Let $A$ and $B$ be compact operators in Hilbert space $H$ and Im$A \subset$ Im$B$.

Can you help me please to prove that if $B\in S_p(H)$ then $A\in S_p(H)$?

I have no idea.

  • 1
    $\begingroup$ What is $S_p(H)$? $\endgroup$ – Robert Israel Jun 2 at 14:48
  • $\begingroup$ @RobertIsrael $S_p(H)$ - Schatten-von Neumann class $\endgroup$ – Gera Slanova Jun 2 at 15:03
  • $\begingroup$ @RobertIsrael and $\| A \|_{S_p} = \| A \| _p = \left( \sum_{k=1}^{\infty} s_k (A)^p \right)^{1/p}$, where $s_n(A) = \sqrt{\lambda_n(A^*A)}$ n=1,2,... $\endgroup$ – Gera Slanova Jun 2 at 15:18
  • $\begingroup$ @RobertIsrael $A \in S_p(H)$ if $\{s_n\}_{n=1}^{\infty} \in l_p, 1 \leq p < \infty $ $\endgroup$ – Gera Slanova Jun 2 at 15:24

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