Let A and B be two Banach algebras and let M be a Banach B-bimodule. Suppose that σ : A → B is a linear mapping and d : A → M is a σ-derivation. We prove several results about automatic continuity of σderivations on Banach algebras. In addition, we define a notion for m-weakly continuous linear mapping and show that, under certain conditions, d and σ are m-weakly continuous. Moreover, we prove that if A is commutative and σ : A → A is a continuous homomorphism such that σ 2 = σ then σdσ(A) ⊆ σ(Q(A)) ⊆ rad(A).
Suppose S and T are adjointable linear operators between Hilbert C*-modules.
It is well known that an operator T? has closed range if and only if its
Moore-Penrose inverse Ty exists. In this paper, we show that (TS)? = S?T?,
where S and T have closed ranges and (ker(T))? = ran(S). Moreover, we
investigate some results related to the polar decomposition of T. We also
obtain the inverse of 1-T?T + T, when T is a self-adjoint operator.
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