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Compact operators have a countable set of singular values $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the Schatten $p$-class. So Schatten $p$-class operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making 2-Schatten class a Hilbert space. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).

Compact operators have a countable set of singular values $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the Schatten $p$-class. So Schatten $p$-class operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making Schatten $2-$class a Hilbert space. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).

Compact operators have a countable set of singular values $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the $p$-Schatten class. So $p$-Schatten operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making 2-Schatten class a Hilbertspace. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).

Compact operators have a countable set of singular values $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the Schatten $p$-class. So Schatten $p$-class operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making 2-Schatten class a Hilbert space. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).

Compact operators have a countable set of eigenvalues $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the $p$-Schatten class. So $p$-Schatten operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making 2-Schatten class a Hilbertspace. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).

Compact operators have a countable set of singular values $\lambda_j$ with the only possible accumulation point being zero, so to say $c_0$-sequences. If the eigenvalues are an $\ell^p$-sequence, the operator is in the $p$-Schatten class. So $p$-Schatten operators are a noncommutative analogue of $\ell^p$ spaces. This carries over to duals, etc, making 2-Schatten class a Hilbertspace. Be careful, the analogue of $\ell^\infty$ is not compact in this setting but should be $B(H)$, the set eigenvalues may not be countable anymore, but still the spectrum is a bounded set (spectral radius $\leq$ the norm).