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Let H denote a separable, infinite-dimensional Hilbert space. We shall consider operators (that is, bounded linear transformations) on/-/. An operator T is said to be in the Schatten class Cp if T is compact and if the eigenvalues of (T'T) t/2 are in 1 p (0<p<oo); the norm in Cp is the I p norm of these eigenvalues. The class C 2 is called the Hilbert-Schmidt class, and C 1 is called the trace class. By C~o we denote the class of all compact operators. For information on these classes see, for example, [23], [4, Chap. XI], [8, Chap. III], [19], [13. Although there is an obvious analogy with the sequence spaces I p, there is a deeper analogy with the function spaces L p on the unit circle (however, the containment relations are reversed from the L p case, namely, Cp c Cr for p < r). This analogy serves as a source for new conjectures in operator theory. Motivated by this we establish an operator analgue of a coefficient inequality for the function space H 1. First we survey some known results in order to emphasize the nature and depth of the analogy. Fix an orthonormal basis {en} (n= l, 2, ...) in H, and let En =span{e I .... , en}. Each operator T on/-/has a matrix representation:

An analogue of a hardy-littlewood-fej�r inequality for upper triangular trace class operators