Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.
The Banach-Lie algebra L(A) of multiplication operators on the JB * -triple A is introduced and it is shown that the hermitian part L(A) h of L(A) is a unital GM-space the base of the dual cone in the dual GL-space (L(A) h ) * of which is affine isomorphic and weak * -homeomorphic to the state space of L(A). In the case in which A is a JBW * -triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space L(A) h satisfy 0 D(u, u) + D(v, v) id A , and that u is a pre-associate of v if and only if D(u, u) D(v, v).
We consider operator A on the reproducing Kernel Hilbert space H = H (Ω) over some set Ω with the reproducing kernel K λ (z) = K (z,λ ) and define Davis-Wielandt-Berezin radius η (A) by the formulawhere A is the Berezin symbol of A defined by A (λ )is the normalized reproducing kernel of H . We prove several inequalities for this new quantity η (A) involving known Dragomir inequalities. Some other Berezin number inequalities are also proved.Mathematics subject classification (2020): 47A12, 47A20.
The aim of this paper is to provide new upper bounds of ω(T), which denotes the numerical radius of a bounded operator T on a Hilbert space (H,⟨·,·⟩). We show the Aczél inequality in terms of the operator |T|. Next, we give certain inequalities about the A-numerical radius ωA(T) and the A-operator seminorm ∥T∥A of an operator T. We also present several results related to the A-numerical radius of 2×2 block matrices of semi-Hilbert space operators, by using symmetric 2×2 block matrices.
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