An operator T is said to be paranormal if ||T2x|| ≥ ||Tx||2 holds for every unit vector x.Several extensions of paranormal operators are considered until now,for example absolute-k-paranormal and p-paranormal introduced in [10],[14],respectively.Yamazaki and Yanagida [38] introduced the class of absolute-(p,r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators.An operator T ∈ B(H) is called absolute-(p,r)-paranormal operator if |||T|p|T*|rx||r ≥|||T*|rx||p+r for every unit vector x ∈ H and for positive real numbers p ＞ 0 and r ＞ 0.The famous result of Browder,that self adjoint operators satisfy Browder's theorem,is extended to several classes of operators.In this paper we show that for any absolute-(p,r)-paranormal operator T,T satisfies Browder's theorem and a-Browder's theorem.It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p,r)-paranormal operator T,then E is self-adjoint if and only if the null space of T-μ,N(T-μ) (∪) N(T*-(-μ)).
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机译：如果|| T2x ||，则算符T被认为是超自然的。 ≥|| Tx || 2对于每个单位向量x都成立。到目前为止，仍考虑超自然算子的几个扩展，例如分别在[10]，[14]中引入的绝对k-超自然和p-超正规。Yamazaki和Yanagida [38]引入了绝对-（p，r）-超正规算子的类，作为对绝对-k-超正规算子和p-超正规算子的类的进一步推广。算子T∈B（H）称为绝对-（对于每个单位向量x∈H和正实数p＞ 0，|||| T | p | T * | rx || r≥||| T * | rx || p + r且r＞0。自伴算子满足Browder定理的Browder著名结果被推广到几类算子。本文证明，对于任何（p，r）超正规算子T，T都满足Browder的定理和a-Browder定理。还表明，如果E是绝对-（p，r）-超正规算子T的谱的非零孤立点μ的Riesz幂等号，则E仅当且仅当是自伴随的如果是空的温泉ce的T-μ，N（T-μ）（∪）N（T *-（-μ））。
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