In mathematics, specifically functional analysis, a pth Schatten-class operator is a bounded linear operator on a Hilbert space with finite pth Schatten norm. The space of pth Schatten-class operators is a Banach space with respect to the Schatten norm.
Via polar decomposition, one can prove that the space of pth Schatten class operators is an ideal in B(H). Furthermore, the Schatten norm satisfies a type of Hölder inequality:
If we denote by the Banach space of compact operators on H with respect to the operator norm, the above Hölder-type inequality even holds for . From this it follows that, is a well-defined contraction. (Here the prime denotes (topological) dual.)
Observe that the 2nd Schatten class is in fact the Hilbert space of Hilbert–Schmidt operators. Moreover, the 1st Schatten class is the space of trace class operators.
Famous quotes containing the word class:
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)