The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,

$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$

We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

In this paper we generalize the notion of the C-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator C, the C-numerical range of such an operator is defined; it is a compact, convex subset of ℂ. We explicitly describe the C-numerical ranges of several operators and classes of operators.

Palmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.

When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

The tracial numerical range of operators on a 2-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained.

Let V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that

⊆ W*(DK(T)) ⊆ W(DK(T)) = con

where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

Every norm $\mathcal{V}$ on ${{\text{C}}^{\text{n}}}$ induces two norm numerical ranges on the algebra ${{M}_{n}}\,\text{of}\,\text{all}\,\text{n}\,\times \,\text{n}$ complex matrices, the spatial numerical range

$$W\left( A \right)=\left\{ {{x}^{*}}Ay:x,y\in {{\text{C}}^{n}},{{v}^{D}}\left( x \right)=v\left( y \right)={{x}^{*}}y=1 \right\},$$

where ${{\mathcal{V}}^{D}}$ is the norm dual to $\mathcal{V}$, and the algebra numerical range

$$V\left( A \right)=\left\{ f\left( A \right):f\in S \right\},$$

where $S$ is the set of states on the normed algebra ${{M}_{n}}$ under the operator norm induced by $\mathcal{V}$. For a symmetric norm $\mathcal{V}$, we identify all linear maps on ${{M}_{n}}$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e., linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\mathcal{V}$ is not the ${{\ell }_{1}},{{\ell }_{2}},\,\text{or}\,{{\ell }_{\infty }}$ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e., of the form $A\mapsto {{Q}^{*}}AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ${{\ell }_{1}}$ and the ${{\ell }_{\infty }}$ norms, the results are quite different.

Let $H$ be a complex Hilbert space, and $\mathcal{H}\left( H \right)$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi :\,\mathcal{H}(H)\,\to \,\mathcal{H}(H)$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps $\phi$ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X\mapsto \xi TX{{T}^{*}}orX\mapsto \xi T{{X}^{t}}{{T}^{*}}$, for a suitable invertible or unitary $T$ and $\xi \,\in \,\{1,\,-1\}$, where ${{X}^{t}}$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I\,+\,K$, where $\alpha$ is a scalar and $K$ is compact.

Westwick's convexity theorem on the numerical range is generalized in the context of compact connected Lie groups.

A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

The bidual of a unital infrabarrelled l.m.c. C* algebra E, equipped with the bidual topology and the regualr Arens product, is always an l.m.c. C*-algebra. On the other hand, a unital l.m.c. *-algebra E has the C*-property if and only if every self-adjoint element x of E is strongly hermitian (x has real numerical range), or the sets of normalized states and normalized continuous positive linear forms of E coincide. Finally, every unital cpmplete l.m.c. C* algebra satisfying, locally, the property ‘the extreme points are dense in that set of continuous positive linear forms” (antiliminal algebra) has the complexes as its only normal elements.

For α ∈ [0, 1] the operator is the operator formally defined on the Hardy space H2 by

If α = 1, then the usual identification of H2 with l2 takes A1 onto the discrete Cesàro operator. Here we see that {Aα: α ∈ [0, 1]} is not arcwise connected, that Re Aα ≥ 0, that Aα is a Hilbert-Schmidt operator if α ∈[0, 1), and that Aα is neither normaloid nor spectraloid if α ∈(0, 1).