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. Several related results have been proved as well. 
Vladimir Muller and Yuri Tomilov
math.FA