Some relations between the self-commutator and the numerical range of operators

25 April 2014, a general meeting of AMU was held. Prof. Levon Gevorgyan from the State Engineering University of Armenia have presented  a talk “Some relations between the self-commutator and the numerical range of operators”.

For a Hilbert space bounded linear operator the norm of its self-commutator     shows «how far» is it from being normal. For some classes of operators different estimates of that norm from above (Putnam’s inequality, Berger-Shaw theorem) and from below (Khavinson, Ferguson ttheorems) are known.In the report similar problems will be discussed for finite-dimensional operators (the tensor product of two Hilbert space elements, 2×2 complex matrices, tri-diagonal Toeplitz matrices, SOR matrices), as well for infinite-dimensional oerators (composition operators, acting in the Dirichlet sace, the Volterra integration operator).It turned out that the above mentioned norm is connected with the numerical range of the operator. Solving an isoperimetric-type problem for compact subsets of the complex plane, the norm of the self-commutator is bounded from above by the multiple of numerical range’s area.|Based on the numerical evidence, two more precise conjectures on the value of that norm are formulated.


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