Singular Traces and Compact Operators

Singular Value Inequalities for Compact Operators

A singular value inequality due to Bhatia and Kittaneh says that if A and B are compact operators on a complex separable Hilbert space such that A is self-adjoint, B 0, and A B; then sj(A) sj(B B) for j = 1; 2; :::We give an equivalent inequality, which states that if A;B; and C are compact operators such that A B B C 0; then sj(B) sj(A C) for j = 1; 2; :::Moreover, we give a sharper inequality...

Compact and strictly singular operators on certain function spaces

Strictly Singular Non-compact Operators on Hereditarily Indecomposable Banach Spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic `1 Banach space. The construction of this operator relies on the existence of transfinite c0-spreading models in the dual of the space.

Traces of Singular Moduli

Introduction. “Singular moduli” is the classical name for the values assumed by the modular invariant j(τ) (or by other modular functions) when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [5], formulas for their norms, and for the norms of their differences, were obtained. Here we obtai...

Compact Sets and Compact Operators

Proof. Properties 2 and 3 are left to the reader. For property 1, assume that S is an unbounded compact set. Since S is unbounded, we may select a sequence {vn}n=1 such that ‖vn‖ → 0 as n→∞. Since S is compact, this sequence will have a convergent subsequence, say {vk}k=1, which will still be unbounded. This sequence is Cauchy, so there is a positive integer K for which ‖v`− vm‖ ≤ 1/2 for all `...