singular values of infinite direct sum of operators

Question

Hi and thanks in advance,

Suppose $\bigoplus_{n\in \mathbb N} T_n$ is a bounded operator in $\mathcal{B}(\bigoplus_{n\in \mathbb N} H_n)$, for some infinite dimensional Hilbert spaces $H_n$.

I know that $\bigoplus_{n\in \mathbb N} T_n$ is compact on $\bigoplus_{n\in \mathbb N} H_n$ iff

1. for every $n\in \mathbb N$ the operator $T_n$ is compact,
2. $\lim_{n\to \infty}\|T_n\|=0$.

I wonder if something similar holds for Schatten classes?

An operator $A$ is in the Schatten class $\mathcal{S}^p(H)$ if the sequence of its singular values $(s_n)_{n\in \mathbb N}$, written in decreasing order and counting multiplicity, is in $\ell^{p}(\mathbb N)$.

Question: In particular, let $(T_n)_{n\in \mathbb N}$ be a sequence of operators $T_n\in \mathcal{B}(H_n)$ such that,

• for every $n\in \mathbb N$ the operator $T_n$ has finite rank,
• $(\|T_n\|_{n\in \mathbb N})\in \ell^{p}(\mathbb N)$.

Then

• Is it true that $\bigoplus_{n\in \mathbb N} T_n\in \mathcal{S}^p(\bigoplus_{n\in \mathbb N} H_n)$?
• Should we care about the growth of the ranks of $T_n$ as $n$ goes to infinity?

Answer 1

The statement as you currently have it is false. In particular, taking $T_n = \frac 1{n^2} \operatorname{id}_{H_n}$ with $H_n = \Bbb C^n$ and $p = 1$ yields a counterexample.