In this paper two-dimensional Vilenkin-like systems will be investigated. We prove the Sunouchi operator is bounded from H to L for (2/3 < q ≤ 1). As a consequence, we prove the Sunouchi operator is L bounded for 1 < s < ∞ and of weak type (H, L).

In this paper we construct a special sort of dilation for an arbitrary polynomially bounded operator. This enables us to show that the problem whether every polynomially bounded operator is similar to a contraction can be reduced to a subclass of it.

For an operator bimodule X over von Neumann algebras A ⊆ B(H) and B ⊆ B(K), the space of all completely bounded A, B-bimodule maps from X into B(K,H), is the bimodule dual of X. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To X a normal operator bimodule Xn is associated so that completely bounded A, B-bimodule maps from...

In this paper, we explore what happens when the same techniques are applied to the problem of estimating eigenvalues of the adjacency operator on finite graphs of bounded degree. In Theorem 7, we show how eigenvalues of the adjacency operator on a finite graph Γ may be bounded in terms of the biggest eigenvalues of the adjacency operator on “geodesic balls” in Γ. We find explicit bounds for the...

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...

We introduce the notion of Stieltjes integral with respect to the spectral measure corresponding to a normal operator. Sufficient conditions for the existence of this integral are given, and estimates for its norm are established. The results are applied to operator Sylvester and Riccati equations. Assuming that the spectrum of a closed densely defined operator A does not have common points wit...

We introduce the concept of Stieltjes integral of an operator-valued function with respect to the spectral measure associated with a normal operator. We give sufficient conditions for the existence of this integral and find bounds on its norm. The results obtained are applied to the Sylvester and Riccati operator equations. Assuming that the entry C is a normal operator, the spectrum of the ent...

We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternio...

The terminology and notation used here are introduced in the following articles: [18], [8], [20], [5], [7], [6], [3], [1], [17], [13], [19], [14], [2], [4], [15], [10], [11], [9], and [12]. One can prove the following propositions: (1) Let X, Y , Z be complex linear spaces, f be a linear operator from X into Y , and g be a linear operator from Y into Z. Then g · f is a linear operator from X in...

in this article we study two different generalizations of von neumann regularity, namely strong topological regularity and weak regularity, in the banach algebra context. we show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. then we consider strong topological regularity of certain concrete algebras. moreover we obtain ...