Schur products of operators and the essential numerical range

Abstract

Let E = { e n } n = 1 ∞ \mathcal {E} = \{ {e_n}\} _{n = 1}^\infty be an orthonormal basis for a Hilbert space H \mathcal {H} . For operators A A and B B having matrices ( a i j ) i , j = 1 ∞ ({a_{ij}})_{i,\;j = 1}^\infty and ( b i j ) i , j ∞ = 1 ({b_{ij}})_{i,\;j}^\infty = 1 , their Schur product is defined to be ( a i j b i j ) i , j ∞ = 1 ({a_{ij}}{b_{ij}})_{i,\:j}^\infty = 1 . This gives B ( H ) \mathcal {B}(\mathcal {H}) a new Banach algebra structure, denoted P E {\mathcal {P}_\mathcal {E}} . For any operator T T it is shown that T T is in the kernel (hull(compact operators)) in some B E {\mathcal {B}_\mathcal {E}} iff 0 0 is in the essential numerical range of T T . These conditions are also equivalent to the property that there is a basis such that Schur multiplication by T T is a compact operator mapping Schatten classes into smaller Schatten classes. Thus we provide new results linking B ( H ) \mathcal {B}(\mathcal {H}) , B E {\mathcal {B}_\mathcal {E}} and B ( B ( H ) ) \mathcal {B}(\mathcal {B}(\mathcal {H})) .