Some generalized theorems onp-hyponormal operators

The Furuta inequality with negative powers

“…, the validity of (1) is equivalent to the validity of (2). By a simple observation, the problem to find all real numbers p, q, r guaranteeing the validity of the Furuta inequality is reduced to the case where 0 < p, 0 < q and r < 0.…”

“…In the operator mean theory, the Furuta inequality was generalized to the grand Furuta inequality [8] and it extends the Ando-Hiai log-majorization theory. Also, the Furuta inequality was used in the p-hyponormal operator theory ( [2], [9]) and the relative entropy theory ( [6], [7]). …”

On upper and lower bounds of the numerical radius and an equality condition

“…The Aluthge transform T of T is defined by T = |T | 1/2 U |T | 1/2 (see [1]). The following properties of T are well known:…”

WEYL'S THEOREM FOR CLASSA(k) OPERATORS

“…An operator T ∈ B(H) is said to be hyponormal if T * T ≥ TT * , where T * is the adjoint of T. As a generalisation of hyponormal operators, p-hyponormal and log hyponormal operators have been introduced in [2] and [13], respectively. An operator T is said to be p-hyponormal if (T * T) p ≥ (TT * ) p for a positive number p and loghyponormal if T is invertible and log(T * T) ≥ log(TT * ).…”