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For an element a in a unital -algebra , let V(a) and v(a) denote the numerical range and the numerical radius of a, respectively. First, we show that a is a unitary central element in if and only if for all normal elements . We then use this result to show that if a and b are two elements in with , then for all normal elements if and only if ba is a unitary central element in and for some positive scalar . If, however, the numerical radius is replaced by the numerical range, we show that such a condition is redundant and prove that for all normal elements if and only if a is invertible such that is a central element in and . Furthermore, we show that for all invertible elements if and only if |a| is in the centre of and is a unitary element. [ABSTRACT FROM AUTHOR]

Letv(a) be the numerical radius of an elementain a unital-algebra. In this paper, we show thatfor allif and only if. We also show thatfor allif and only ifais a unitary element in the centreof. Furthermore, for two fixed elementsaandbin, we investigate whenfor all, and derive a number of consequences. [ABSTRACT FROM PUBLISHER]

Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$(X) (resp. ${\mathcal B}$(Y)) be the algebra of all bounded linear operators on X (resp. on Y). For an operator T ∈ ${\mathcal B}$(X) and a vector x ∈ X, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0 ∈X and y0 ∈ Y, we show that a map ϕ from ${\mathcal B}$(X) onto ${\mathcal B}$(Y) satisfies $ \sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)), $ if and only if there exists a bijective bounded linear mapping A from X into Y such that Ax0 = y0 and either ϕ(T) = ATA−1 or ϕ(T) = -ATA−1 for all T ∈ ${\mathcal B}$(X). [ABSTRACT FROM AUTHOR]

Letdenote the algebra of all bounded linear operators on a Banach space. Letandbe infinite-dimensional complex Banach spaces, and fix two nonzero vectorsand. Then we show that a surjective mapsatisfiesif and only if there exists a bijective mappingsuch thatandwhereis a third root of unity. [ABSTRACT FROM AUTHOR]