Almost 𝜶-Hyponormal Operators with Weyl Spectrum of Area Zero

Abstract

We define the class of almost 𝛼-hyponormal operators and prove that for an operator 𝑇 in this class, (𝑇^∗𝑇)^𝛼−(𝑇𝑇^∗)^𝛼 is trace-class and its trace is zero when 𝛼∈(0,1] and the area of the Weyl spectrum is zero.

This note is dedicated to Professor Carl M. Pearcy with the occasion of his 75th birthday.

Let ℋ be a complex, separable, infinite-dimensional Hilbert space, and let 𝐿(ℋ) denote the algebra of all linear bounded operators on ℋ, and for 1≤𝑝<∞, let 𝒞_𝑝(ℋ) denote the 𝑝-Schatten class on ℋ. For 𝐾∈𝒞_𝑝(ℋ), the expression ||𝐾||_𝑝∶=(∑^∞𝑛=1𝜇_𝑛(𝐾)^𝑝)^1/𝑝, where 𝜇₁(𝐾)≥𝜇₂(𝐾)≥⋯ are the singular values of 𝐾, is a norm for 𝑝≥1, and is only a quasinorm for 0<𝑝<1 (it does not satisfy the triangle inequality). Nevertheless, the latter case will be used in what follows.

For 𝑇∈𝐿(ℋ),𝜎(𝑇) and 𝜎_𝑤(𝑇) will denote the spectrum and the Weyl spectrum, respectively. Recall that Weyl spectrum is the union of the essential spectrum, 𝜎_𝑒(𝑇), and all bounded components of ℂ⧵𝜎_𝑒(𝑇) associated with nonzero Fredholm index. An operator 𝑇∈𝐿(ℋ) is called (𝒞_𝑝,𝛼)-normal (notation: 𝑇∈𝑁^𝛼𝑝(ℋ)) if 𝐶^𝛼𝑇∶=(𝑇^∗𝑇)^𝛼−(𝑇𝑇^∗)^𝛼 belongs to 𝒞_𝑝(ℋ), and 𝑇 is called (𝒞_𝑝,𝛼)-hyponormal (notation: 𝑇∈𝐻^𝛼𝑝(ℋ)) if 𝐶^𝛼𝑇 is the sum of a positive definite operator and an operator in 𝒞_𝑝(ℋ), or equivalently, (𝐶^𝛼𝑇)₋ (the negative part of 𝐶^𝛼𝑇) belongs to 𝒞_𝑝(ℋ), where 𝛼 is a positive number. This note will be concerned with the particular class 𝐻^𝛼1(ℋ), which by some parallelism with some terminology used in [1], would be appropriate to be referred as almost 𝛼-hyponormal operators.

Voiculescu's [1] generalization of Berger-Shaw inequality gives an estimate for the trace of 𝐶^1𝑇. The result was extended in [2]. The combination of these results will be stated after recalling some terminology and notation. The rational cyclic multiplicity of an operator 𝑇 in 𝐿(ℋ), denoted by 𝑚(𝑇), is the smallest cardinal number 𝑚 with the property that there are 𝑚 vectors 𝑥₁,…,𝑥_𝑚 in ℋ such that ∨𝑓(𝑇)𝑥_𝑗∣1≤𝑗≤𝑚,𝑓∈Rat(𝜎(𝑇))=ℋ,(1) where Rat(𝜎(𝑇)) is the algebra of complex-valued rational functions with poles off 𝜎(𝑇).

For a Borel subset 𝐸⊆ℂ and 𝛼>0, denote 𝜇_𝛼(𝐸)=(𝛼/2)∬_𝐸𝜌^𝛼−1𝑑𝜌𝑑𝜃. In particular, 𝜇₂ is the planar Lebesgue measure.

Theorem A (see [1, 2]). Suppose 𝑇∈𝐻¹¹(ℋ). If there exists 𝐾∈𝒞₂(ℋ) such that either 𝑚(𝑇+𝐾)<∞ or 𝜇₂(𝜎(𝑇+𝐾))=0, then 𝑇∈𝑁¹¹(ℋ). Moreover, when 𝑚(𝑇+𝐾)<∞, tr𝐶^1𝑇≤𝑚(𝑇+𝐾)𝜋⋅𝜇²(𝜎(𝑇+𝐾)),(2) and when 𝜇₂(𝜎(𝑇+𝐾))=0,tr(𝐶^1𝑇)≤0,   and consequently, tr(𝐶^1𝑇)=0.

