A NOTE ON WEYL'S THEOREM FOR *-PARANORMAL OPERATORS

Abstract

In this note we investigate Weyl's theorem for �-paranormal operators on a separable infinite dimensional Hilbert space. We prove that if T is a �-paranormal operator satisfying Property (E) - (TI)HT({�}) is closed for each � 2 C, where HT({�}) is a local spectral subspace T, then Weyl's theorem holds for T. Let H denote an infinite dimensional separable Hilbert space. Let B(H) and K(H) denote the algebra bounded linear operators and the ideal compact operators on H, respectively. If T 2 B(H) write N(T) and R(T) for the null space and range T; �(T) := dimN(T); �(T) := dimN(T � ); �(T) for the spectrum T; �ap(T) for the approximate point spectrum T; �0(T) for the set eigenvalues T. An operator T 2 B(H) is called Fredholm if it has closed range with finite dimensional null space and its range finite co-dimension. The index a Fredholm operator T 2 B(H) is given by ind(T) := �(T) − �(T). An operator T 2 B(H) is called Weyl if it is Fredholm index zero. An operator T 2 B(H) is called Browder if it is Fredholm of finite ascent and descent: equivalently ((11, Theorem 7.9.3)) if T is Fredholm and T − �I is invertible for sufficiently small� 6 0 in C. The essential spectrume(T), the Weyl spectrum !(T) and the Browder spectrumb(T) T 2 B(H) are defined by ((10), (11), (12))