Fuglede-Putnam type theorems via the generalized second Aluthge transform.

Abstract

Let T = U|T| and S = V |S| be the polar decompositions of T ∈ B(H) and S ∈ B(K). A pair (T, S) is said to have the FP-property if TX = XS implies T∗X = XS∗ for any X. Let ˆ Z(s,t) := | ˜ Z|sW| ˜ Z|t denote the generalized second Aluthge transform of a bounded linear operator Z such that ˜ Z = W| ˜ Z| is the polar decomposition of ˜ Z where ˜ Z denotes the first Aluthge transform of operator Z. We show that (i) if T is class A(s, t) and S∗ is invertible class A(s, t) operators with s + t = 1 such that TX = XS for some Hilbert Schmidt operator X, then T∗X = XS∗; (ii) if ˜TX = X ˜ S ⇒ ( ˜ T)∗X = X( ˜ S)∗ for any X, then ˜TX = X ˜ S ⇒ ˆ T(s,t)X = X ˆ S(s,t) for any X, furthermore, if T is invertible, then ˆ T(s,t)X = X ˆ S(s,t) ⇒ ˜TX = X ˜ S. Finally, if Re(U| ˜ T|s) ≥ a > 0 and Re(V | ˜ S|s) ≥ a > 0 and X is an operator such that U∗X = XV , then we prove that ( ˆ T(s,s))∗X −X ˆ S(s,t) p ≥ 2a| ˜ T |sX −X| ˜ S|sp for any 1 ≤ p≤∞ such that 0 < s < 1.