**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^{*n}A^{n}=(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^{*s}A^{s}=(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.

- P. Pietrzycki: The single equality A
^{*n}A^{n}=(A*A)^{n}does not imply the quasinormality of

weighted shifts on rootless directed trees,*J. Math. Anal. Appl*435 (2016), 338-348. - P. Pietrzycki: Reduced commutativity of moduli of operators
*arXiv preprint arXiv:1802.01007*, 2018.