For a positive integer k , an operator T ∈ B(H ) is called k -quasi-paranormal if T k+1 x 2 T k+2 x T k x for all x ∈ H , which is a common generalization of paranormal and quasi-paranormal. In this paper, firstly we prove that if T is a contraction of k -quasi-paranormal operators, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator D λ = T .k (|T 2 | 2 . 2 λ |T | 2 + λ 2 I )T k for 0 λ 1 is a strongly stable contraction; secondly we prove that k -quasi-paranormal operators are not supercyclic; at last we prove that the spectrum is continuous on the class of all k -quasi-paranormal operators.
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机译：对于正整数k，如果对于所有x∈H，T k + 1 x 2 T k + 2 x T kx，则算符T∈B（H）称为k-准超自然数，这是超自然和准常态的常见概括-超自然的。在本文中，首先我们证明，如果T是k个拟超正规算子的压缩，则T具有非平凡不变子空间或T是适当的压缩，而非负算子Dλ= T .k（| T 2 | 0 <λ1时的2。2λ| T | 2 +λ2 I）T k是一个非常稳定的收缩。其次，我们证明k-准超正规算子不是超循环的;最后我们证明了该谱在所有k个拟超正规算子的类别上是连续的。
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