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Properties of the set of operators with a given part of the numerical range

Let $H$ be a complex Hilbert space. The numerical range of a bounded linear operator A is W(A)={ < Ax,x>: x in H and ||x||=1}. This is a non-empty bounded set of complex numbers. Some basic properties of the numerical range will be presented, for instance its convexity. In the second part of the talk, we will discuss the properties of sets of operators which have a given set of numbers in the numerical range, i.e., W_E={ A in B(H);~E\subseteq \overline{W(A)}}, where E\subseteq C$ is a given set. Interesting results are also obtained in the particular case when E={0}.
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