ON THE EXISTENCE OF NON-NORM-ATTAINING OPERATORS

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent: (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ; (ii) Every operator from E into F attains its norm; (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ , where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

Some properties of D-weak operator topology

Let 𝓖D(𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D-weak operator topology (introduced by other authors) on 𝓖D(𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖D(𝓗) is the whole 𝓖D(𝓗). The closure of the set of all unbounded elements of 𝓖D(𝓗) is also the set 𝓖D(𝓗). If Q is arbitrary unbounded element of 𝓖D(𝓗), it determines an interval in 𝓖D(𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D-weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

Banach spaces whose algebras of operators have a large group of unitary elements

We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra $\mathcal L (X)$ (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on $\mathcal L (X)$ satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and $\mathcal L (X)$ is $w_{op}'$-unitary, where $w'_{op}$ stands for the dual weak-operator topology.

ON THE ENTANGLED ERGODIC THEOREM

Let U be a unitary operator acting on the Hilbert space [Formula: see text], and α: {1, …, m} ↦ {1, …, k} a partition of the set {1, …, m}. We show that the ergodic average [Formula: see text] converges in the weak operator topology if the Aj belong to the algebra of all the compact operators on [Formula: see text]. We write esplicitly the formula for these ergodic averages in the case of pair-partitions. Some results without any restriction on the operators Aj are also presented in the almost periodic case.

On the almost sure approximation of self-adjoint operators in L2 (0, 1)

There are several important theorems concerning the almost sure convergence of (monotone) sequences of orthogonal projections in L2-spaces. Let us mention here the martingale convergence theorems or the results on the developments of functions with respect to orthogonal systems. On the other hand every self-adjoint operator with the spectrum on the interval [0, 1] is a limit of some sequence of orthogonal projections in the weak operator topology (see [1]). This paper is devoted to a problem of approximation of a self-adjoint operator A acting in L2 (0, 1) by a sequence Pn of orthogonal projections in the sense that

The Type I Part of the Regular Representation

Let G be a discrete group and let H = L2(G), with norm | |. Let B(H) be the ring of bounded operators on H with the normThe right regular representation of G on H induces an injection ρ : C[G] → B(H), and W(G) is the closure of the image of ρ in the weak operator topology on B(H) (C = complex numbers). Using ρ, we identify C[G] with its image in W(G).

On the algebra generated by a Hermitian operator

The purpose of this note is to solve a problem of Dr A. M. Sinclair. Denote by Aw(I, T) the algebra with identity generated by a bounded linear operator T in the weak operator topology. We prove the following result.