Abstract. The normalized numerical range of an operator A is defined as the set F N (A) of all the values Ax,x / Ax attained by unit vectors x / ∈ ker A . We prove that F N (A) is simply connected, establish conditions for it to be star-shaped with the center at zero, to be open, closed, and to have empty interior. For some classes of operators (weighted shifts, isometries, essentially Hermitian) the complete description of F N (A) is obtained.Mathematics subject classification (2010): 15A60, 47A12, 47B15.
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