An asymmetric Fuglede-Putnam’s Theorem for w -hyponormal operators and dominant operators is proved, as a consequence of this result, we obtain that the range of the generalized derivation induced by the above classes of operators is orthogonal to its kernel. Mathematics subject classification (2010): 47B47, 47A30, 47B20.

An asymmetric Fuglede-Putnam’s Theorem for w−hyponormal operators and class Y operators is proved, as a consequence of this result, we obtain that the range of the generalized derivation induced by the above classes of operators is orthogonal to its kernel.

We propose some refinements for the second inequality in $$ \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, where A ∈ B(H). In particular, if is hyponormal, then, by refining Young with Kantorovich constant K K(⋅, ⋅), we show that \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left.{A}^{\ast}\right\Vert \righ...