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An operator T ∈ B(H) is called p-quasi-n-hyponormal if T*(T*nTn)pT > T*(TnT*n)pT for a positive number 0 < p < 1 and a positive integer n, which is a further generalization of normal operator. In this paper we introduce the class of p-quasi-n-hyponormal operators and show its structural properties via Hansen inequality and L?owner-Heinz inequality. As important applications, we obtain that every p-quasi-n-hyponormal operator has a scalar extension. In addition, we prove that if T is a quasiaffine transform of p-quasi-n-hyponormal, then T satisfies Weyl's theorem. Finally some examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, we give the method to calculate the generalized Gao's constant under the absolute normalized norms in R². Using this method, we can compute the exact values of the generalized Gao's constant of some concrete Banach spaces easily, such as Banach lattice, Lorentz sequence spaces etc. [ABSTRACT FROM AUTHOR]

An operator T ∈ B(H) is called square-p -quasihyponormal if T*(T*²T²)pT ≥ T*(T²T*²)pT for p ∈ (0,1], which is a further generalization of normal operator. In this paper, we give a sufficient condition for an injective square-p-quasihyponormal operator to be self-adjoint, and we obtain that every square-p-quasihyponormal operator has a scalar extension. As a consequence, we prove that if T is a quasiaffine transform of square-p -quasihyponormal, then T satisfies Weyl's theorem. Finally some examples are presented. [ABSTRACT FROM AUTHOR]