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 ${\omega }_m\left(t\right)/\omega ₙ\left(t\right)\to \infty$ 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...

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

Let $\mathscr{H}$ be a separable infinite dimensional complex Hilbert space, and let $ℒ\left(\mathscr{H}\right)$ denote the algebra of all bounded linear operators on $\mathscr{H}$ into itself. Let $A=(A_1,A_2,\cdots ,A_n)$, $B=(B_1,B_2,\cdots ,B_n)$ be $n$-tuples of operators in $ℒ\left(\mathscr{H}\right)$; we define the elementary operators ${\Delta }_{A,B}ℒ\left(\mathscr{H}\right)\mapsto ℒ\left(\mathscr{H}\right)$ by ${\Delta }_{A,B}\left(X\right)={\sum }_{i=1}^n{A}_{i}X{B}_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in ℒ\left(\mathscr{H}\right)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in ℒ\left(\mathscr{H}\right)$ such that ${\sum }_{i=1}^n{B}_{i}T{A}_i=T$ implies ${\sum }_{i=1}^n{A}_i^{*}T{B}_i^{*}=T$ for all $T\in {𝒞}_1\left(\mathscr{H}\right)$ (trace class operators). The main result is the equivalence between this property and the fact that...

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.

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

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.

Let $L\left(H\right)$ denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space $H$ into itself. Given $A\in L\left(H\right)$, we define the elementary operator ${\Delta }_A:L\left(H\right)⟶L\left(H\right)$ by ${\Delta }_A\left(X\right)=AXA-X$. In this paper we study the class of operators $A\in L\left(H\right)$ which have the following property: $ATA=T$ implies $A{T}^{*}A=T^{*}$ for all trace class operators $T\in C_1\left(H\right)$. 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 ${\Delta }_A$ is closed under taking...