Generalized Putnam-Fuglede theorems and the range-kernel orthogonality of an elementary operator in C _p classes

Abstract

Let

be commuting operators on a Hilbert space H , where T is w -hyponormal with ker  T ⊆ ker  T * and S is normal. Consider the elementary operator ψ _{T , S} defined on

L ( H ) $\mathcal{L}\left(H\right)$

by ψ _{T , S} ( X ) = TXT * − SXS *, for all

X ∈ L ( H ) $X\in \mathcal{L}\left(H\right)$

, and let

C p ${\mathcal{C}}_{p}$

denote the Schatten p -class. It is shown that:

1. For all

, TXT * = SXS * implies T * XT = S * XS , i.e.,

ker ( ψ T , S ∣ C p ) ⊆ ker ψ T * , S * ∣ C p $\mathrm{ker}\left({\psi }_{T,S}\mid {\mathcal{C}}_{p}\right)\subseteq \mathrm{ker}\left({\psi }_{{T}^{{\ast}},{S}^{{\ast}}}\mid {\mathcal{C}}_{p}\right)$

; and

2. For all

with

Y ∈ ker ( ψ T , S ∣ C p ) $Y\in \mathrm{ker}\left({\psi }_{T,S}\mid {\mathcal{C}}_{p}\right)$

, we have

T X T * − S X S * + Y p ≥ ‖ Y ‖ p , $${{\Vert}TX{T}^{{\ast}}-SX{S}^{{\ast}}+Y{\Vert}}_{p}\ge {\Vert}Y{{\Vert}}_{p},$$