On zero-diagonal operators

Let H be a complex, separable, infinite-dimensional Hilbert space and let B(H) be the Banach algebra of bounded linear operators on H. An operator T in B(H) is called a zero-diagonal operator if there exists an orthonormaJLbasis {en}∞n=1 for H such that =0 for all n. An operator T is called a Hilbert-Schmidt operator if there exists an oo orthonormal basis {en}∞=1 for H, such that∑ <∞>,and T is n=l called a trace class operator if for every O.N.B (en} n=l n∞= for H, ∑||<∞. One of the main goals of this thesis is to give a comprehensive study on zero-diagonal operators and the relation between them with Hilbert-Schmidt operators and trace class operators. We start this thesis by basic properties of zero-diagonal operators and some of the related concepts. We illustrate these concepts by examples, and we give details of the proofs of known results, and some times different proofs. We also add few seemingly new results.