If you are interested in closed ideals, then

• for separable $H$ the only ideal is $K(H)$

• for non-separable $H$ complete characterization may be read here

If you are inrested in all ideals, then they are between $F(H)$ and $K(H)$ and consisist of operators whose singular values belong to some order ideal in $c_0^+$.

For details see section I.8.7 here.

If you are interested in closed ideals, then

• for separable $H$ the only ideal is $K(H)$

• for non-separable $H$ complete characterization may be read here

If you are inrested in all ideals, then they are between $F(H)$ and $K(H)$ and consisist of operators whose singular values belong to some order ideal in $c_0^+$.

For details see section I.8.7 here.

If you are interested in closed ideals, then

• for separable $H$ the only ideal is $K(H)$

• for non-separable $H$ complete characterization may be read here

If you are inrested in all ideals, then

• if it contained in $K(H)$, then it is $F(H)$ - the ideal of finite rank operators.

• if it contains $K(H)$, then it consisist of operators whose singular values belong to some order ideal in $c_0^+$.

For details see section I.8.6 here.

If you are interested in closed ideals, then

• for separable $H$ the only ideal is $K(H)$

• for non-separable $H$ complete characterization may be read here

If you are inrested in all ideals, then they are between $F(H)$ and $K(H)$ and consisist of operators whose singular values belong to some order ideal in $c_0^+$.

For details see section I.8.7 here.