A bounded linear operator between Banach spaces is called completely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from L1 into an arbitrary Banach space, namely, the operator from L1 into l∞ defined by

Let Ai (i = 1,2, · · · ,k) be bounded linear operators on a Hilbert space. This paper aims to show a characterization of operator order Ak Ak−1 · · · A2 A1 > 0 in terms of operator inequalities. Afterwards, an application of the characterization is given to operator equalities due to Douglas’s majorization and factorization theorem. Mathematics subject classification (2010): 47A63.

We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator A', diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) o...

Abstract. We study a factorization of bounded linear maps from an operator space A to its dual space A∗. It is shown that T : A −→ A∗ factors through a pair of a column Hilbert spaces Hc and its dual space if and only if T is a bounded linear form on A ⊗ A by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map ...

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...

Abstract. Let T and V be two Hilbert space contractions and let X be a linear bounded operator. It was proved by C. Foias and J.P. Williams that in certain cases the operator block matrix R(X ;T, V ) (equation (1.1) below) is similar to a contraction if and only if the commutator equation X = TZ−ZV has a bounded solution Z. We characterize here the similarity to contractions of some operator ma...

A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT ∗C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ |T |, where J is an auxiliary conjugation commuting with |T | = √ T ∗T . We consider numerous examples, including the Poincaré-Neumann singular integral (bounded) operator and the Jordan model operator (compr...

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown th...

Let X be a Banach space and T be a bounded linear operator from X to itself (T ∈ B(X).) An operator D ∈ B(X) is a Drazin inverse of T if TD = DT , D = TD and T k = T D for some nonnegative integer k. In this paper we look at the Jörgens algebra, an algebra of operators on a dual system, and characterise when an operator in that algebra has a Drazin inverse that is also in the algebra. This resu...

Here we look at strong and weak operator topologies on spaces of bounded linear mappings, and convergence of sequences of operators with respect to these topologies in particular.