On the infinite product of operators in Hilbert space

We give a necessary and sufficient condition for a certain set of infinite products of linear operators to be zero. We shall investigate also the case when this set of infinite products converges to a non-zero operator. The main device in these results is a weighted version of the König Lemma for infinite trees in graph theory.

     

Factoring operators through Hilbert space

We give an example of a pair of Banach spacesE andF so that neitherE′ norF has cotype 2, but every bounded linear operator fromE intoF factors through Hilbert space.     

On operators generating conditional bases in a Hilbert space

A spectral characterization is given to the linear operators which in a Hilbert space transform some complete orthonormal system into a conditional basis.Translated from Matematicheskie Zametki, Vol. 12, No, 1, pp. 73–84, July, 1972.     

On some integral properties of multidimensional Hilbert operators

In the present paper, we study the integral properties of multidimensional Hilbert transforms.     

Irregular vectors of Hilbert space operators

A vector x in a Hilbert space H is called irregular for an operator provided that supn‖Tnx‖=∞ and infn‖Tnx‖=0. We establish some basic properties of operators having irregular vectors and present examples that highlight the relationship, or lack thereof, between irregularity and hypercyclicity.     

On the eigenvalues of compact operators in a Hilbert space

Summary This paper contains several remarks concerning the behavior of the eigenvalues of compact operators in a Hilbert space. The results are related to the theorem for a finite matrix obtained as Theorem 1 in my previous paper[5].This work was partially supported by a research grant of the Sakkokai Foundation.     

On interpolation approximation of differentiable operators in a Hilbert space

In a Hilbert space, we construct an interpolation approximation of the Taylor polynomial for differentiable operators. By using this approximation, we obtain estimates of accuracy for analytic operators that strengthen previously known results and for operators containing finitely many Fréchet derivatives. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 554–563, April, 2006.     

Hilbert space structure and positive operators

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the nonsymmetric case.     

Commutator inequalities for Hilbert space operators

We prove singular value inequalities for positive operators. Some of these inequalities generalize recent results for commutators due to Bhatia-Kitttaneh, Kittaneh, and Wang-Du. Applications of our results are given.     

Bohr's inequalities for Hilbert space operators

The classical Bohr's inequality states that

²|z+w|?p²|z|+q²|w|