A new characterization of Anderson’s inequality in C ₁-classes

Abstract

Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A ₁, A ₂,..., A _n), B = (B ₁, B ₂,..., B _n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ _A,B : ℒ(H) ↦ ℒ(H) by

$$\Delta _{A,B} (X) = \sum\limits_{i = 1}^n {A_i XB_i - X} .$$

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In this paper, we characterize the class of pairs of operators A, B ∈ ℒ(H) satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators A,B ∈ ℒ(H) such that $\sum\limits_{i = 1}^n {B_i TA_i = T}$ implies $\sum\limits_{i = 1}^n {A_i^* TB_i^* = T}$ for all T ∈ C ₁ (H) (trace class operators). The main result is the equivalence between this property and the fact that the ultraweak closure of the range of the elementary operator Δ_A,B is closed under taking adjoints. This leads us to give a new characterization of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel in C ₁ classes.