operators on Fourier algebras of locally compact amenable groups. We show that such operators are given by weighted homomorphisms induced by piecewise affine proper maps. © 2011 Elsevier Inc.

On the weak amenability of β(X)

We investigate the weak amenability of the Banach algebra ß(X) of all bounded linear operators on a Banach space X. Sufficient conditions are given for weak amenability of this and other Banach operator algebras with bounded one-sided approximate identities.

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 ? ? R+ × C which is not of zero three-dimensional Lebesgue measure, the family {a T + b I : (a, b) ? ?} 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 D = {z ? C : | z | < 1} and f ? H8 (D) is non-constant, then the family {z Mf{star operator} : b- 1 < | z | < a- 1} has a common hypercyclic vector, where Mf : H2 (D) ? H2 (D), Mf f = f f, a = inf {| f (z) | : z ? D} and b = sup {| f (z) | : | z | ? D}, providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family {a Tb : a, b ? C {set minus} {0}} has a common hypercyclic vector, where Tb f (z) = f (z - b) acts on the Fréchet space H (C) of entire functions on one complex variable.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Entanglement between collective operators in a linear harmonic chain

We investigate entanglement between collective operators of two blocks of oscillators in an infinite linear harmonic chain. These operators are defined as averages over local operators (individual oscillators) in the blocks. On the one hand, this approach of "physical blocks" meets realistic experimental conditions, where measurement apparatuses do not interact with single oscillators but rather with a whole bunch of them, i.e., where in contrast to usually studied "mathematical blocks" not every possible measurement is allowed. On the other, this formalism naturally allows the generalization to blocks which may consist of several noncontiguous regions. We quantify entanglement between the collective operators by a measure based on the Peres-Horodecki criterion and show how it can be extracted and transferred to two qubits. Entanglement between two blocks is found even in the case where none of the oscillators from one block is entangled with an oscillator from the other, showing genuine bipartite entanglement between collective operators. Allowing the blocks to consist of a periodic sequence of subblocks, we verify that entanglement scales at most with the total boundary region. We also apply the approach of collective operators to scalar quantum field theory.