Commutants of nest algebras modulo the compact operators

Any operatorx which commutes modulo the compact operators with a nest algebra is of the form λI+C, where λ is a scalar andC is a compact operator. Any derivation from a nest algebra on a Hilbert spaceH into the compact operators onH is implemented by a compact operator. Any derivation on a quasitriangular operator algebra is inner.     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Generalized Essential Norm of Weighted Composition Operators on some Uniform Algebras of Analytic Functions

We compute the essential norm of a composition operator relatively to the class of Dunford-Pettis operators or weakly compact operators, on some uniform algebras of analytic functions. Even in the context of H^∞ (resp. the disk algebra), this is new, as well for the polydisk algebras and the polyball algebras. This is a consequence of a general study of weighted composition operators.      

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Deddens Algebras and Shift

We consider Deddens algebras associated to operators of the form S−λI, where S is the unilateral shift and λ is a complex number. We show that such an algebra properly contains the commutant of S and that it is always weakly dense in L(H){{\mathcal L}({\mathcal H})}. Yet, it contains no rank one operators, unless λ = 0, in which case it equals L(H){{\mathcal L}({\mathcal H})}.     

Grüss-type and Ostrowski-type inequalities in approximation theory

We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér operator.     

Lie triple derivations on nest algebras

Let δ be a Lie triple derivation from a nest algebra ?? into an ??‐bimodule ??. We show that if ?? is a weak* closed operator algebra containing ?? then there are an element S ∈ ?? and a linear functional f on ?? such that δ (A) = SA – AS + f (A)I for all A ∈ ??, and if ?? is the ideal of all compact operators then there is a compact operator K such that δ (A) = KA – AK for all A ∈ ??. As applications, Lie derivations and Jordan derivations on nest algebras are characterized. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.     

Topological Structure of the Set of Weighted Composition Operators on Weighted Bergman Spaces of Infinite Order

We consider the topological space of all weighted composition operators on weighted Bergman spaces of infinite order endowed with the operator norm. We show that the set of compact weighted composition operators is path connected. Furthermore, we find conditions to ensure that two weighted composition operators are in the same path connected component if the difference of them is compact. Moreover, we compare the topologies induced by L(H^∞) and L(H^∞_v) on the space of bounded composition operators and give a sufficient condition for a composition operator to be isolated.     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.     

Weyl’s theorem and thin spectra

Using the notion of thin sets we prove a theorem of Weyl type for the Wolf essential spectrum ofT ∈β (H). *Further we show that Weyl’s theorem holds for a restriction convexoid operator and consequently modify some results of Berberian. Finally we show that Weyl’s theorem holds for a paranormal operator and that a polynomially compact paranormal operator is a compact perturbation of a diagnoal normal operator. A structure theorem for polynomially compact paranormal operators is also given.     

Composition operators within singly generated composition <Emphasis Type="Italic">C</Emphasis>*-algebras

Let ϕ be a linear-fractional self-map of the open unit disk D, not an automorphism, such that ϕ(ζ) = η for two distinct points ζ,η in the unit circle ∂D. We consider the problem of determining which composition operators, acting on the Hardy space H ², lie in C*(C _ϕ ,K), the unital C*-algebra generated by the composition operator C _ϕ and the ideal K of compact operators. This necessitates a companion study of the unital C*-algebra generated by the composition operators induced by all parabolic non-automorphisms with common fixed point on the unit circle.     

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW ^*-algebras of finite type; i.e., with minor restrictions, compact operators onH* _A can be diagonalized overA. We show that ifB is a weakly denseC ^*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromH _B toH* _A ⊃H _B of a compact operator can be diagonalized so that the diagonal elements belong to the originalC ^*-algebraB. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 865–870, December, 1997. Translated by O. V. Sipacheva     

Extension of derivations

Any derivation of a properly infinite von Neumann algebra on a Hilbert space into the algebra of bounded operators on this space is implemented by a bounded operator.     

Maps preserving numerical radius or cross norms of products of self-adjoint operators

Let H be a complex Hilbert space with dimH ≥3, Bs（H） the （real） Jordan algebra of all self-adjoint operators on H. Every surjective map Ф ： Bs（H）→13s（H） preserving numerical radius of operator products （respectively, Jordan triple products） is characterized. A characterization of surjective maps on Bs （H） preserving a cross operator norm of operator products （resp. Jordan triple products of operators） is also given.     

A remark on the range of elementary operators

Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L(H), we define the elementary operator Δ _A : L(H) → L(H) by Δ _A (X) = AXA − X. In this paper we study the class of operators A ∈ L(H) which have the following property: ATA = T implies AT*A = T* for all trace class operators T ∈ C ₁(H). Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ _A is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators.     

Density of algebras generated by Toeplitz operators on Bergman spaces

In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, theC ^*-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spacesA ²(Ω) for a wide class of plane domains Ω⊂C, and in Fock spacesA ²(C ^N),N≧1.