On the Isolated Points of the Spectrum of Paranormal Operators

Abstract

For paranormal operator T on a separable complex Hilbert space H, we show that (1) Weyl's theorem holds for T , i.e., σ(T ) \ w(T ) = π00(T ) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T ) is self-adjoint(i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ) * .

Riesz Projections for a Non-Hyponormal Operator

“…This result has since been generalized by many mathematicians ( [1], [6], [4], [13]). In particular we should recall Duggal's result ( [2]; [3]) for an extended class of non-hyponormal operators.…”

A Note on ∗-Paranormal Operators and Related Classes of Operators

“…In [9], Uchiyama showed that if T is paranormal, i.e., T ∈ P(2), then Weyl's theorem holds for T , and if λ is a non-zero isolated point of σ(T ), then the Riesz idempotent E λ of T with respect to λ is self-adjoint and…”

“…In [9], Uchiyama proved that for every paranormal operator T and each isolated point λ of σ(T ) the Riesz idempotent E λ satisfies that…”

“…It is well-known that every normal, hyponormal, p-hyponormal, w-hyponormal, class A, or paranormal operator satisfies Weyl's theorem and has the SVEP (see [9], [10] for definitions). We shall show that every * -paranormal operator satisfies Weyl's theorem.…”

Spectrum of Class wF(p,r,q) Operators

“…We cite [8,[14][15][16][17][18][19][20][21][22]. In this paper, we will discuss several spectral properties of class wF(p,r, q) for p > 0, r > 0, p + r ≤ 1, and q ≥ 1.…”