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We formulate general conditions which imply that $L(X,Y)$, the space ofoperators from a Banach space $X$ to a Banach space $Y$, has $2^{\mathfrak c}$closed ideals where $\mathfrak c$ is the cardinality of the continuum. Theseresults are applied to classical sequence spaces and Tsirelson type spaces. Inparticular, we prove that the cardinality of the set of closed ideals in$L(\ell_p\oplus\ell_q)$ is exactly $2^{\mathfrak c}$ for all $1<p<q<\infty$,which in turn gives an alternate proof of the recent result of Johnson andSchechtman that $L(L_p)$ also has $2^{\mathfrak c}$ closed ideals for $1<p\neq2<\infty$.

Banach spaces for which the space of operators has $2^{\mathfrak c}$ closed ideals