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.     

Interpolation of polynomial operators in a Hilbert space

An interpolant with orthonormal nodes is constructed for a polynomial operator of a given degree defined on an abstract Hilbert space. An estimate for the interpolation accuracy is derived in the metric of the space of operator values, while in the case where nodes are elements of a basis, the pointwise convergence of the interpolational operator process in this metric is proved as the number of nodes increases. Bibliography: 7 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp. 3–9     

Approximation of unbounded operators by bounded operators in a Hilbert space

We determine the best approximation of an arbitrary power A ^k of an unbounded self-adjoint operator A in a Hilbert space H on the class {x ∈ D(A ^r ) : ∥A ^r x∥ ≤ 1}, k < r.     

Closure in a Hilbert space of a prehilbert space Chebyshev set

Let E denote the real inner product space that is the union of all finite dimensional Euclidean spaces. There is a bounded nonconvex set S, that is a subset of E, such that each point of E has a unique nearest point in S. Let H denote the separable Hilbert space that is the completion of space E. A condition is given in order that a point in H have a unique nearest point in the closure of S. We shall also provide an example where the condition fails.     

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.     

Generalized random linear operators on a Hilbert space

In this paper, we are concerned with generalized random linear operators on a separable Hilbert space. Generalized random linear bounded operators, generalized random linear normal operators and generalized random linear self-adjoint operators are defined and investigated. The spectral theorems for generalized random linear normal operators and generalized random linear self-adjoint operators are obtained.     

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.     

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|     

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.     

Minimization of convex functionals over frame operators

We present results about minimization of convex functionals defined over a finite set of vectors in a finite-dimensional Hilbert space, that extend several known results for the Benedetto-Fickus frame potential. Our approach depends on majorization techniques. We also consider some perturbation problems, where a positive perturbation of the frame operator of a set of vectors is realized as the frame operator of a set of vectors which is close to the original one.     

On typical properties of Hilbert space operators

We study the typical behavior of bounded linear operators on infinite-dimensional complex separable Hilbert spaces in the norm, strong-star, strong, weak polynomial and weak topologies. In particular, we investigate typical spectral properties, the problem of unitary equivalence of typical operators, and their embeddability into C ₀-semigroups. Our results provide information on the applicability of Baire category methods in the theory of Hilbert space operators.