Image numérique simultanée d'une famille d'opérateurs sur l'espace de Hilbert. (Simultaneous numerical range of a family of operators in Hilbert space) (Q750842)

The main results are as follows: Let \(A\subset L(H)\) be an ultraweakly closed subalgebra of the full operator algebra L(H) on a separable Hilbert space on which the Gelfand transform is an isometry. Theorem 1: For any countable family \(R=(R_ k)_{k\geq 0}\subset A\) the closure of the simultaneous range \[ W(R)=\{<R_ kx| x>| \| x\| =1,\quad k\geq 0\}\subset {\mathbb{C}}^{{\mathbb{N}}} \] is convex and coincides with the simultaneous essential numerical range \(W_ e(R)=\{\rho (R_ k+K(H))| k\geq 0\), \(\rho\in ((L(H)/K(H))\); \(\| \rho \| =\rho (1)=1\}\) if A does not contain a finite dimensional subalgebra. Theorem 2: For a finite family \(R\subset A\) one has int(W(R)) convex where the interior is taken with respect to the affine subspace generated by R.