On the range of elementary operators
Aleksej Turnšek · Publicationes Mathematicae Debrecen · 2003
Let H be a separable infinite dimensional Hilbert space and let B(H) denote the algebra of operators on H into itself.For a two-sided ideal J ⊂ B(H) with unitarily invariant norm and an elementary operator E : B(H) → B(H) we consider the question when ran(E| J ) J = ran E ∩ J J .We prove that this holds when: (i) E(X) = AXB and J is separable: (ii) E(X) = AXB + CXD, where A and C, respectively B and D are commuting normal operators and J = C p with p > 1; (iii) E(X) = A i XB i , where A = (A 1 , . . ., A n ) and B = (B 1 , . . ., B n ) are n-tuples of mutually commuting normal operators and J = C 2 .Finally, as an application of (iii) we prove some results about double operator integrals.