Kichi-Suke Saito

Coordinates for triangular operator algebras

On demontre un theoreme de structure pour des algebres sous diagonales contenant une sous algebre de Cartan. On etudie les isomorphismes pour ces algebres

Uniform convexity of ψ-direct sums of Banach spaces

Abstract Let X and Y be Banach spaces and ψ a continuous convex function on the unit interval [0,1] satisfying certain conditions. Let X ⊕ ψ Y be the direct sum of X and Y equipped with the associated norm with ψ . We show that X ⊕ ψ Y is uniformly convex if and only if X , Y are uniformly convex and ψ is strictly convex. As a corollary we obtain that the l p , q -direct sum X⊕ p,q Y, 1⩽q⩽p⩽∞ (not p = q =1 nor ∞), is uniformly convex if and only if X , Y are, where l p , q is the Lorentz sequence space. These results extend the well-known fact for the l p -sum X⊕ p Y, 1 . Some other examples are also presented.

On Ψ direct sums of Banach spaces and convexity

Let X1; X2;:::;X N be Banach spaces and a continuous convex function with some appropriate conditions on a certain convex set in N 1 . Let.X1 X2 X N/ be the direct sum of X1; X2;:::;X N equipped with the norm associated with . We characterize the strict, uniform, and locally uniform convexity of .X1 X2 X N/ by means of the convex function . As an application these convexities are characterized for the ‘ p;q-sum .X1 X2

On Sharp Triangle Inequalities in Banach Spaces II

Sharp triangle inequality and its reverse in Banach spaces were recently showed by Mitani et al. (2007). In this paper, we present equality attainedness for these inequalities in strictly convex Banach spaces.

Toeplitz operators associated with analytic crossed products II (invariant subspaces and factorization)

In [16], we introduced the notion of Toeplitz operators associated with analytic crossed products. In this paper, we study the structure of invariant subspaces with respect to the analytic crossed products. We also investigate the inner-outer factorization problems for analytic Toeplitz operators, the factorization problem for non-negative Toeplitz operators and Szegös infimum problem.

Invariant Subspaces and Cocycles in Nonselfadjoint Crossed Products

Let G be a compact abelian group with the archimedean totally ordered dual Γ and let L be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a one-parameter group {αγ}γϵΓ of trace preserving ∗-automorphisms of M. In this paper, we investigate the structure of invariant subspaces and cocycles for the subalgebra L+ of L consisting of those operators whose spectrum with respect to the dual automorphism group {βg}gϵG on L is nonnegative. Our main result asserts that if M is a factor, then L+ is maximal among the σ-weakly closed subalgebras of L.

On the calculation of the Dunkl-Williams constant of normed linear spaces

Recently, Jiménez-Melado et al. [Jiménez-Melado A., Llorens-Fuster E., Mazcuñán-Navarro E.M., The Dunkl-Williams constant, convexity, smoothness and normal structure, J. Math. Anal. Appl., 2008, 342(1), 298–310] defined the Dunkl-Williams constant DW(X) of a normed linear space X. In this paper we present some characterizations of this constant. As an application, we calculate DW(ℓ2-ℓ∞) in the Day-James space ℓ2-ℓ∞.

Noncommutative L p -spaces with 0 p < 1

The noncommutative Lp-spaces (1 ≤ p ≤ ∞) of unbounded operators associated with a regular gauge space (a von Neumann algebra equipped with a faithful normal semifinite trace) are studied by many authors ((4), (5) and (7)). It is well-known that the noncommutative L p -spaces (1 ≤ P L p is L q (1 ≤ p p + 1/ q = 1) by means of a Radon-Nikodym theorem.

Every n -dimensional normed space is the space R n endowed with a normal norm

Recently, Alonso showed that every two-dimensional normed space is isometrically isomorphic to a generalized Day-James space introduced by Nilsrakoo and Saejung. In this paper, we consider the result of Alonso for n-dimensional normed spaces.MSC:46B20.

On wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system acting on a separable Hilbert space H into itself. It is proved that the wandering vector multipliers for a unitary group form a group, which gives a positive answer for a problem of Han and Larson. Furthermore, non-abelian unitary groups of order 6 are considered. We prove that the wandering vector multipliers of such a unitary group can not generate B(H), This negatively answers another of their problems.