Operator Theory, Complex Analysis and Applications Seminar 

22/11/2012, 16:30 — 17:30 — Room P3.10, Mathematics Building
Janko Bracic, University of Ljubljana, Slovenia

Numerical ranges and hyperreflexivity

Let $\mathscr{H}$ be a complex Hilbert space and let ${𝒮}_{\mathscr{H}}=\{x\in \mathscr{H};\parallel x\parallel =1\}$ be the unit sphere of $\mathscr{H}$. Every bounded linear operator $A$ on $\mathscr{H}$ defines a quadratic form as follows $$q_A:\begin{array}{c}{𝒮}_{\mathscr{H}}⟶ℂ\\ x\mapsto ⟨Ax,x⟩,\end{array}$$ where $⟨\cdot ,\cdot ⟩$ denotes the inner product of $\mathscr{H}$. The image of $q_A$ is the numerical range $W(A)\subset ℂ$ of $A$. It is not hard to see that the spectrum of an operator is a subset of the closure of the numerical range, which means that numerical ranges are a useful tool in locating the spectrum. Some classical results about numerical ranges will be presented; for instance, the Toeplitz-Hausdorff Theorem and the Hildebrandt's Theorem. The hyperreflexivity of sets of operators determined by the numerical range will be discussed, as well.