Operators Related To Log-ω-hyponormality And P-ω-hyponormality

Let X(or H) be a separable complex Banach(or Hilbert) space of infinite dimension.Denote by B(X)(B(H)) the space of bounded linear operators on X(H). We shall denote by K(X)(K(H)), C the set of all compact operators on X(H), complex numbers respectively.Suppose T∈B(H) with polar decomposition T = U|T| , define f(?)= (?) as the Aluthge transform of T.An invertible operator T in B(H) is said to be log-ω-hyponormal ifFor 0 < p≤1, an operator T in B(H) is said to be p-ω-hyponormal ifFor latter convenience, we denote:(l-ω-H)={T∈B(H) : T is log-ω-hyponormal},(p-ω-H) = {T∈B(H) :T is p-ω-hyponormal and keiT (?) kerT^*}. For A, B∈B(H), defineσ(A,B)(â–³(a,b)) : B{H)→B(H) asLet d_A,,B denoteδ_A,B orâ–³_A,BiIt is well known that if A and B are normal, thenδ_A,B satisfies the Putnam-Fugledecommutativity property ker(δ_A,B) (?) ker(δ_A^*B^*.). In 1987, M. Radjabalipour proved that if A and B^* are hyponormal, then ker(δ_A,B) (?) ker(δ_A^*B^*). Many researchers extended this result. Duggal proved that p-hyponormal operators satisfy the Putnam-Fugledecommutativity property in 1996. Later Jeon, Tahanashi and Uchiyama proved that if A and B^* are log-hyponormal operators, then ker(δ_A,B) (?) ker(δ_A^*B^*). In 2008, Duggal proved that if A and B* are in the set of log-hyponormal operators and p-hyponormal operators, then ker(d_A,B-λ) (?) ker(d_a^*B^* - (?)),(?)λ∈C.We first prove the following theorem.Theorem: If T G (l-ω-H)∪(p-ω-H), then T is completely nonnormal if and only if T is completely nonnormal.Using the theorem above, we proved the following result.Theorem: If A,B^*∈(l-ω-H)∪(p-ω-H), then ker ker(d_A,B-λ) (?) ker(d_a^*B^*,(?)λ∈C.remark: We know that log-hyponormal operators and p-hyponormal operators are all in (l-ω-H), therefore this theorem generalizes Duggal's result.In 1909, Weyl studied the spectra of all compact perturbations T + K of a Hermitian operator T acting on a Hilbert space and showed thatλ∈C belongs toσ_ω(T) precisely when A is not an isolated point of finite multiplicity inσ(T). For T∈B(H), the set(?)σ(T + K) = {λ∈C : T -λis not a Fredholm operator of zero index},denoted byσ_ω(T), is called the Weyl spectrum of T. Hence Weyl's conclusion can be reformulated aswhere(?)Today this result is known as Weyl's theorem, and it has been extended from Hermitian operators to hyponormal operators by Coburn. As the development of theory, the class of operators satisfying Weyl's theorem expands unceasingly. For example, in 1997, Ch(?), Itoh and Oshiro proved that p-hyponormal operators satisfy Weyl's theorem and Young Min Han etc proved thatω-hyponormal operators satisfy Weyl's theorem in 2005. Later, people began to study the elementary operators d_A,B with entries in the above classes. In 2002, Duggal proved that when A, B^* are hyponormal and f is a analytic function on a neighborhood ofσ(d_A,B), then f(d_A,B) satisfies Weyl' theorem. In 2007, Lombarkia and Bachir proved when A, B* are log-hyponormal, the same conclusionstill holds. Furthermore, Duggal extended it to the case that A and B^* being log-hyponormal or p-hyponormal in 2008.We proved the following theorem:Theorem: If A, B^*∈(l-ω-H)∪(p-ω-H), f is a analytic function on a neighborhood ofσ(d_a,b), then f(d_A,B) satisfies Weyl' theorem.In fact, by using techniques in Duggal's paper, we obtain a better result. It is formulated as follows.Theorem: If A, B^*∈(l-ω-H)∪(p-ω-H), then H₀(d_A,B-λ) = ker(d_A,B-λ), (?)λ∈isoσ(d_A,B).Here, given T∈B(X), H₀(T) := {x∈X : (?)→0}.