Operator cotype describes how a bounded operator between Banach spaces acts on finite
random sums of vectors. The two
principal versions are Gaussian cotype, which
uses independent and identically
distributed random variables with a standard
normal distribution, and Rademacher cotype,
which uses independent random
variables taking the values 
and 1 with probability 
each. For a real
number 
,
each version is quantified by the least nonnegative constant comparing the 
-norm of the sequence
of image norms with the expectation
value of the norm of the corresponding random sum,
uniformly over all finite sequences
of vectors in the source. The constant
is assigned the value 
when no finite constant works (Talagrand 2021, p. 606).
For an identity operator, operator cotype gives the corresponding cotype property of the underlying Banach space. Comparing the Gaussian and Rademacher constants together with a related summing norm gives Talagrand's operator cotype problem.
