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A class of weighted convolution Fréchet algebras

Thomas Vils Pedersen (2010)

Banach Center Publications

For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ω m ( t ) / ω ( t ) as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...

A Kleinecke-Shirokov type condition with Jordan automorphisms

Matej Brešar, Ajda Fošner, Maja Fošner (2001)

Studia Mathematica

Let φ be a Jordan automorphism of an algebra . The situation when an element a ∈ satisfies 1 / 2 ( φ ( a ) + φ - 1 ( a ) ) = a is considered. The result which we obtain implies the Kleinecke-Shirokov theorem and Jacobson’s lemma.

A new characterization of Anderson’s inequality in C 1 -classes

S. Mecheri (2007)

Czechoslovak Mathematical Journal

Let be a separable infinite dimensional complex Hilbert space, and let ( ) denote the algebra of all bounded linear operators on into itself. Let A = ( A 1 , A 2 , , A n ) , B = ( B 1 , B 2 , , B n ) be n -tuples of operators in ( ) ; we define the elementary operators Δ A , B ( ) ( ) by Δ A , B ( X ) = i = 1 n A i X B i - X . In this paper, we characterize the class of pairs of operators A , B ( ) satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators A , B ( ) such that i = 1 n B i T A i = T implies i = 1 n A i * T B i * = T for all T 𝒞 1 ( ) (trace class operators). The main result is the equivalence between this property and the fact that...

A note on compact semiderivations

Matej Brešar, Yuri Turovskii (2005)

Banach Center Publications

Let 𝓐 be a Banach algebra without nonzero finite dimensional ideals. Then every compact semiderivation on 𝓐 is a quasinilpotent operator mapping 𝓐 into its radical.

A note on the commutator of two operators on a locally convex space

Edvard Kramar (2016)

Commentationes Mathematicae Universitatis Carolinae

Denote by C the commutator A B - B A of two bounded operators A and B acting on a locally convex topological vector space. If A C - C A = 0 , we show that C is a quasinilpotent operator and we prove that if A C - C A is a compact operator, then C is a Riesz operator.

A note on the range of generalized derivation.

Mohamed Amouch (2006)

Extracta Mathematicae

Let L(H) denote the algebra of bounded linear operators on a complex separable and infinite dimensional Hilbert space H. For A, B ∈ L(H), the generalized derivation δA,B associated with (A, B), is defined by δA,B(X) = AX - XB for X ∈ L(H). In this note we give some sufficient conditions for A and B under which the intersection between the closure of the range of δA,B respect to the given topology and the kernel of δA*,B* vanishes.

A remark on the range of elementary operators

Said Bouali, Youssef Bouhafsi (2010)

Czechoslovak Mathematical Journal

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 ) = A X A - X . In this paper we study the class of operators A L ( H ) which have the following property: A T A = T implies A T * 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...

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