In fact, it was observed in [2] that the inequality can be improved by replacing 𝑚(𝑇+𝐾) with 𝜏(𝑇+𝐾), where 𝜏(𝑆)∶=liminf[ran𝑘(𝐼−𝑃)𝑆𝑃],(3) and the liminf is taken over all sequences of finite-rank orthogonal projections such that 𝑃→𝐼 in the strong operator topology.

Corollary B (see [2]). Let 𝑇∈𝐻¹¹(ℋ) such that 𝜇₂(𝜎_𝑤(𝑇))=0. Then 𝑇∈𝑁¹¹(ℋ) and tr(𝐶^1𝑇)=0.

On the other hand, Berger-Shaw inequality was extended to operators in 𝐻^𝛼1(ℋ) using similar circle of ideas used in [1]. This was done in [3] for the case 𝛼∈[(1/2),1] and later on in [4] for the case 𝛼∈(0,(1/2)].

Theorem C (see [3, 4]). Let 0<𝛼≤1, and let 𝑇∈𝐻^𝛼1(ℋ) and 𝐾∈𝒞_2𝛼(ℋ) with 𝑚(𝑇+𝐾)<∞. Then 𝑇∈𝑁^𝛼1(ℋ) and tr𝐶^𝛼𝑇≤𝑚(𝑇+𝐾)𝜋⋅𝜇^2𝛼(𝜎(𝑇+𝐾)).(4)

The case in which 𝑚(𝑇+𝐾)=∞ and 𝜇_2𝛼(𝜎(𝑇+𝐾))=0 was not discussed in [4] or [3]. It is the goal of this note to make some progress towards this case. We have the following.

Theorem 1. Let 𝛼∈(0,1) and let 𝑇∈𝐻^𝛼1(ℋ) and 𝐾∈𝒞_𝛼(ℋ) with 𝜇_2𝛼(𝜎(𝑇+𝐾))=0. Then 𝑇∈𝑁^𝛼1(ℋ) and tr(𝐶^𝛼𝑇)=0.

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

Before we prove Theorem 1, we extract a similar consequence to Corollary B.

Corollary 2. Let 𝛼∈(0,1] and let 𝑇∈𝐻^𝛼1(ℋ) such that 𝜇₂(𝜎_𝑤(𝑇))=0. Then 𝑇∈𝑁^𝛼1(ℋ) and tr(𝐶^𝛼𝑇)=0.

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.

The proof of Theorem 1 makes use of the following three inequalities.

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.

The following is a consequence of Theorem 3.4 of [8].

