Schur products of operators and the essential numerical range

Abstract:

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