Dedicated to Professor Daoxing Xia on his 77th birthday with respect and affectionRecommended by Jozsef Szabados This paper discusses some spectral properties of class wF(p,r, q) operators for p > 0, r > 0, p + r ≤ 1, and q ≥ 1. It is shown that if T is a class wF(p,r, q) operator, then the Riesz idempotent E λ of T with respect to each nonzero isolated point spectrum λ is selfadjoint and* . Afterwards, we prove that every class wF (p,r, q) operator has SVEP and property (β), and Weyl's theorem holds for f (T) when f ∈ H(σ(T)).

Let n be a positive integer, an operator T belongs to class A(n) if |T 1+n | 2/(1+n) ≥ |T | 2 , which is a generalization of class A and a subclass of n-paranormal operators, i.e., T 1+n x 1/(1+n) ≥ T x for unit vector x. It is showed that if T is a class A(n) or n-paranormal operator, then the spectral mapping theorem on Weyl spectrum of T holds. If T belongs to class A(n), then the nonzero points of its point spectrum and joint point spectrum are identical, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical.

The entanglement entropy generated by quantum transport, similar to any physical observable quantity, is a stochastic variable that has its distribution and can fluctuate. The fundamental question is how to define the entanglement entropy operator which allows one to discuss entanglement entropy fluctuation. By introducing the entanglement entropy operator, we develop a theoretical framework to calculate the entanglement entropy fluctuation as well as its higher order cumulants generated by electronic transport in open systems. The distribution of entanglement entropy generated by opening or closing a quantum point contact (QPC) is solved exactly. When the transmission coefficient of QPC is one-half, the entanglement entropy is maximized and fluctuationless. We also establish a general relation between the generated entanglement entropy fluctuation and charge fluctuation. We apply our theory to electronic transport through a quantum dot and study the generated entanglement entropy in the transient regime. Universal behavior is found for the cumulants of entanglement entropy at short times.

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