On a convergence principle in the Hilbert space

On the moment convergence rates of LIL in Hilbert space

Let {X, Xn; n ≥ 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance Σ and set Sn = ∑n k=1 Xk , n ≥ 1. Let σ 2 i , i ≥ 1 be the eigenvalues of Σ arranged in non-increasing order and take into account the multiplicities. Let l be the dimension of the corresponding eigenspace of largest eigenvalue σ 2 = σ 2 1 . Let lo...

A Note on Quadratic Maps for Hilbert Space Operators

In this paper, we introduce the notion of sesquilinear map on Β(H) . Based on this notion, we define the quadratic map, which is the generalization of positive linear map. With the help of this concept, we prove several well-known equality and inequality...  

On iterative fixed point convergence in uniformly convex Banach space and Hilbert space

Some fixed point convergence properties are proved for compact and demicompact maps acting over closed, bounded and convex subsets of a real Hilbert space. We also show that for a generalized nonexpansive mapping in a uniformly convex Banach space the Ishikawa iterates converge to a fixed point. Finally, a convergence type result is established for multivalued contractive mappings acting on clo...

Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space

Let C be a nonempty closed convex subset of a Hilbert spaceH, T a self-mapping of C. Recall that T is said to be nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ C. Construction of fixed points of nonexpansive mappings via Mann’s iteration 1 has extensively been investigated in literature see, e.g., 2–5 and reference therein . But the convergence about Mann’s iteration and Ishikawa’s iterati...

On the Rate of Convergence of the Conjugate Gradient Method for Linear Operators in Hilbert Space