Almost α-Hyponormal Operators with Weyl Spectrum of Area Zero

Theorem A (see [1], [2].)Suppose $$. If there exists K ∈ 𝒞₂(ℋ) such that either m(T + K) < ∞ or μ₂(σ(T + K)) = 0, then $$. Moreover, when m(T + K) < ∞,

and when μ₂(σ(T + K)) = 0,$$,   and consequently, $$.

Corollary B (see [2].)Let $$ such that μ₂(σ_w(T)) = 0. Then $$ and $$.

Theorem C (see [3], [4].)Let 0 < α ≤ 1, and let $$ and K ∈ 𝒞_2α(ℋ) with m(T + K) < ∞. Then $$ and

Theorem 1. Let α ∈ (0,1) and let $$ and K ∈ 𝒞_α(ℋ) with μ_2α(σ(T + K)) = 0. Then $$ and $$.

Remark. It would have been desirable that Theorem 1 be proved with the hypothesis that K ∈ 𝒞_2α(ℋ).

Corollary 2. Let α ∈ (0,1] and let $$ such that μ₂(σ_w(T)) = 0. Then $$ and $$.

Proof. If α = 1, then conclusion holds according to Corollary B. Let α ∈ (0,1). First, a careful inspection of the proof of a result of Stampfli [5] leads to the following. For T ∈ L(ℋ) and α > 0, there exists K_α ∈ 𝒞_α(ℋ) such that σ(T + K_α)∖σ_w(T) consists of a countable set which clusters only on σ_w(T). Therefore μ₂(σ(T + K_α)) = 0 and thus Theorem 1 applies.

Proposition D (Hansen′s inequality [6]). If A, B ∈ L(ℋ), A ≥ 0, | | B|| ≤ 1, and α ∈ (0,1], then B^*A^αB ≤ (B^*AB) ^α.

Proposition E (Lowner′s inequality [7]). If A, B ∈ L(ℋ), A ≥ B ≥ 0, and α ∈ (0,1], then A^α ≥ B^α.

Proposition F (Jocic′s inequality [8]). Let A, B ∈ L(ℋ), A,   B ≥ 0, α ∈ (0,1], and 1 ≤ p < ∞. If A − B ∈ 𝒞_αp(ℋ), then A^α − B^α ∈ 𝒞_p(ℋ) and | | B^α − A^α | |_p≤| |    | B − A|^α   | |_p.

Proof of Theorem 1. Let α ∈ (0,1),  $$, and K ∈ 𝒞_α(ℋ) with μ_2α(σ(T + K)) = 0, and assume m(T + K) = ∞, otherwise Theorem C implies $$.

Let {e_n} _n∈ℕ be an orthonormal basis of ℋ and let

Assume that with respect to the decomposition $$, operators T and K are written as Since ℋ_n is a rationally invariant subspace for T + K, we have T_3n + K_3n = 0, and thus T_3n = −K_3n ∈ 𝒞_α(ℋ_n)⊆𝒞_2α(ℋ_n), and σ(T_1n + K_1n)⊆σ(T + K), which implies μ_2α(σ(T_1n + K_1n)) = 0.

Let P_n be the orthogonal projection onto ℋ_n, and thus P_n↑I strongly. We will prove next that $$ by first establishing that

where $$ is positive semidefinite and $$.

Assuming that equality (7a) was already proved and writing $$ with Q ≥ 0 and K ∈ 𝒞₁(ℋ), then we have

that is, $$ is the sum of $$, which is a positive semidefinite operator, and of $$, which is a trace-class operator.

Indeed, the expression $$ can be written as D₁ − D₂, where

We can write $$, where which according to Hansen′s inequality is a positive semidefinite operator, and which according to Jocic′s inequality is a trace-class operator that satisfies Concerning operator D₂, we can write $$, where which according to Lowner′s inequality is a positive semidefinite operator, and which is also a trace-class operator since and according to Jocic′s inequality Therefore, and consequently, $$, where $$ is positive semidefinite and $$ is trace-class, which establishes equality (7a).

According to (7b), $$, and since m(T_1n + K_1n) ≤ n and σ(T_1n + K_1n)⊆σ(T + K), Theorem C implies that $$,   and furthermore, by replacing T_1n with $$,    $$. Furthermore, equality (7a) implies

which further implies Similar utilization of Lowner′s and Hansen′s inequalities implies that $$ and $$ are positive semidefinite, and thus so is $$. Therefore Since T_3n = −K_3n ∈ 𝒞_p(ℋ_n) and K_3n → 0 weakly and both |T_3n| and $$,   we have | | T_3n | |_α → 0, and thus $$. Replacing T with T^* we conclude that $$.