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Product of operators and numerical range preserving maps

Chi-Kwong Li, Nung-Sing Sze (2006)

Studia Mathematica

Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of A ₁ , . . . , A k ∈ V by A ₁ * ⋯ * A k = A i ₁ ⋯ A i ₘ . This includes the usual product A ₁ * ⋯ * A k = A ₁ ⋯ A k and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying μ m = 1 such that ϕ: V → V has the form A...

Products of Toeplitz operators and Hankel operators

Yufeng Lu, Linghui Kong (2014)

Studia Mathematica

We first determine when the sum of products of Hankel and Toeplitz operators is equal to zero; then we characterize when the product of a Toeplitz operator and a Hankel operator is a compact perturbation of a Hankel operator or a Toeplitz operator and when it is a finite rank perturbation of a Toeplitz operator.

Produits finis de commutateurs dans les C * -algèbres

Pierre de La Harpe, Georges Skandalis (1984)

Annales de l'institut Fourier

Soient A une C * -algèbre approximativement finie simple avec unité, G L 1 ( A ) le groupe des inversibles et U 1 ( A ) le groupe des unitaires de A . Nous avons défini dans un précédent travail un homomorphisme Δ T , appelé déterminant universel de A , de G L 1 ( A ) sur un groupe abélien associé à A . Nous montrons ici que, pour qu’un élément x dans G L 1 ( A ) ou dans U 1 ( A ) soit produit d’un nombre fini de commutateurs, il (faut et il) suffit que x ∈ Ker ( Δ T ) . Ceci permet en particulier d’identifier le noyau de la projection canonique K 1 ( A ) → K 1 top ( A ) . On établit aussi...

Projections onto the spaces of Toeplitz operators

Marek Ptak (2005)

Annales Polonici Mathematici

Projections onto the spaces of all Toeplitz operators on the N-torus and the unit sphere are constructed. The constructions are also extended to generalized Toeplitz operators and applied to show hyperreflexivity results.

Properties of derivations on some convolution algebras

Thomas Pedersen (2014)

Open Mathematics

For all convolution algebras L 1[0, 1); L loc1 and A(ω) = ∩n L 1(ωn), the derivations are of the form D μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D μ to a natural dual space is weak-star continuous.

Properties of two variables Toeplitz type operators

Elżbieta Król-Klimkowska, Marek Ptak (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

The investigation of properties of generalized Toeplitz operators with respect to the pairs of doubly commuting contractions (the abstract analogue of classical two variable Toeplitz operators) is proceeded. We especially concentrate on the condition of existence such a non-zero operator. There are also presented conditions of analyticity of such an operator.

Pseudo shift operators with large images

M. C. Calderón-Moreno (2002)

Colloquium Mathematicae

We give suitable conditions for the existence of many holomorphic functions f on a disc such that the image of any nonempty open subset under the action of pseudo shift operators on f is arbitrarily large. This generalizes an earlier result about images of derivatives and completes another one on infinite order differential operators.

Pseudocomplémentation dans les espaces de Banach

Patric Rauch (1991)

Studia Mathematica

This paper introduces the following definition: a closed subspace Z of a Banach space E is pseudocomplemented in E if for every linear continuous operator u from Z to Z there is a linear continuous extension ū of u from E to E. For instance, every subspace complemented in E is pseudocomplemented in E. First, the pseudocomplemented hilbertian subspaces of L ¹ are characterized and, in L p with p in [1, + ∞[, classes of closed subspaces in which the notions of complementation and pseudocomplementation...

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