In this note we describe the dual and the completion of the space of finite linear combinations of (p,∞)-atoms, 0 < p ≤ 1. As an application, we show an extension result for operators uniformly bounded on (p,∞)-atoms, 0 < p < 1, whose analogue for p = 1 is known to be false. Let 0 < p < 1 and let T be a linear operator defined on the space of finite linear combinations of (p,∞)-atoms, 0 < p < 1...

in this paper we proved that every g-riesz basis for hilbert space $h$ with respect to $k$ by adding a condition is a riesz basis for hilbert $b(k)$-module $b(h,k)$. this is an extension of [a. askarizadeh,m. a. dehghan, {em g-frames as special frames}, turk. j. math., 35, (2011) 1-11]. also, we derived similar results for g-orthonormal and orthogonal bases. some relationships between dual fram...

In this paper we introduce the concept of entropy operator for continuous systems of finite topological entropy. It is shown that it generates the Kolmogorov entropy as a special case. If $phi$ is invertible then the entropy operator is bounded with the topological entropy of $phi$ as its norm.

If m is a positive integer, let ǫk = e 2kπi/m for k = 1, . . . ,m. Then ǫk = 1 (k = 1, . . . ,m), ǫk 6= 1 (k = 1, . . . ,m− 1) and ǫm = 1 . If a ∈ A is invertible, define the bounded linear operator Ta : A → A by Tax = a xa (x ∈ A) . Proposition 1. Suppose that a ∈ A is invertible, m is a positive integer and that σ(a) is irrotational (mod 2π/m). Let the bounded linear operator S : A → A be giv...

By B(X), we also denote the set of all bounded linear operators on X into itself. If X is any Banach space and T ∈ B(X), then the adjoint T∗ of T is a bounded linear operator on the dual X∗ of X defined by (T∗ f )(x)= f (Tx) for all f ∈ X∗ and x ∈ X . LetX = {θ} be a nontrivial complex normed space and T : (T)→ X a linear operator defined on a subspace (T)⊆ X . We do not assume thatD(T) is dens...

Let 0 < p ≤ q ≤ +∞. Let T be a bounded sublinear operator from a Banach space X into an Lp(Ω,μ) and let ∇T be the set of all linear operators ≤ T . In the present paper, we will show the following. Let C be a positive constant. For all u in ∇T , Cpq(u) ≤ C (i.e., u admits a factorization of the form X ũ → Lq(Ω,μ) Mgu → Lp(Ω,μ), where ũ is a bounded linear operator with ‖ũ‖ ≤ C , Mgu is the bo...

in this paper, a complete description concerning linear operators of banach spaces with range in lipschitz algebras $lip_al(x)$ is provided. necessary and sufficient conditions are established to ensure boundedness and (weak) compactness of these operators. finally, a lower bound for the essential norm of such operators is obtained.