# Compact operators and $S_p(H)$

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.

• What is $S_p(H)$? – Robert Israel Jun 2 at 14:48
• @RobertIsrael $S_p(H)$ - Schatten-von Neumann class – Gera Slanova Jun 2 at 15:03
• @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,... – Gera Slanova Jun 2 at 15:18
• @RobertIsrael $A \in S_p(H)$ if $\{s_n\}_{n=1}^{\infty} \in l_p, 1 \leq p < \infty$ – Gera Slanova Jun 2 at 15:24