| Popis: |
We study properties of the numerical ranges of Foguel operators F T = [ S ⁎ 0 T S ] , where S is the simple unilateral shift and T is some operator, both acting on l 2 . Among other things, we show that (1) if T is nonzero compact, then the numerical radius w ( F T ) is strictly less than 1 + ( ‖ T ‖ / 2 ) , (2) if T is a diagonal unitary operator, then 5 / 2 w ( F T ) ≤ 3 / 2 , and (3) if T is a scalar operator aI, then the numerical range W ( F T ) is open and is not a circular disc unless a = 0 . |
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