Some Connections between an Operator and Its Aluthge Transform

Associated with T = U|T | (polar decomposition) in L(H) is a related operator T̃ = |T | 1 2U|T | 1 2 , called the Aluthge transform of T . In this paper we study some connections betweenT and T̃ , including the following relations; the single valued extension property, an analogue of the single valued extension property onWm(D,H), Dunford’s property (C) and the property (β). 2000 Mathematics Subject Classification. 47B20, 47A15. LetH be a complex Hilbert space, and denote by L(H) the algebra of all bounded linear operators onH. If T ∈ L(H), we write σ (T), σap(T), and σp(T) for the spectrum, the approximate point spectrum, and the point spectrum of T , respectively. An arbitrary operator T ∈ L(H) has a unique polar decomposition T = U|T |, where |T | = (T∗T) 1 2 and U is the appropriate partial isometry satisfying kerU = ker|T | = kerT and kerU∗ = kerT∗. AssociatedwithT is a related operator |T | 1 2U|T | 2 , called the Aluthge transform of T , and denoted throughout this paper by T̃ . An operator T ∈ L(H) is said to be p-hyponormal, where 0 < p ≤ 1, if (T∗T)p ≥ (TT∗)p, where T∗ is the adjoint of T . In particular, if p = 1, T is called hyponormal. There is a vast literature concerning p-hyponormal operators. An operator T ∈ L(H) is said to satisfy the single-valued extension property if for any open subset V in C, the function T − λ : O(V,H) −→ O(V,H) defined by the obvious pointwise multiplication, is one-to-one. Here O(V,H) denotes theFréchet space ofH-valued analytic functions onV with respect touniform topology. If T has the single valued extension property, then for any x ∈ H there exists a unique maximal open set ρT (x) (⊃ ρ(T), the resolvent set) and a unique H-valued analytic function f defined in ρT (x) such that (T − λ) f (λ) = x (λ ∈ ρT (x)). ∗The second author is supported by a grant (R14-2003-006-01000-0) from Korea Science and Engineering Foundation. at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089504002149 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 12 Sep 2017 at 19:46:45, subject to the Cambridge Core terms of use, available 168 MEE-KYOUNG KIM AND EUNGIL KO In the following theorem we show that Aluthge transforms preserve the single valued extension property. THEOREM 1.1. An operator T with polar decomposition U|T | has the single valued extension property if and only if T̃ has. Proof.Assume that T has the single valued extension property. Suppose thatW is an open subset ofC and f : W → H is an analytic function satisfying (T̃ − λ) f (λ) = 0, for each λ ∈ W . Since T(U|T | 1 2 ) = (U|T | 1 2 )T̃ , (T − λ)U|T | 1 2 f (λ) = U|T | 1 2 (T̃ − λ) f (λ) = 0, for each λ ∈ W . Since T has the single valued extension property, U|T | 1 2 f (λ) = 0 for each λ ∈ W . Since T̃ = |T | 1 2U|T | 1 2 , T̃f (λ) = 0 for each λ ∈ W . Since (T̃ − λ) f (λ) = 0 for each λ ∈ W , λ f (λ) = 0 for each λ ∈ W . Since f (λ) = 0 on W\{0} and is analytic onW , f is identically 0 on W . Therefore, T̃ has the single valued extension property. The proof of the converse implication is similar. The following corollary shows the relationships between the local spectra of T and T̃ . COROLLARY 1.2. If an operator T with polar decomposition U|T | has the single valued extension property, then σT̃ (|T | 1 2 x) ⊂ σT (x) and σT(U|T | 1 2 x) ⊂ σT̃ (x). Proof. For λ ∈ ρT (x), we have (T − λ)x(λ) ≡ x, where λ → x(λ) is the analytic function defined on ρT (x). Since |T | 1 2T = T̃ |T | 1 2 , (T̃ − λ)|T | 1 2 x(λ) = |T | 1 2 (T − λ)x(λ) ≡ |T | 1 2 x. Hence ρT (x) ⊂ ρT̃ (|T | 1 2 x), so that σT̃ (|T | 1 2 x) ⊂ σT (x). Similarly, we can prove the second inclusion. COROLLARY 1.3. If an operator T with polar decomposition U|T | has the single valued extension property, then |T | 1 2HT (F) ⊆ HT̃ (F) and U|T | 1 2HT̃ (F) ⊆ HT (F), where HT (F) = {x ∈ H : σT (x) ⊆ F} for F ⊂ C. Proof. If x ∈ HT (F), then σT (x) ⊆ F . By Corollary 1.2, we get σT̃ (|T | 1 2 x) ⊆ F . Hence |T | 1 2 x ∈ HT̃ (F). Thus |T | 1 2HT (F) ⊆ HT̃ (F). Similarly, we get U|T | 12HT̃ (F) ⊆ HT (F). Our next result shows that the Aluthge transform preserves an analogue of the single valued extension property for Wm(D,H) and an operator T on H; that is, T − λ : Wm(D,H) → Wm(D,H) is one-to-one if and only if T̃ − λ is. First of all, let us define a special Sobolev type space. Let D be a bounded open subset of C and m a fixed non-negative integer. The vector valued Sobolev space Wm(D,H) with respect to ∂̄ and order m will be the space of those functions f ∈ L2(D,H) whose derivatives ∂̄f, · · · , ∂̄mf in the sense of distributions still belong to L2(D,H). at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089504002149 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 12 Sep 2017 at 19:46:45, subject to the Cambridge Core terms of use, available AN OPERATOR AND ITS ALUTHGE TRANSFORM 169 Endowed with the norm