Matrix regular operator space and operator system

We establish a relationship between Schreiner's matrix regular operator space and Werner's (nonunital) operator system. For a matrix ordered operator space V with complete norm, we show that V is completely isomorphic and complete order isomorphic to a matrix regular operator space if and only if both V and its dual space V^∗ are (nonunital) operator systems.     

Nuclearity for dual operator spaces

In this short paper, we study the nuclearity for the dual operator space V* of an operator space V. We show that V* is nuclear if and only if V*** is injective, where V*** is the third dual of V. This is in striking contrast to the situation for general operator spaces. This result is used to prove that V** is nuclear if and only if V is nuclear and V** is exact.     

Generation and commutation properties of the Volterra operator

Let V be the classical Volterra operator on L ₂(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert?CSchmidt operators and as well in the space ${\mathcal{B}(L_2(0,1))}$ equipped with the weak operator topology. Moreover, the algebra generated by V ² and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.     

Asymptotic type for sectorial operators and an integral of fractional powers

There is a standard notion of type for a sectorial linear operator acting in a Banach space. We introduce a notion of asymptotic type for a linear operator V, involving estimates on the resolvent ⁻¹(λI+V) as λ→0. We show, for example, that if V is sectorial and of asymptotic type ω then the fractional power Vα is of asymptotic type αω for a suitable range of positive α. Moreover, we establish various properties of the operator ; in particular, this operator is of asymptotic type 0, for a sectorial operator V. This result has an application to the construction of operators satisfying the well-known Ritt resolvent condition.     

Spectrum and continuum eigenfunctions of Schrödinger operators

We consider Schrödinger operators $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ for a large class of potentials. V. We show that if H? = E? has a polynomially bounded solution ? then E is in the spectrum of H. This is accomplished by proving that the spectrum of H as an operator on L² is identical to its spectrum as an operator on the weighted L² space, L²_δ.     

Lifting bounded approximation properties from Banach spaces to their dual spaces

Based on a new reformulation of the bounded approximation property, we develop a unified approach to the lifting of bounded approximation properties from a Banach space X to its dual X^*. This encompasses cases when X has the unique extension property or X is extendably locally reflexive. In particular, it is shown that the unique extension property of X permits to lift the metric A-approximation property from X to X^*, for any operator ideal A, and that there exists a Banach space X such that X,X**,… are extendably locally reflexive, but X^*,X***,… are not.     

Topological uniform descent and Weyl type theorem

The generalized Weyl’s theorem holds for a Banach space operator T if and only if T or T^∗ has the single valued extension property in the complement of the Weyl spectrum (or B-Weyl spectrum) and T has topological uniform descent at all λ which are isolated eigenvalues of T. Also, we show that the generalized Weyl’s theorem holds for analytically paranormal operators.     

The approximation property in terms of the approximability of weak∗-weak continuous operators

By a well-known result of Grothendieck, a Banach space X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous compact operator T:X∗→Y can be uniformly approximated by finite rank operators from X⊗Y. We prove the following “metric” version of this criterion: X has the approximation property if and only if, for every Banach space Y, every weak∗-weak continuous weakly compact operator T:X∗→Y can be approximated in the strong operator topology by operators of norm ?‖T‖ from X⊗Y. As application, easier alternative proofs are given for recent criteria of approximation property due to Lima, Nygaard and Oja.     

Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C^∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?     

Quadratic forms on completely symmetric spaces

Let V be an n-dimensional Euclidean vector space, and let V(^m) be the corresponding m-th completely symmetric space over V equipped with the induced inner product. The purpose of this paper is to prove the following conjecture of H.A. Robinson: if T is a linear operator on V(^m) and (Tz, z) = 0 for every decomposable element z of V(^m), then T is skew-symmetric.