Gudder–Nagy’s Theorem for Hilbert <i>K</i> ( <i>H</i> )‐Modules

We show in this paper Gudder–Nagy’s theorem for operators on Hilbert C ∗ ‐modules over C ∗ ‐algebra of compact operators. Let H be a complex Hilbert space with dim H &gt; 1, and K ( H ) the C ∗ ‐algebra of compact operators on H . For bounded K ( H )‐linear operators A , B and C on Hilbert C ∗ ‐module X over K ( H ), we show that 〈 A x , x 〉〈 B x , x 〉 = 〈 x , x 〉〈 C x , x 〉, for all x ∈ X , if and only if there is a complex number λ such that A = λ I and C = λ B .