On $ (n_1, \cdots, n_m) $-hyponormal tuples of Hilbert space operators

This paper introduces a new class of multivariable operators called $ (n_1, \cdots, n_m) $-hyponormal tuples, which combine joint normal and joint hyponormal operators. A tuple of operators $ \mathcal{Q} = (\mathcal{Q}_1, \; \cdots, \mathcal{Q}_m) $ is said to be an $ (n_1, \cdots, n_m) $-hyponormal tuple for some $ (n_1, \cdots, n_m)\in \mathbb{N}^m $ if$ \sum\limits_{1\leq k,\;l\leq m}\big\langle[\mathcal{Q}_k^{*n_k}, \;\mathcal{Q}_l^{n_l}]\omega_k\mid \omega_l\big\rangle\geq 0, \quad \forall\; (\omega_k)_{1\leq k\leq m}\in {\mathcal K}^m. $We show several properties of this class that correspond to the properties of joint hyponormal operators.