Some results on ∗-differential identities in prime rings

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

WEYL TYPE THEOREMS FOR ALGEBRIACALLY CLASS $p$-$wA(s,t)$ OPERATORS

In this paper, we study Weyl type theorems for $f(T)$, where $T$ is algebraically class $p$-$wA(s, t)$ operator with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$ and $f$ is an analytic function defined on an open neighborhood of the spectrum of $T$. Also we show that if $A , B^{*} \in B(\mathcal{H}) $ are class $p$-$wA(s, t)$ operators with $0 < p \leq 1$ and $0 < s, t, s + t \leq 1$,then generalized Weyl's theorem , a-Weyl's theorem, property $(w)$, property $(gw)$ and generalized a-Weyl's theorem holds for $f(d_{AB})$ for every $f \in H(\sigma(d_{AB})$, where $ d_{AB}$ denote the generalized derivation $\delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\delta_{AB}(X)=AX-XB$ or the elementary operator $\Delta_{AB}:B(\mathcal{H})\rightarrow B(\mathcal{H})$ defined by $\Delta_{AB}(X)=AXB-X$.

Generalized derivations, quasiderivations and centroids of Bihom-Lie triple system

In this paper, we give some properties of the generalized derivation algebra [Formula: see text] of a Bihom-Lie triple system [Formula: see text]. In particular, we prove that [Formula: see text], the sum of the quasiderivation algebra and the quasicentroid. We also prove that [Formula: see text] can be embedded as derivations in a larger Bihom-Lie triple system.

Some Identities in Rings and Near-Rings with Derivations

In the present paper we investigate commutativity in prime rings and 3-prime near-rings admitting a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings and 3-prime near-rings have been generalized.

Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

A classification of generalized derivations in rings with involution

Let R be a ring. An additive mapping F : R ? R is called a generalized derivation if there exists a derivation d of R such that F(xy) = F(x)y + xd(y) for all x,y ? R. The main purpose of this paper is to characterize some specific classes of generalized derivations of rings. Precisely, we describe the structure of generalized derivations of noncommutative prime rings with involution that belong to a particular class of generalized derivations. Consequently, some recent results in this line of investigation have been extended. Moreover, some suitable examples showing that the assumed hypotheses are crucial, are also given.

Higher-order commutators with power central values on rings and algebras involving generalized derivations

Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

Generalized derivation, SVEP, finite ascent, range closure

Let X be an infinite complex Banach space and consider two bounded linear operators A,B ? L(X). Let LA ? L(L(X)) and RB ? L(L(X)) be the left and the right multiplication operators, respectively. The generalized derivation ?A,B ? L(L(X)) is defined by ?A,B(X) = (LA-RB)(X) = AX-XB. In this paper we give some sufficient conditions for ?A,B to satisfy SVEP, and we prove that ?A,B-?I has finite ascent for all complex ?, for general choices of the operators A and B, without using the range kernel orthogonality. This information is applied to prove some necessary and sufficient conditions for the range of ?A,B-?I to be closed. In [18, Propostion 2.9] Duggal et al. proved that, if asc(?A,B-?)? 1, for all complex ?, and if either (i) A* and B have SVEP or (ii)?* A,B has SVEP, then ?A,B-? has closed range for all complex ? if and only if A and B are algebraic operators, we prove using the spectral theory that, if asc(?A,B-?) ? 1, for all complex ?, then ?A,B-? has closed range, for all complex ? if and only if A and B are algebraic operators, without the additional conditions (i) or (ii).

On continuous linear generalized derivation in rings and Banach algebras

In this paper, we investigate the commutativity of a prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivation [Formula: see text] associated with continuous linear derivation [Formula: see text] such that either [Formula: see text] or [Formula: see text] for intergers [Formula: see text] and [Formula: see text] and sufficiently many [Formula: see text]. Further, similar results are also obtained for unital prime Banach algebra [Formula: see text] which admits a nonzero continuous linear generalized derivations [Formula: see text] satisfying either [Formula: see text] or [Formula: see text] for an integer [Formula: see text] and sufficiently many [Formula: see text].

ON mTH-COMMUTATORS AND ANTI-COMMUTATORS INVOLVING GENERALIZED DERIVATIONS IN PRIME RINGS

In this manuscript, we study the $m$-th commutator and anti-commutator involving generalized derivations on some suitable subsets of rings. We attain the information about the structure of rings and the behaviour of generalized derivation in form of multiplication by some specific element of Utumi quotient ring which satisfies certain differential identities.