The higher rank numerical range is convex

In this short note we provide the final step in showing that the higher rank numerical range is convex. The previous steps appear in the paper “Geometry of Higher-Rank Numerical Ranges” by Choi, M.-D., Giesinger, M., Holbrook, J.A. and Kribs, D.W.     

Numerical ranges of the powers of an operator

The numerical range W(A) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form (Av,v) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W(A), inclusion regions are obtained for W(Ak) for positive integers k, and also for negative integers k if A−1 exists. Related inequalities on the numerical radius and the Crawford number are deduced.     

The finite rank perturbations of the product of Hankel and Toeplitz operators

In this paper we completely characterize when the product of a Hankel operator and a Toeplitz operator on the Hardy space is a finite rank perturbation of a Hankel operator, and when the commutator of a Hankel operator and a Toeplitz operators has finite rank.     

Pairs of quaternionic selfadjoint operators with numerical range in a halfplane

The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool.     

Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series

Using a Laurent series representation, we present a detailed discussion of necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space that are singular in the unperturbed state.     

Circles in the spectrum and the geometry of orbits: A numerical ranges approach

We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is obtained via the study of numerical ranges of operator tuples where several new results are also obtained. As consequences of our numerical ranges approach, we derive in particular wide generalizations of Arveson's theorem as well as show that the weak convergence of operator powers implies the uniform convergence of their compressions on an infinite-dimensional subspace.