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),  𝑇∈𝐻^𝛼1(ℋ), and 𝐾∈𝒞_𝛼(ℋ) with 𝜇_2𝛼(𝜎(𝑇+𝐾))=0, and assume 𝑚(𝑇+𝐾)=∞, otherwise Theorem C implies 𝑇∈𝑁^𝛼1(ℋ).
Let {𝑒_𝑛}_𝑛∈ℕ be an orthonormal basis of ℋ and let ℋ_𝑛=∨𝑟(𝑇+𝐾)𝑒_𝑗∣𝑗=1,…,𝑛,𝑟∈Rat(𝜎(𝑇+𝐾)).(5) Assume that with respect to the decomposition ℋ=ℋ_𝑛⊕ℋ^⟂𝑛, operators 𝑇 and 𝐾 are written as 𝑇=𝑇^1𝑛𝑇_2𝑛𝑇_3𝑛𝑇_4𝑛,𝐾=𝐾^1𝑛𝐾_2𝑛𝐾_3𝑛𝐾_4𝑛.(6) 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 𝑇_1𝑛∈𝐻^𝛼1(ℋ_𝑛) by first establishing that 𝑃_𝑛𝐶^𝛼𝑇𝑃_𝑛−𝐶^𝛼𝑇_1𝑛=−𝑄^𝑛+𝐾^𝑛,(7a) where 𝑄^𝑛∈𝐿(ℋ_𝑛) is positive semidefinite and 𝐾^𝑛∈𝒞₁(ℋ_𝑛).
Assuming that equality (7a) was already proved and writing 𝐶^𝛼𝑇=𝑄+𝐾 with 𝑄≥0 and 𝐾∈𝒞₁(ℋ), then we have 𝐶^𝛼𝑇_1𝑛=𝑃_𝑛𝑄𝑃_𝑛+𝑃_𝑛𝐾𝑃_𝑛+𝑄^𝑛−𝐾^𝑛,(7b)that is, 𝐶^𝛼𝑇_1𝑛 is the sum of 𝑃_𝑛𝑄𝑃_𝑛+𝑄^𝑛, which is a positive semidefinite operator, and of 𝑃_𝑛𝐾𝑃_𝑛−𝐾^𝑛, which is a trace-class operator.
Indeed, the expression 𝑃_𝑛𝐶^𝛼𝑇𝑃_𝑛−𝐶^𝛼𝑇_1𝑛 can be written as 𝐷₁−𝐷₂, where 𝐷𝐷₁₂=𝑃_𝑛𝑇^∗𝑇^𝛼𝑃_𝑛−𝑇^∗1𝑛𝑇_1𝑛^𝛼,=𝑃_𝑛𝑇𝑇^∗^𝛼𝑃_𝑛−𝑇_1𝑛𝑇^∗1𝑛^𝛼.(8) We can write 𝐷₁=−𝑄^𝑛+𝐾^𝑛, where 𝑄^𝑛=𝑃_𝑛𝑇^∗𝑇𝑃_𝑛^𝛼−𝑃_𝑛𝑇^∗𝑇^𝛼𝑃_𝑛,(9) which according to Hansen's inequality is a positive semidefinite operator, and 𝐾^𝑛=𝑃_𝑛𝑇^∗𝑇𝑃_𝑛^𝛼−𝑃_𝑛𝑇^∗𝑃_𝑛𝑇𝑃_𝑛^𝛼,(10) which according to Jocic's inequality is a trace-class operator that satisfies ‖‖𝐾^𝑛‖‖₁≤‖‖||𝑃_𝑛𝑇^∗𝑇𝑃_𝑛−𝑃_𝑛𝑇^∗𝑃_𝑛𝑇𝑃_𝑛||^𝛼‖‖₁=‖‖(𝑇^∗3𝑛𝑇_3𝑛)^𝛼‖‖₁=‖‖𝑇^∗3𝑛𝑇_3𝑛‖‖^𝛼𝛼≤‖‖𝑇^∗3𝑛‖‖^𝛼⋅‖‖𝑇_3𝑛‖‖^𝛼𝛼≤‖𝑇‖^𝛼⋅‖‖𝑇_3𝑛‖‖^𝛼𝛼.