6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Paweł Pietrzycki

Jensen’s inequality for operators and commutativity
In 1973 M. R. Embry published a very influential paper studying the Halmos-Bram criterion for subnormality. In particular, she gave a characterization of the class of quasinormal operators in terms of powers of operators. Namely, bounded operator in Hilbert space is quasinormal if and only if the following condition holds

A*nAn=(A*A)n for all n∈N.

This leads to the following question:
is it necessary to assume that the above equality holds for all n∈N? To be more precise we ask for which subset S⊂N the following system of operator equations:

A*sAs=(A*A)s for all s∈S

implies the quasinormality of A.

We will prove that operator A is quasinormal if and only if it satisfies the system of the above equations with S={p,m,m+p,n,n+p}. This theorem generalizes Embry’s characterization of quasinormality of bounded
operators. We obtain a new characterization of the normal operators which resembles that for the quasinoraml operators.

  1. P. Pietrzycki: The single equality A*nAn=(A*A)n does not imply the quasinormality of
    weighted shifts on rootless directed trees, J. Math. Anal. Appl 435 (2016), 338-348.
  2. P. Pietrzycki: Reduced commutativity of moduli of operators arXiv preprint arXiv:1802.01007, 2018.
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