(11) Concerning operator 𝐷₂, we can write 𝐷₂=𝑄^{𝑛}+𝐾^{𝑛}, where 𝑄^{𝑛}=𝑃_𝑛𝑇𝑇^∗^𝛼𝑃_𝑛−𝑃_𝑛𝑇𝑃_𝑛𝑇^∗^𝛼𝑃_𝑛,(12) which according to Lowner's inequality is a positive semidefinite operator, and 𝐾^{𝑛}=𝑃_𝑛𝑇𝑃_𝑛𝑇^∗^𝛼𝑃_𝑛−𝑃_𝑛𝑇𝑃_𝑛𝑇^∗𝑃_𝑛^𝛼=𝑃_𝑛𝑇𝑃_𝑛𝑇^∗^𝛼−𝑃_𝑛𝑇𝑃_𝑛𝑇^∗𝑃_𝑛^𝛼𝑃_𝑛,(13) which is also a trace-class operator since 𝑇𝑃_𝑛𝑇^∗−𝑃_𝑛𝑇𝑃_𝑛𝑇^∗𝑃_𝑛==𝑇𝑇𝑃𝑃_𝑛𝑛𝑇^∗−𝑇𝑃_𝑛𝑇^∗𝑃_𝑛+𝑇𝑃_𝑛𝑇^∗𝑃_𝑛−𝑃_𝑛𝑇𝑃_𝑛𝑇^∗𝑃_𝑛𝑇^∗𝐼−𝑃_𝑛+𝐼−𝑃_𝑛𝑇𝑃_𝑛𝑇^∗𝑃_𝑛=𝑇𝑇^∗3𝑛+𝑇_3𝑛𝑇^∗𝑃_𝑛∈𝒞_𝛼(ℋ),(14) and according to Jocic's inequality ‖‖𝐾^{𝑛}‖‖₁≤‖‖(𝑇𝑃_𝑛𝑇^∗)^𝛼−(𝑃_𝑛𝑇𝑃_𝑛𝑇^∗𝑃_𝑛)^𝛼‖‖₁≤‖‖||𝑇𝑇^∗3𝑛+𝑇_3𝑛𝑇^∗𝑃_𝑛||^𝛼‖‖₁=≤𝐶‖‖𝑇𝑇‖𝑇^∗3𝑛‖+_𝛼𝑇_3𝑛𝑇^∗𝑃_𝑛‖‖^𝛼𝛼≤𝐶‖‖𝑇𝑇^∗3𝑛‖‖^𝛼𝛼+‖‖𝑇_3𝑛𝑇^∗𝑃_𝑛‖‖^𝛼𝛼‖‖𝑇^∗3𝑛‖‖^𝛼𝛼+‖‖𝑇_3𝑛‖‖^𝛼𝛼=2𝐶‖𝑇‖^𝛼‖‖𝑇_3𝑛‖‖^𝛼𝛼.(15) Therefore, 𝐷₂=𝑄^{𝑛}+𝐾^{𝑛},with𝑄^{𝑛}≥0,𝐾^{𝑛}∈𝐶₁(𝐻),(16) and consequently, 𝐷₁−𝐷₂=(−𝑄^𝑛+𝐾^𝑛)−(𝑄^{𝑛}+𝐾^{𝑛})=−(𝑄^𝑛+𝑄^{𝑛})+(𝐾^𝑛−𝐾^{𝑛}), where 𝑄^𝑛+𝑄^{𝑛}=∶𝑄^𝑛 is positive semidefinite and 𝐾^𝑛−𝐾^{𝑛}=∶𝐾^𝑛 is trace-class, which establishes equality (7a).
According to (7b), 𝑇_1𝑛∈𝐻^𝛼1(ℋ_𝑛), and since 𝑚(𝑇_1𝑛+𝐾_1𝑛)≤𝑛 and 𝜎(𝑇_1𝑛+𝐾_1𝑛)⊆𝜎(𝑇+𝐾), Theorem C implies that tr(𝐶^𝛼𝑇_1𝑛)≤0,   and furthermore, by replacing 𝑇_1𝑛 with 𝑇^∗1𝑛,    tr(𝐶^𝛼𝑇_1𝑛)=0. Furthermore, equality (7a) implies 𝑃_𝑛𝐶^𝛼𝑇𝑃_𝑛≤𝐶^𝛼𝑇_1𝑛+𝐾^𝑛,(17) which further implies tr𝑃_𝑛𝐶^𝛼𝑇𝑃_𝑛≤tr𝐾^𝑛.(18) Similar utilization of Lowner's and Hansen's inequalities implies that 𝐾^𝑛 and −𝐾^{𝑛} are positive semidefinite, and thus so is 𝐾^𝑛=𝐾^𝑛−𝐾^{𝑛}. Therefore tr𝐾^𝑛≤‖‖𝐾^𝑛‖‖₁+‖‖𝐾^{𝑛}‖‖₁≤(1+2𝐶)‖𝑇‖^𝛼‖‖𝑇_3𝑛‖‖^𝛼𝛼.(19) Since 𝑇_3𝑛=−𝐾_3𝑛∈𝒞_𝑝(ℋ_𝑛) and 𝐾_3𝑛→0 weakly and both |𝑇_3𝑛| and |𝑇^∗3𝑛|≤||𝑇||𝐼,   we have ||𝑇_3𝑛||_𝛼→0, and thus tr(𝐶^𝛼𝑇)≤0. Replacing 𝑇 with 𝑇^∗ we conclude that tr(𝐶^𝛼𝑇)=0.